US20210197029A1 - Polyhedra golf ball with lower drag coefficient - Google Patents

Abstract

A golf ball having an outer surface with a pattern forming a polyhedron. The pattern can be flat faces forming sharp edges and sharp points therebetween. In one embodiment, the polyhedron is a Goldberg polyhedron.

Description

RELATED APPLICATIONS
• This application claims the benefit of U.S. Provisional Application No. 62/616,861, filed Jan. 12, 2018, the entire contents of which are incorporated herein by reference.
BACKGROUND Technical Field
• The present invention relates to golf balls having a polyhedra design that can yield lower drag coefficient than a dimpled sphere. The drag reduction is applicable to other sports equipment in general having a bluff body, such as the head of a golf club or a bike helmet.
Background of the Related Art
• For the past 100 years the vast majority of commercial golf balls have used designs with dimples. For the purpose of this invention, a dimple generally refers to any curved or spherical depression in the face or outer surface of the ball. The traditional golf ball, as readily accepted by the consuming public, is spherical with a plurality of dimples, where the dimples are generally depressions on the outer surface of a sphere. The vast majority of commercial golf balls use dimples that have a substantially spherical shape. Some examples of such golf balls can be found in U.S. Pat. Nos. 6,290,615, 6,923,736, and U.S. Publ. No. 20110268833.
• It is well established that such depressions can lower the drag coefficient of a ball compared to that of a smooth sphere at the same speed. The drag coefficient is a dimensionless parameter that is used to quantify the drag force of resistance of an object in a fluid. The drag force is always opposite to the direction in which the object travels. The drag coefficient for a golf ball is defined as C_D=2*Fd/(ρ*U²*A), where Fd is the drag force, p is the density of the fluid in which the object is moving, U is the speed of the object and A is the cross sectional area. For a sphere the cross-sectional area is πD²/4, where D is the diameter of the ball.
• FIG. 1 shows the variation of the drag coefficient, C_D, as a function of Reynolds number, Re, for smooth and dimpled spheres. The data were obtained by performing wind tunnel experiments of non-spinning spheres. The Reynolds number is a dimensionless parameter used in fluid mechanics and is defined as Re=U*D/v, where v is the kinematic viscosity in which the object moves. For a smooth sphere the drag coefficient (shown by the solid black line) remains constant (C_D˜0.5) until the Reynolds number approaches a critical value (Re_cr˜300,000). At this point, which is usually referred to as drag crisis, C_D decreases rapidly and hits a minimum, which is an order of magnitude lower C_D˜0.08. With further increase in the Reynolds number the flow enters the post-critical regime characterized by turbulent boundary layers on the surface of the sphere. In this regime the drag coefficient rises slowly with increasing Reynolds number.
• In dimpled spheres (shown by dashed and dotted black lines) the drag crisis happens at a much lower critical Reynolds number (Re<100,000). In general, a dimpled sphere will have a 50% or more reduction in drag compared to a smooth sphere in the range of Reynolds number from 100,000 to 250,000. A golf ball in flight during a driver shot can experience a Reynolds number ranging from 220,000 at the beginning of the flight to 60,000 at the end of the flight. It is therefore very important that a golf ball design can achieve a very low drag coefficient in that range. The critical value of the Reynolds number, as well as the attained minimum drag coefficient in the post-critical regime depend on the dimple geometry and arrangement. In general, as the aggregate dimple volume, measured as the sum of a dimple volume of each dimple of the plurality of dimples, decreases the critical Reynolds is increased and the drag coefficient in the post-critical regime increases. As the aggregate dimple volume approaches zero the drag curve approaches that of a smooth sphere. Drag is one of the two forces (the other being lift) that influence the aerodynamic performance of a golf ball. To increase the carry distance on a driver shot, the distance a golf ball travels during the flight, the dimple design is a balance between achieving the lowest critical Reynolds number and the lowest drag coefficient in the post-critical regime.
• However, dimples are the only configuration known to provide a golf ball having a reduced drag coefficient.
SUMMARY
• Accordingly, it is an object of the invention to provide a golf ball having minimal drag coefficient. It is a further object of the invention to provide a golf ball having reduced drag coefficient compared to a golf ball having only dimples. It is another object to provide a golf ball with minimal drag coefficient that is non-dimpled. In one embodiment, the golf ball has a plurality of flat faces with sharp edges and points that collectively form a polyhedron. These and other objects of the invention, as well as many of the intended advantages thereof, will become more readily apparent when reference is made to the following description, taken in conjunction with the accompanying drawings.
BRIEF DESCRIPTION OF THE FIGURES
• FIG. 1 shows a plot of the drag coefficient C_D vs Reynolds number Re for smooth and dimpled spheres. The solid black line represents a smooth sphere (Achenbach, 1972); the double-dashed lines represent a dimpled sphere (J. Choi, 2006); and the dash-dot lines represent a dimpled sphere (Harvey, 1976). The shaded area represents the typical range of Reynolds experience by a golf ball in flight during a driver shot (50,000-200,000).
• FIG. 2(a) shows a golf ball in accordance with one embodiment of the invention.
• FIG. 2(b) shows an outline of a golf ball.
• FIG. 2(c) shows an outline of a golf ball with non-sharp rounded edges.
• FIG. 2(d) shows an icosahedron, a well-known Platonic solid used to derive the golf ball 100.
• FIG. 2(e) shows an example of splitting a hexagonal face into 6 triangular faces.
• FIG. 3 shows the Goldberg polyhedron with 192 faces.
• FIG. 4 is a graph of the drag coefficient versus Reynolds for the Goldberg polyhedra with 162 faces and 192 faces.
• FIG. 5 is a graph showing the C_D versus Reynolds for a Goldberg polyhedron with 192 faces and a dimpled sphere.
• FIG. 6 is a graph showing the C_D versus Reynolds for a Goldberg polyhedron with 162 faces and a dimpled sphere.
• FIG. 7 shows a geodesic polyhedron made from 320 triangles.
• FIG. 8 shows a geodesic cube with 174 faces.
• FIG. 9 shows a polyhedron with 162 faces and 162 dimples.
• FIG. 10 is a graph showing the drag coefficient C_D versus Reynolds number Re for a Goldberg polyhedron with 162 faces and a Goldberg polyhedron with 162 faces and 162 dimples.
• FIG. 11 is a comparison of drag coefficient C_D versus Reynolds number for one of the embodiments of the present invention shown in FIG. 9 against a commercial ball Callaway Superhot.
• FIG. 12 is an alternative embodiment of a golf ball based on an icosahedron with 312 faces and 312 spherical dimples.
• FIG. 13 is a comparison of drag coefficient C_D versus Reynolds number Re for one of the embodiments of the present invention based on a polyhedron with 312 faces and 312 spherical dimples of FIG. 12 against a commercial golf ball Bridgestone Tour.
DETAILED DESCRIPTION
• In describing certain illustrative, non-limiting embodiments of the invention illustrated in the drawings, specific terminology will be resorted to for the sake of clarity. However, the invention is not intended to be limited to the specific terms so selected, and it is to be understood that each specific term includes all technical equivalents that operate in similar manner to accomplish a similar purpose. Several embodiments of the invention are described for illustrative purposes, it being understood that the invention may be embodied in other forms not specifically shown in the drawings.
• The present invention is directed to a golf ball design based on polyhedra that can have reduced drag coefficient compared to a dimpled sphere. In one embodiment, a family of golf ball designs are made up of convex polyhedra whose vertices lie on a sphere. A polyhedron is a solid in three dimensions with flat polygonal faces, straight sharp edges and sharp corners or vertices.
• FIG. 2(a) shows a golf ball 100 in accordance with one embodiment of the present invention. The golf ball 100 has a body 110 with an inner core and an outer shell with an outer surface 112. A plurality of faces 120 are formed in the outer surface, creating a pattern 116. All the faces are formed in the outer surface and each of such faces is flat and lies in a plane. The faces are bound by straight or linear edges 124. Here, the golf ball 100 is a polyhedron with 162 polygons.
• The body 110 defines a circumscribed sphere 102, which is the smallest sphere containing the polyhedron, and is shown with a dashed line about the outermost points 104 of the body 110. The sphere and the diameter provide a reference for the size of the golf ball. The Rules of Golf, jointly governed by the R&A and the USGA, state that the diameter of a “conforming” golf ball cannot be any smaller than 1.680 inches. For the purpose of a golf ball the diameter of the circumscribed sphere is at least 1.68 in. The vertices 122 a, 122 b of the polyhedron are the only points 104 on the polyhedron that lie on the sphere. Any point along the edges 124 a, 124 b or on the faces 120 a, 120 b of the polygons lies below the surface of the circumscribed sphere.
• The golf ball body 110 is a polyhedron that is made from first faces 120 a and second faces 120 b. As shown, the first faces 120 a have a first shape, namely pentagons, and the second faces 120 b have a second shape different than the first shape, namely hexagons. All of the faces are flat and each such face forms a single plane. In one embodiment, there are 12 pentagons 120 a and 150 hexagons 120 b (a hexagon-to-pentagon ratio of 12.5:1), each having corners or points 122 a, 122 b connected by boundaries such as straight lines or edges 124 a, 124 b. In various other embodiments, other quantities and/or ratios of such pentagons 120 a and hexagons 120 b can be used. However, the number of polygons and the angle between them determine when the drag coefficient will start to drop and how low it will become. In general, as the number of faces is increased the drag crisis occurs at higher Reynolds number and the drag coefficient decreases. The first and second faces 120 a, 120 b form the pattern 116.
• The edges 124 are sharp, in that the faces are at an angle with respect to one another. FIG. 2(b) shows a cross sectional cut through the body 110 of ball 100 of FIG. 2(a) along the line 150. In this embodiment the edges 124 are sharp, in that the radius of curvature 140 of an edge is less than 0.001 D, where D is the diameter of the circumscribed sphere. Ideally all edges should be the sharpest possible. A radius of curvature of 0 is the case with the sharpest possible edges, though golf ball edges might become a little rounded during manufacture or follow use. FIG. 2(c) shows a cross section cut of two faces with rounded edges, whose radius of curvature is more than 0.001 D. The resulting edge is not sharp and the reduction in drag is not maximized, which can be detrimental to the aerodynamic performance of the golf ball as the shape would approach that of a smooth sphere. Both a sharp edge and a non-sharp edge is shown in that embodiment for illustrative purposes. The angle θ formed between two adjacent flat/planar faces 120 is always smaller than 180 degrees. The geometric shape of the embodiment illustrated in FIG. 2(a) falls into the class of convex polyhedral where the angle between any pair of adjacent faces is always less than 180 degrees. In this embodiment, the angle between a pentagon face 120 a and an adjacent hexagonal face 120 b is 166.215 degrees. The angle between two adjacent hexagon faces 120 b varies from 161.5 degrees to 162.0 degrees.
• Each face 120 is immediately adjacent and touching a neighboring face 120, such that each edge 124 forms a border between two neighboring faces 120 and each point 122 is at the intersection of three neighboring faces 120. And, each point 122 is at an opposite end of each linear edge 124 and is at the intersection of three linear edges 124. Accordingly, there is no gap or space between adjacent neighboring faces 120, and the faces 120 are contiguous and form a single integral, continuous outer surface 112 of the ball 100.
• A golf ball usually has a rubber core and at least one more layer surrounding the core. The pattern 116 is formed on the outermost layer. The pattern is based on an icosahedron shown in FIG. 2(d). The icosahedron 170 is a well-known convex polyhedron made up of 20 equilateral triangle faces 180, 12 vertices 182 and 30 edges 184. An icosahedron is one of the five regular Platonic solids, the other four being the cube, the tetrahedron, the octahedron and the dodecahedron (see https://en.wikipedia.org/wiki/Platonic solid).
• In the icosahedron 170 five equilateral triangles 180 meet at each of its twelve vertices 182. The 12 pentagons 120 a of the golf ball 100 shown in FIG. 2(a) are centered on the vertices of an icosahedron. Therefore, a pair of 3 pentagons 120 a forms an equilateral triangular pattern 180. Along each of the edges of the triangles 180 there are 3 hexagons 120 b. Finally, inside each triangular pattern 180 there are three hexagons 120 b. The pentagons 120 a are all equilateral, that is the 5 edges 124 a all have the same length equal to 0.151 D, where D is the diameter of the circumscribed sphere. The hexagons 120 b are not equilateral and the lengths of the edges 124 b vary from 0.151 D to 0.1834 D.
• Table 1 lists the coordinates of all the vertices and how each face is formed, which defines the geometry of the golf ball. Table 1 lists the coordinates x, y, and z of all of the vertices 122 of golf ball 100. Faces are constructed by connecting the vertices with straight lines. The coordinates x, y, z are normalized by the diameter D of the circumscribed sphere 104 and are with respect to the center of the sphere. The golf ball 100 contains 320 vertices 122, 480 straight edges 120 and 162 polygonal faces 120.
• The particular polyhedron can be any suitable configuration, and in one embodiment is a class of solids called Goldberg polyhedra that comprises convex polyhedra that are made entirely from a combination of rectangles, pentagons and hexagons. Goldberg polyhedra are derived from either an icosahedron (a convex polyhedron made from 12 pentagons), or an octahedron (a convex polyhedron made from 8 triangles) or a tetrahedron (a convex polyhedron made from 4 triangles). An infinite number of Goldberg polyhedra exist, as shown in https://en.wikipedia.org/wiki/List_of_geodesic_polyhedra_and_Goldberg_polyhedra.
• However, it will be recognized that the invention can utilize any convex polyhedron (that is, a polyhedron made up of polygons whose angle is less than 180 degrees), though in one embodiment on such polyhedron is a convex polyhedron with sharp edges, flat faces forming a single plane, and adjacent faces having an angle of less than 180 degrees between them.
• The particular configuration (Goldberg polyhedral 162 faces and icosahedral symmetry) can only be achieved with a combination of hexagons and pentagons. However other geometries with around 162 faces may be possible to do using only pentagons or only hexagons. Other embodiments of the invention can include a pattern with various geometric configurations. For example, the pattern can be comprised of more or fewer of hexagons and pentagons than shown. Or it can comprise all hexagons, all pentagons, or no hexagons or pentagons but instead one or more other shapes or polyhedrals having flat faces and sharp edges. One other shape can be formed, for example, by splitting each hexagon into 6 triangles or each pentagon into 5 triangles, which provides a similar drag coefficient. One embodiment can include any of the Goldberg polyhedra with a combination of pentagons and hexagons or even a convex polyhedral made of triangles or squares.
• It is further noted that flat faces give lower drag and have the uniqueness of not being dimples (curved indentations). The flat faces only provide points of the faces that lie on the circumscribed sphere. The sharp edges are defined by the angle between two adjacent faces. In addition, the edges forming the boundaries between the two adjacent faces are flat and not excessively rounded. An example of a non-sharp edge is shown in FIG. 2(c). The angle between the two edges is the same as is in FIG. 2(b) but it is rounded such that the edge is not sharp.
• As the number of polygons increases (i.e. from 162 faces to 312 faces) the angle between the faces increases too and approaches 180 degrees. For the embodiments that are based purely on polyhedral shape such as those shown in FIGS. 2(a), 2(b) with 162 and 192 faces the range of angles is between 160 and 165 degrees. For the other embodiments in which dimples are added inside each face it is possible to go to as many as 312 faces and the angle between the faces can increase to 172 degrees. In one embodiment, the maximum angle could be close to 175 degrees and a range of angles between 160 and 175 degrees may be suitable for the purpose of a golf ball. Thus, a convex polyhedra is one having faces with angles substantially with the values and in the ranges noted herein.
• In FIG. 2(a), the ratio of pentagons to hexagons is 12:150, though any suitable ratio can be provided. For example, out of the 150 hexagons one of the hexagons can be split into 6 triangles and a have a polyhedron with 12 pentagons, 149 hexagons and 6 triangles and obtain substantially the same drag coefficient. FIG. 2(e) illustrates how such a splitting can be performed on one of the hexagonal faces 120. A vertex 140 can be chose anywhere inside the hexagon 120. For illustrative purposes, the vertex 140 is near the center of the hexagon although any other location can be used. Six new edges 142 can be formed by connecting the each of the vertices 122 with the new vertex 140. A triangular face 144 is formed by one edge 124 of the hexagon and two adjacent edges 142. The exact shape of the faces making up the polyhedral can vary but one important feature of the polyhedral pattern is the angle between faces.
• FIG. 3 shows an example of a golf ball in accordance with another embodiment of the invention. The golf ball 200 has a body 210 with an inner core and an outer shell with an outer surface 212. A plurality of faces 220 are formed in the outer surface, creating a pattern 216. All the faces are formed in the outer surface and each of such faces is flat and lies in a plane. The faces are bound by straight or linear edges 224. Here, the golf ball 200 is a polyhedron with 192 polygons.
• The body 210 defines a circumscribed sphere 202, which is the smallest sphere containing the polyhedron, and is shown with a dashed line about the outermost points 204 of the body 210. The vertices 222 a, 222 b of the polyhedron are the only points 204 on the polyhedron that lie on the sphere. Any point along the edges 224 a, 224 b or on the face of the polygons lies below the surface of the circumscribed sphere.
• The golf ball body 210 is a polyhedron that is made from first faces 220 a and second faces 220 b. As shown, the first faces 220 a have a first shape, namely pentagons, and the second faces 220 b have a second shape different than the first shape, namely hexagons. All of the faces are flat and each such face forms a single plane. In one embodiment, there are 12 pentagons 220 a and 180 hexagons 220 b (a hexagon-to-pentagon ratio of 15:1), each having corners or points 222 a, 222 b connected by boundaries such as straight lines or edges 224 a, 224 b. In various other embodiments, other quantities and/or ratios of such pentagons 220 a and hexagons 220 b can be used. The first and second faces 220 a, 220 b form the pattern 216. The edges 224 are sharp, in that the radius of curvature of an edge is less than 0.001 D, where D is the diameter of the circumscribed sphere. Ideally all edges should be the sharpest possible. A radius of curvature of 0 is the case with the sharpest possible edges, though golf ball edges might become a little rounded during manufacture or follow use.
• The geometric shape of the embodiment illustrated in FIG. 3 falls into the class of convex polyhedral where the angle between any pair of adjacent faces is always less than 180 degrees. In this embodiment, the angle between a pentagon face 220 a and an adjacent hexagonal face 220 b is 167.6 degrees. The angle between two adjacent hexagon faces 120 b varies from 163.4 degrees to 164.2 degrees. When comparing this embodiment with the golf ball 100 illustrated in FIG. 2 it is obvious that as the number of faces on a convex polyhedron increases the angle between faces increases too.
• Each face 220 is immediately adjacent and touching a neighboring face 220, such that each edge 224 forms a border between two neighboring faces 220 and each point 222 is at the intersection of three neighboring faces 220. And, each point 222 is at an opposite end of each linear edge 224 and is at the intersection of three linear edges 224. Accordingly, there is no gap or space between adjacent neighboring faces 220, and the faces 220 are contiguous and form a single integral, continuous outer surface 212 of the ball 200.
• The pattern is based on an icosahedron shown in FIG. 2(c). In the icosahedron 170 five equilateral triangles 180 meet at each of its twelve vertices 182. The 12 pentagons 220 a of the golf ball 200 shown in FIG. 3 are centered on the vertices of an icosahedron. Therefore, any pair of 3 pentagons 220 a form an equilateral triangle 280. The pentagons 220 a are all equilateral, that is the 5 edges 224 a all have the same length equal to 0.136 D, where D is the diameter of the circumscribed sphere. The hexagons 220 b are not equilateral and the lengths of the edges 224 b vary from 0.136 D to 0.168 D.
• Table 2 lists the coordinates of all the vertices and how each face is formed, which defines the geometry of the golf ball. Table 1 lists the coordinates x, y, and z of all of the vertices 220 of polyhedron of golf ball 200. Faces are constructed by connecting the vertices with straight lines. The coordinates x, y, z are normalized by the diameter D of the circumscribed sphere 204 and are with respect to the center of the sphere. The golf ball 200 contains 380 vertices 222, 570 straight edges 220 and 192 polygonal faces 220.
• From a visual perspective the above designs of FIGS. 2, 3 have the unique characteristics of not having any dimples. From a utility perspective the behavior of the drag coefficient is very interesting. FIG. 4 shows a graph of the drag coefficient, C_D, versus the Reynolds number, Re, for the polyhedron with 162 and 192 faces. The drag coefficient was obtained by wind tunnel experiments of non-spinning models. Overall the drag curve is qualitatively very similar to that of a dimpled sphere. Namely there is a drag crisis that occurs around Re=60,000. For the polyhedron with 162 faces C_D reaches a minimum value of 0.16 at Re=90,000 and remains almost constant as the Reynolds increases. For the polyhedron with 192 faces C_D reaches a minimum value of 0.14 at Re=110,000 and remains almost constant as the Reynolds increases.
• The graph reveals that as the number of faces increases the drag crisis shifts towards a higher Reynolds number and the C_D in the post-critical regime decreases. This feature can be taken into advantage when designing a golf ball to tailor the needs of a golfer. For an amateur golfer a golf ball in flight during a driver shot can experience a Reynolds number ranging from 180,000 at the beginning of the flight to 60,000 at the end of the flight. For a professional golfer a golf ball in flight during a driver shot can experience a Reynolds number ranging from 220,000 at the beginning of the flight to 80,000 at the end of the flight.
• In other words, the range of Reynolds number experienced by a golf ball for an amateur golfer is lower than that for a professional golfer. A golf ball with more polygon faces might suit the needs of a professional golfer while a golf ball with less polygon faces might suit the needs of an amateur golfer.
• The advantage of a design based on a convex polyhedron such as golf ball 200 with respect to a dimpled golf ball is now discussed. A comparison of the drag curve of the polyhedron with 192 faces, namely golf ball 200, against a dimpled sphere is shown in FIG. 5. The dimpled sphere has 322 spherical dimples and is representative of a commercial golf ball. The drag crisis for the polyhedron with 192 faces, namely golf ball 200, occurs at approximately the same range of Reynolds numbers as the dimpled sphere. The minimum C_D for both balls is reached at Re=110,000. For the dimpled sphere C_D=0.16 while for the golf ball 200 C_D=0.14, that is 12.5% drag reduction. At Re=140,000 C_D=0.174 for the dimpled sphere while for the golf ball 200 C_D=0.147, that is 15% drag reduction. Indeed, the drag coefficient for golf ball 200 illustrated in FIG. 3 is consistently lower than that of a dimpled golf ball in the range of Re=90,000-220,000. A lower drag coefficient is important because it plays an important role in achieving more carry distance during a driver shot.
• A comparison of the drag curve of the golf ball 100 against the same dimpled sphere is shown in FIG. 6. While C_D in the post-critical regime is almost identical for the two balls, the drag crisis for the golf ball 200 occurs at a lower Reynolds number. As a result, C_D for Re<110,000 is consistently lower for the polyhedron with 192 faces than for the dimpled sphere. Thus, the embodiment of FIG. 2(a) has a lower drag coefficient than a dimpled sphere. A lower drag coefficient is important because it plays an important role in achieving more carry distance during a driver shot.
• FIGS. 7, 8 are additional non-limiting embodiments of the invention. Those golf balls 300, 400 have similar structure as the embodiments of FIGS. 2, 3, and those structures have similar purpose. Those structures have been assigned a similar reference numeral and similar structure with the differences noted below. For example, FIG. 7 shows an example of a golf ball 300 with a body 310 with an inner core and an outer shell with an outer surface 312. A plurality of faces 320 are formed in the outer surface, creating a pattern 316. All the faces are formed in the outer surface 312 and each of such faces is flat and lies in a plane. The faces are bound by straight or linear edges 324 and corners vertices 322. The body 310 defines a circumscribed sphere 302, which is the smallest sphere containing the polyhedron, and is shown with a dashed line about the outermost points 304 of the body 310. And FIG. 8 shows an example of a golf ball 400 with a body 410 with an inner core and an outer shell with an outer surface 412. A plurality of faces 420 are formed in the outer surface, creating a pattern 416. All the faces are formed in the outer surface 412 and each of such faces is flat and lies in a plane. The faces are bound by straight or linear edges 424 and corners or vertices 422. The body 410 defines a circumscribed sphere 402, which is the smallest sphere containing the polyhedron, and is shown with a dashed line about the outermost points 404 of the body 410.
• The embodiment shown in FIGS. 7, 8 are included for illustrative purposes as examples of convex polyhedral that are not made from pentagons or hexagons. Other convex polyhedra not made up of pentagons or hexagons can also be used for the design of a golf ball. In another embodiment of the present invention a convex polyhedron is shown in FIG. 7. The polyhedron belongs to a class of solids called Geodesic polyhedron which are derived from an icosahedron by subdividing each face into smaller faces using a triangular grid, and then applying a canonicalization algorithm to make the result more spherical. (see https://en.wikipedia.org/wiki/Geodesic_polyhedron). The vertices of the polyhedron are the only points on the polyhedron that lie on a sphere. Any point along the edges or on the face of the triangles lies below the surface of a circumscribed sphere. The polyhedron in FIG. 7 is made from 320 triangles but any geodesic polyhedron with an arbitrary number of triangles can be used as the design of a golf ball.
• In another embodiment of the present invention a convex polyhedron is shown in FIG. 8. The polyhedron belongs to a class of solids called Geodesic cubes which are made from rectangular faces. A geodesic cube is a polyhedron derived from a cube by subdividing each face into smaller faces using a square grid, and then applying a canonicalization algorithm to make the result more spherical (see http://dmccooey.com/polyhedra/GeodesicCubes.html). The vertices of the polyhedron are the only points on the polyhedron that lie on a sphere. Any point along the edges or on the face of the rectangular faces lies below the surface of a circumscribed sphere. The polyhedron in FIG. 8 is made from 174 rectangular flat faces but any geodesic cube with an arbitrary number of faces can be used as the design of a golf ball.
• It is important to note that the polyhedra described above and shown in FIGS. 2, 3, 7, 8 do not contain any dimples (i.e., curved or spherical depressions or indents), but instead have flat surfaces that lie in a plane. However, the polyhedra provide enhanced aerodynamic characteristics, drag coefficient being one of them, that can help increase the carry distance of the golf ball.
• However, golf ball geometries that are based on any convex polyhedra and either include or omit dimples are contemplated. In particular, for any convex polyhedra at least one of the faces may include one or more dimples. For example, FIG. 9 shows an embodiment of a golf ball 500 that is based on a convex polyhedron with a plurality of polygonal faces 520. The convex polyhedron is identical to the one shown in FIG. 2, and includes a body 510 with an inner core and an outer shell with an outer surface 512. A plurality of faces 520 are formed in the outer surface, creating a pattern 516. All the faces are formed in the outer surface 512 and each of such faces is flat and lies in a plane. The faces are bound by straight or linear edges 524 and corners or vertices 522. The body 510 defines a circumscribed sphere 502, which is the smallest sphere containing the polyhedron, and is shown with a dashed line about the outermost points 504 of the body 510.
• However, in this embodiment each face 520 of the polyhedron contains one dimple 560. The dimples 560 have a substantially spherical shape and are created by subtracting spheres 570 from each of the faces 510 of the polyhedron 500. Table 3 lists the coordinates x, y and z of the center of the spheres 570 along with the diameter d of the spheres 570. The coordinates x, y, z and the diameter of the spheres d are normalized by the diameter D of the circumscribed sphere 504. The coordinates x, y, and z are with respect to the center of the circumscribed sphere. The total dimple volume ratio, defined as the aggregate volume removed from the surface of the polyhedron divided by the volume of the polyhedron, is 0.317%.
• As shown, the dimples 560 are each located at the center of each 510. However, the dimples 560 can be positioned at another location such as offset within each face 510, or overlapping two or more faces 510. In addition, while spherical dimples 560 are shown, the dimples 560 can have any suitable size and shape. For example, non-spherical depressions can be utilized, such as triangles, hexagons, pentagons. And the dimples need not all have the same size and shape, for example there can be dimples with more than one size and more than one shape.
• The effect that the addition of dimples has on the drag coefficient is now discussed. A comparison of the drag curve of this embodiment against that of golf ball 100 is shown in FIG. 10. There are two important observations. First when dimples are added to the faces of a polyhedron, the drag crisis occurs at a lower Reynolds number range. That is, the drag coefficient starts to drop at a lower Reynolds number. Second, the drag coefficient in the post-critical regime increases. This effect may be desirable when designing a golf ball for players with lower swing speeds such as an amateur golf player where the range of Reynolds number that the golf ball experiences during a driver shot is reduced. As the total dimple volume approaches zero the drag curve of golf ball 500 would approach that of golf ball 100. Therefore, one can fine tune the exact behavior of the drag curve by adjusting the total dimple volume. Each dimple is formed by subtracting a sphere for the face of the polyhedron as explained above. The dimple volume is the amount of volume that each sphere subtracts from the volume of the polyhedron.
• FIG. 11 compares the drag curve of the golf ball 500 shown in FIG. 9 against a commercial golf ball, namely the Callaway SuperHot, which is a dimpled ball marketed as having a low drag coefficient and shown in U.S. Pat. No. 6,290,615. The drag coefficient for both balls were obtained by performing wind tunnel experiments using the actual commercial golf ball and a 3D printed prototype of the golf ball 500. The drag crisis for the golf ball 500 happens earlier, that is the drag coefficient starts to drop at lower Reynolds number. Clearly C_D for golf ball 500 is consistently lower than that of the commercial golf ball for the range of Reynolds numbers 60,000-160,000. A lower drag coefficient can help a golf ball achieve longer carry distances. The golf ball 500 is the exact same polyhedron of golf ball 100, however the golf ball 500 has a dimple on each face. The polyhedron which golf ball 500 is based on is identical to the polyhedron of golf ball 100 in FIG. 2.
• FIG. 12 shows an example of a golf ball based on a convex polyhedron with dimples in accordance with another embodiment of the invention. The golf ball 600 has a body 610 with an inner core and an outer shell with an outer surface 612. A plurality of faces 620 are formed in the outer surface, creating a pattern 616. All the faces are formed in the outer surface and each of such faces is flat and lies in a plane. The faces are bound by straight or linear edges 624. Here, the golf ball 600 is based on a polyhedron with 312 polygons.
• The body 610 defines a circumscribed sphere 602, which is the smallest sphere containing the polyhedron, and is shown with a dashed line about the outermost points 604 of the body 610. The vertices 622 a, 622 b of the polyhedron are the only points 604 on the polyhedron that lie on the sphere. Any point along the edges 624 or on the face of the polygons lies below the surface of the circumscribed sphere.
• The golf ball body 610 is a polyhedron that is made from first faces 620 a and second faces 620 b. As shown, the first faces 620 a have a first shape, namely pentagons, and the second faces 620 b have a second shape different than the first shape, namely hexagons. All of the faces are flat and each such face forms a single plane. In one embodiment, there are 12 pentagons 620 a and 300 hexagons 620 b (a hexagon-to-pentagon ratio of 25:1), each having corners or points 622 a, 622 b connected by boundaries such as straight lines or edges 624 a, 624 b. The first and second faces 620 a, 620 b form the pattern 616. The edges 624 are sharp, in that the radius of curvature of an edge is less than 0.001 D, where D is the diameter of the circumscribed sphere. Ideally all edges should be the sharpest possible. A radius of curvature of 0 is the case with the sharpest possible edges, though golf ball edges might become a little rounded during manufacture or follow use.
• The geometric shape of the embodiment illustrated in FIG. 12 falls into the class of convex polyhedral where the angle between any pair of adjacent faces is always less than 180 degrees. In this embodiment, the angle between a pentagon face 620 a and an adjacent hexagonal face 620 b is 170.2 degrees. The angle between two adjacent hexagon faces 620 b varies from 167.1 degrees to 168.2 degrees.
• Each face 620 is immediately adjacent and touching a neighboring face 620, such that each edge 624 forms a border between two neighboring faces 620 and each point 622 is at the intersection of three neighboring faces 620. And, each point 622 is at an opposite end of each linear edge 624 and is at the intersection of three linear edges 624. Accordingly, there is no gap or space between adjacent neighboring faces 620, and the faces 620 are contiguous and form a single integral, continuous outer surface 612 of the ball 600.
• The pattern is based on an icosahedron shown in FIG. 2(c). In the icosahedron 170 five equilateral triangles 180 meet at each of its twelve vertices 182. The 12 pentagons 620 a of the golf ball 600 shown in FIG. 12 are centered on the vertices of an icosahedron. Therefore, any pair of 3 pentagons 620 a form an equilateral triangle 680 shown with dashed line in FIG. 12. The pentagons 620 a are all equilateral, that is the 5 edges 624 a all have the same length equal to 0.102 D, where D is the diameter of the circumscribed sphere. The hexagons 620 b are not equilateral and the lengths of the edges 624 b vary from 0.102 D to 0.132 D.
• Table 4 lists the coordinates of all the vertices and how each face is formed, which defines the geometry of the polyhedron. Table 4 lists the coordinates x, y, and z of all of the vertices 622 of polyhedron of golf ball 600. Faces are constructed by connecting the group of vertices with straight lines. The coordinates x, y, z are normalized by the diameter D of the circumscribed sphere 604 and are with respect to the center of the sphere. The golf ball 600 contains 620 vertices 622, 930 straight edges 624 and 312 polygonal faces 620.
• Each of the face 620 of the polyhedron contains one dimple 690. The dimples 690 have a substantially spherical shape and are created by subtracting spheres 670 from the face 620 of the polyhedron. Table 5 lists the coordinates x, y and z of the center of the spheres 670 along with the diameter d of the spheres 670. The coordinates x, y, z and the diameter of the spheres d are normalized by the diameter D of the circumscribed sphere 604. The coordinates x, y, and z are with respect to the center of the circumscribed sphere.
• A comparison of the drag curve of this embodiment against that of a commercial golf ball, namely the Bridgestone Tour is shown in FIG. 13, which is a dimpled ball marketed as having a low drag so that the ball travels further, and described in U.S. Pat. No. 7,503,857. The graphs shows the invention having a lower drag coefficient. The drag coefficient for both balls were obtained by performing wind tunnel experiments using the actual commercial golf ball and a 3D printed prototype of the embodiment. The drag crisis for both golf balls occurs at approximately the same range of Reynolds number, namely from Re=50,000-80,000. At Reynolds number of 100,000 C_D for the Bridgestone Tour ball is 0.195 while C_D for the current embodiment is 0.162, a drag reduction of 17%. Overall C_D for the current embodiment is consistently lower than that of the commercial golf ball for the range of Reynolds numbers 80,000-160,000. A lower drag coefficient can help a golf ball achieve longer carry distances.
• Other sizes and shapes of dimples are contemplated by the present invention, and any suitable size and shape dimple can be utilized in the golf balls of FIGS. 2-3, 7-9, 12. While FIG. 2 employs a convex polyhedron that belongs to a class of Goldberg polyhedral, FIGS. 9, 12 are revisions to the Goldberg principle in that each face of the polyhedron contains a dimple.
• The following documents are incorporated herein by reference. Achenbach, E. (1972). Experiments on the flow past spheres at high Reynolds numbers. Journal of Fluid Mechanics. Harvey, P. W. (1976). Golf ball aerodynamics. Aeronautical Quarterly. J. Choi, W. J. (2006). Mechanism of drag reduction by dimples on a sphere. Physics of Fluids, 149-167. Ogg, S. S. (2001).
• It is further noted that the description and claims use several positional, geometric or relational terms, such as neighboring, circular, spherical, round, and flat. Those terms are merely for convenience to facilitate the description based on the embodiments shown in the figures. Those terms are not intended to limit the invention. Thus, it should be recognized that the invention can be described in other ways without those geometric, relational, directional or positioning terms. In addition, the geometric or relational terms may not be exact. For instance, edges may not be exactly linear, hexagonal, pentagonal or spherical, but still be considered to be substantially linear, hexagonal, pentagonal or spherical, and faces may not be exactly flat or planar but still be considered to be substantially flat or planar because of, for example, roughness of surfaces, tolerances allowed in manufacturing, etc. And, other suitable geometries and relationships can be provided without departing from the spirit and scope of the invention.
• In addition, while the invention has been shown and described with boundaries formed by straight edges 124, 224, 324, 424, 524, 624 and points 122, 222, 322, 422, 522, other suitable boundaries can be provided such as curves with or without rounded points. And the boundaries need not be sharped, but can be curved. And other suitable shapes can be utilized for the faces. And, while the invention has been described with a certain number of faces and/or dimples, other suitable numbers of faces and/or dimples, more or fewer, can also be provided within the spirit and scope of the invention. Within this specification, the various sizes, shapes and dimensions are approximate and exemplary to illustrate the scope of the invention and are not limiting. The faces need not all be the same shape and/or size, and there can be multiple sizes and shapes of faces.
• The sizes and the terms “substantially” and “about” mean plus or minus 15-20%, and in one embodiment plus or minus 10%, and in other embodiments plus or minus 5%, and plus or minus 1-2%. In addition, while specific dimensions, sizes and shapes may be provided in certain embodiments of the invention, those are simply to illustrate the scope of the invention and are not limiting. Thus, other dimensions, sizes and/or shapes can be utilized without departing from the spirit and scope of the invention.
• Use of the term optional or alternative with respect to any element of a claim means that the element is required, or alternatively, the element is not required, both alternatives being within the scope of the claim. Use of broader terms such as comprises, includes, and having may be understood to provide support for narrower terms such as consisting of, consisting essentially of, and comprised substantially of. Accordingly, the scope of protection is not limited by the description set out above but is defined by the claims that follow, that scope including all equivalents of the subject matter of the claims. Each and every claim is incorporated as further disclosure into the specification and the claims are embodiment(s) of the present disclosure.

• TABLE 1
  Vertex	x/D	y/D	z/D	Face	Group of vertices

  1	0.0000	0.0000	0.9778	1	301	300	69	296	295	20
  2	0.6519	0.0000	0.7288	2	312	311	74	302	301	20
  3	−0.3260	0.5646	0.7288	3	295	294	79	313	312	20
  4	−0.3259	−0.5646	0.7288	4	299	300	301	302	303	304
  5	0.7288	0.5646	0.3259	5	293	294	295	296	297	298
  6	0.7288	−0.5645	0.3259	6	310	311	312	313	314	315
  7	−0.8533	0.3489	0.3259	7	300	299	70	292	291	69
  8	0.1245	0.9135	0.3259	8	303	302	74	308	307	73
  9	0.1245	−0.9135	0.3259	9	297	296	69	291	290	68
  10	−0.8533	−0.3489	0.3260	10	311	310	75	309	308	74
  11	0.8533	0.3489	−0.3260	11	294	293	80	320	319	79
  12	0.8533	−0.3489	−0.3259	12	314	313	79	319	318	78
  13	−0.7288	0.5645	−0.3259	13	288	289	290	291	292
  14	−0.1245	0.9135	−0.3259	14	305	306	307	308	309
  15	−0.1245	−0.9135	−0.3259	15	316	317	318	319	320
  16	−0.7288	−0.5646	−0.3259	16	244	243	70	299	304	17
  17	0.3259	0.5646	−0.7288	17	304	303	73	251	250	17
  18	0.3260	−0.5646	−0.7288	18	298	297	68	268	267	18
  19	−0.6519	0.0000	−0.7288	19	261	260	80	293	298	18
  20	0.0000	0.0000	−0.9778	20	278	277	75	310	315	19
  21	0.1755	0.3040	0.9230	21	315	314	78	285	284	19
  22	0.4845	0.3040	0.8050	22	288	292	70	243	242	66
  23	0.5210	0.5716	0.6140	23	252	251	73	307	306	72
  24	0.2345	0.7370	0.6140	24	269	268	68	290	289	67
  25	0.0210	0.5716	0.8050	25	305	309	75	277	276	71
  26	−0.3510	0.0000	0.9230	26	316	320	80	260	259	76
  27	−0.5055	0.2676	0.8050	27	286	285	78	318	317	77
  28	−0.7555	0.1654	0.6140	28	183	182	67	289	288	66
  29	−0.7555	−0.1654	0.6140	29	206	205	72	306	305	71
  30	−0.5055	−0.2676	0.8050	30	229	228	77	317	316	76
  31	0.1755	−0.3040	0.9230	31	250	249	53	245	244	17
  32	0.0210	−0.5716	0.8050	32	267	266	59	262	261	18
  33	0.2345	−0.7370	0.6140	33	284	283	63	279	278	19
  34	0.5210	−0.5716	0.6140	34	242	243	244	245	246	247
  35	0.4845	−0.3040	0.8050	35	248	249	250	251	252	253
  36	0.8770	0.0000	0.4540	36	265	266	267	268	269	270
  37	0.9135	−0.2676	0.2630	37	276	277	278	279	280	281
  38	0.9725	−0.1654	−0.0460	38	259	260	261	262	263	264
  39	0.9725	0.1654	−0.0460	39	282	283	284	285	286	287
  40	0.9135	0.2676	0.2630	40	184	183	66	242	247	11
  41	−0.4385	0.7595	0.4540	41	253	252	72	205	204	14
  42	−0.2250	0.9249	0.2630	42	270	269	67	182	181	12
  43	−0.3430	0.9249	−0.0460	43	207	206	71	276	281	13
  44	−0.6295	0.7595	−0.0460	44	230	229	76	259	264	15
  45	−0.6885	0.6573	0.2630	45	287	286	77	228	227	16
  46	−0.4385	−0.7595	0.4540	46	246	245	53	239	238	52
  47	−0.6885	−0.6573	0.2630	47	249	248	54	240	239	53
  48	−0.6295	−0.7595	−0.0460	48	266	265	60	258	257	59
  49	−0.3430	−0.9249	−0.0460	49	263	262	59	257	256	58
  50	−0.2250	−0.9249	0.2630	50	280	279	63	273	272	62
  51	0.6295	0.7595	0.0460	51	283	282	64	274	273	63
  52	0.6885	0.6573	−0.2630	52	179	180	181	182	183	184
  53	0.4385	0.7595	−0.4540	53	202	203	204	205	206	207
  54	0.2250	0.9249	−0.2630	54	225	226	227	228	229	230
  55	0.3430	0.9249	0.0460	55	247	246	52	188	187	11
  56	0.6295	−0.7595	0.0460	56	198	197	54	248	253	14
  57	0.3430	−0.9249	0.0460	57	175	174	60	265	270	12
  58	0.2250	−0.9249	−0.2630	58	281	280	62	211	210	13
  59	0.4385	−0.7595	−0.4540	59	264	263	58	234	233	15
  60	0.6885	−0.6573	−0.2630	60	221	220	64	282	287	16
  61	−0.9725	0.1654	0.0460	61	237	238	239	240	241
  62	−0.9135	0.2676	−0.2630	62	254	255	256	257	258
  63	−0.8770	0.0000	−0.4540	63	271	272	273	274	275
  64	−0.9135	−0.2676	−0.2630	64	187	186	39	179	184	11
  65	−0.9725	−0.1654	0.0460	65	181	180	38	176	175	12
  66	0.7555	0.1654	−0.6140	66	204	203	43	199	198	14
  67	0.7555	−0.1654	−0.6140	67	210	209	44	202	207	13
  68	0.5055	−0.2676	−0.8050	68	233	232	49	225	230	15
  69	0.3510	0.0000	−0.9230	69	227	226	48	222	221	16
  70	0.5055	0.2676	−0.8050	70	189	188	52	238	237	51
  71	−0.5210	0.5716	−0.6140	71	197	196	55	241	240	54
  72	−0.2345	0.7370	−0.6140	72	254	258	60	174	173	56
  73	−0.0210	0.5716	−0.8050	73	235	234	58	256	255	57
  74	−0.1755	0.3040	−0.9230	74	212	211	62	272	271	61
  75	−0.4845	0.3040	−0.8050	75	220	219	65	275	274	64
  76	−0.2345	−0.7370	−0.6140	76	180	179	39	171	170	38
  77	−0.5210	−0.5716	−0.6140	77	203	202	44	194	193	43
  78	−0.4845	−0.3040	−0.8050	78	226	225	49	217	216	48
  79	−0.1755	−0.3040	−0.9230	79	185	186	187	188	189	190
  80	−0.0210	−0.5716	−0.8050	80	196	197	198	199	200	201
  81	0.2485	0.4305	0.8677	81	173	174	175	176	177	178
  82	0.3932	0.4305	0.8124	82	208	209	210	211	212	213
  83	0.4103	0.5558	0.7230	83	231	232	233	234	235	236
  84	0.2762	0.6332	0.7230	84	219	220	221	222	223	224
  85	0.1762	0.5558	0.8124	85	237	241	55	105	104	51
  86	0.0023	0.3047	0.9342	86	160	159	57	255	254	56
  87	−0.0798	0.4470	0.8714	87	271	275	65	134	133	61
  88	−0.2538	0.4396	0.8361	88	186	185	40	172	171	39
  89	−0.3472	0.2926	0.8714	89	177	176	38	170	169	37
  90	−0.2650	0.1503	0.9342	90	200	199	43	193	192	42
  91	−0.0895	0.1550	0.9616	91	209	208	45	195	194	44
  92	0.2627	−0.1544	0.9342	92	232	231	50	218	217	49
  93	0.4270	−0.1544	0.8714	93	223	222	48	216	215	47
  94	0.5077	0.0000	0.8361	94	190	189	51	104	109	5
  95	0.4270	0.1544	0.8714	95	106	105	55	196	201	8
  96	0.2627	0.1544	0.9342	96	161	160	56	173	178	6
  97	0.1790	0.0000	0.9616	97	236	235	57	159	158	9
  98	0.8203	0.1503	0.5196	98	213	212	61	133	138	7
  99	0.8397	0.2926	0.4181	99	135	134	65	219	224	10
  100	0.7466	0.4396	0.4540	100	168	169	170	171	172
  101	0.6405	0.4470	0.5963	101	191	192	193	194	195
  102	0.6211	0.3047	0.6979	102	214	215	216	217	218
  103	0.7078	0.1550	0.6571	103	100	99	40	185	190	5
  104	0.5551	0.7856	0.2006	104	201	200	42	113	112	8
  105	0.4028	0.8735	0.2006	105	178	177	37	165	164	6
  106	0.2971	0.8664	0.3433	106	129	128	45	208	213	7
  107	0.3477	0.7781	0.4890	107	152	151	50	231	236	9
  108	0.5000	0.6902	0.4890	108	224	223	47	142	141	10
  109	0.6018	0.6905	0.3433	109	104	105	106	107	108	109
  110	−0.0467	0.6902	0.6979	110	156	157	158	159	160	161
  111	0.0669	0.7781	0.5963	111	133	134	135	136	137	138
  112	0.0075	0.8664	0.4540	112	168	172	40	99	98	36
  113	−0.1664	0.8735	0.4181	113	166	165	37	169	168	36
  114	−0.2800	0.7856	0.5196	114	114	113	42	192	191	41
  115	−0.2196	0.6905	0.6571	115	191	195	45	128	127	41
  116	−0.4971	0.0000	0.8677	116	214	218	50	151	150	46
  117	−0.5694	0.1253	0.8124	117	143	142	47	215	214	46
  118	−0.6865	0.0774	0.7230	118	109	108	23	101	100	5
  119	−0.6865	−0.0774	0.7230	119	112	111	24	107	106	8
  120	−0.5694	−0.1253	0.8124	120	164	163	34	156	161	6
  121	−0.2650	−0.1503	0.9342	121	138	137	28	130	129	7
  122	−0.3472	−0.2926	0.8714	122	158	157	33	153	152	9
  123	−0.2538	−0.4396	0.8361	123	141	140	29	136	135	10
  124	−0.0798	−0.4470	0.8714	124	98	99	100	101	102	103
  125	0.0023	−0.3047	0.9342	125	110	111	112	113	114	115
  126	−0.0895	−0.1550	0.9616	126	162	163	164	165	166	167
  127	−0.5403	0.6353	0.5196	127	127	128	129	130	131	132
  128	−0.6733	0.5809	0.4181	128	150	151	152	153	154	155
  129	−0.7540	0.4267	0.4540	129	139	140	141	142	143	144
  130	−0.7073	0.3312	0.5963	130	167	166	36	98	103	2
  131	−0.5744	0.3855	0.6979	131	115	114	41	127	132	3
  132	−0.4881	0.5354	0.6571	132	144	143	46	150	155	4
  133	−0.9579	0.0879	0.2007	133	108	107	24	84	83	23
  134	−0.9579	−0.0880	0.2007	134	157	156	34	148	147	33
  135	−0.8988	−0.1759	0.3433	135	137	136	29	119	118	28
  136	−0.8477	−0.0880	0.4890	136	102	101	23	83	82	22
  137	−0.8477	0.0880	0.4890	137	111	110	25	85	84	24
  138	−0.8988	0.1759	0.3433	138	163	162	35	149	148	34
  139	−0.5744	−0.3855	0.6979	139	131	130	28	118	117	27
  140	−0.7073	−0.3312	0.5963	140	154	153	33	147	146	32
  141	−0.7540	−0.4267	0.4540	141	140	139	30	120	119	29
  142	−0.6733	−0.5809	0.4181	142	103	102	22	95	94	2
  143	−0.5403	−0.6353	0.5196	143	94	93	35	162	167	2
  144	−0.4881	−0.5354	0.6571	144	88	87	25	110	115	3
  145	0.2485	−0.4305	0.8677	145	132	131	27	89	88	3
  146	0.1762	−0.5558	0.8124	146	155	154	32	124	123	4
  147	0.2762	−0.6332	0.7230	147	123	122	30	139	144	4
  148	0.4103	−0.5558	0.7230	148	81	82	83	84	85
  149	0.3932	−0.4305	0.8124	149	145	146	147	148	149
  150	−0.2800	−0.7856	0.5196	150	116	117	118	119	120
  151	−0.1664	−0.8735	0.4181	151	96	95	22	82	81	21
  152	0.0075	−0.8664	0.4540	152	81	85	25	87	86	21
  153	0.0669	−0.7781	0.5963	153	145	149	35	93	92	31
  154	−0.0467	−0.6902	0.6979	154	90	89	27	117	116	26
  155	−0.2196	−0.6905	0.6571	155	125	124	32	146	145	31
  156	0.5000	−0.6902	0.4890	156	116	120	30	122	121	26
  157	0.3477	−0.7781	0.4890	157	92	93	94	95	96	97
  158	0.2971	−0.8664	0.3433	158	86	87	88	89	90	91
  159	0.4028	−0.8735	0.2007	159	121	122	123	124	125	126
  160	0.5551	−0.7856	0.2007	160	97	96	21	86	91	1
  161	0.6018	−0.6905	0.3433	161	126	125	31	92	97	1
  162	0.6211	−0.3047	0.6979	162	91	90	26	121	126	1
  163	0.6405	−0.4470	0.5963
  164	0.7466	−0.4396	0.4540
  165	0.8397	−0.2926	0.4181
  166	0.8203	−0.1503	0.5196
  167	0.7078	−0.1550	0.6571
  168	0.9490	0.0000	0.3154
  169	0.9660	−0.1253	0.2259
  170	0.9937	−0.0774	0.0813
  171	0.9937	0.0774	0.0813
  172	0.9660	0.1253	0.2259
  173	0.7492	−0.6353	0.0271
  174	0.7806	−0.5809	−0.1372
  175	0.8647	−0.4267	−0.1643
  176	0.9247	−0.3312	−0.0271
  177	0.8934	−0.3855	0.1372
  178	0.8019	−0.5354	0.1643
  179	0.9579	0.0880	−0.2007
  180	0.9579	−0.0879	−0.2007
  181	0.8988	−0.1759	−0.3433
  182	0.8477	−0.0880	−0.4890
  183	0.8477	0.0880	−0.4890
  184	0.8988	0.1759	−0.3433
  185	0.8934	0.3855	0.1372
  186	0.9247	0.3312	−0.0271
  187	0.8647	0.4267	−0.1643
  188	0.7805	0.5809	−0.1372
  189	0.7492	0.6353	0.0271
  190	0.8019	0.5354	0.1643
  191	−0.4745	0.8218	0.3154
  192	−0.3745	0.8993	0.2259
  193	−0.4298	0.8993	0.0813
  194	−0.5639	0.8218	0.0813
  195	−0.5915	0.7740	0.2259
  196	0.1756	0.9664	0.0271
  197	0.1128	0.9664	−0.1372
  198	−0.0628	0.9622	−0.1643
  199	−0.1756	0.9664	−0.0271
  200	−0.1128	0.9664	0.1372
  201	0.0628	0.9622	0.1643
  202	−0.5551	0.7856	−0.2007
  203	−0.4028	0.8735	−0.2007
  204	−0.2971	0.8664	−0.3433
  205	−0.3477	0.7781	−0.4890
  206	−0.5000	0.6902	−0.4890
  207	−0.6018	0.6905	−0.3433
  208	−0.7806	0.5809	0.1372
  209	−0.7492	0.6353	−0.0271
  210	−0.8019	0.5354	−0.1643
  211	−0.8934	0.3855	−0.1372
  212	−0.9247	0.3312	0.0271
  213	−0.8647	0.4267	0.1643
  214	−0.4745	−0.8218	0.3154
  215	−0.5915	−0.7740	0.2259
  216	−0.5639	−0.8218	0.0813
  217	−0.4298	−0.8993	0.0813
  218	−0.3745	−0.8993	0.2259
  219	−0.9247	−0.3312	0.0271
  220	−0.8934	−0.3855	−0.1372
  221	−0.8019	−0.5354	−0.1643
  222	−0.7492	−0.6353	−0.0271
  223	−0.7805	−0.5809	0.1372
  224	−0.8647	−0.4267	0.1643
  225	−0.4028	−0.8735	−0.2006
  226	−0.5551	−0.7856	−0.2006
  227	−0.6018	−0.6905	−0.3433
  228	−0.5000	−0.6902	−0.4890
  229	−0.3477	−0.7781	−0.4890
  230	−0.2971	−0.8664	−0.3433
  231	−0.1128	−0.9664	0.1372
  232	−0.1756	−0.9664	−0.0271
  233	−0.0628	−0.9622	−0.1643
  234	0.1128	−0.9664	−0.1372
  235	0.1756	−0.9664	0.0271
  236	0.0628	−0.9622	0.1643
  237	0.5639	0.8218	−0.0813
  238	0.5915	0.7740	−0.2259
  239	0.4745	0.8218	−0.3154
  240	0.3745	0.8993	−0.2259
  241	0.4298	0.8993	−0.0813
  242	0.7073	0.3312	−0.5963
  243	0.5744	0.3855	−0.6979
  244	0.4881	0.5354	−0.6571
  245	0.5403	0.6353	−0.5196
  246	0.6733	0.5809	−0.4181
  247	0.7540	0.4267	−0.4540
  248	0.1664	0.8735	−0.4181
  249	0.2800	0.7856	−0.5196
  250	0.2196	0.6905	−0.6571
  251	0.0467	0.6902	−0.6979
  252	−0.0669	0.7781	−0.5963
  253	−0.0075	0.8664	−0.4540
  254	0.5639	−0.8218	−0.0813
  255	0.4298	−0.8993	−0.0813
  256	0.3745	−0.8993	−0.2259
  257	0.4745	−0.8218	−0.3154
  258	0.5915	−0.7740	−0.2259
  259	−0.0669	−0.7781	−0.5963
  260	0.0467	−0.6902	−0.6979
  261	0.2196	−0.6905	−0.6571
  262	0.2800	−0.7856	−0.5196
  263	0.1664	−0.8735	−0.4181
  264	−0.0075	−0.8664	−0.4540
  265	0.6733	−0.5809	−0.4181
  266	0.5403	−0.6353	−0.5196
  267	0.4881	−0.5354	−0.6571
  268	0.5744	−0.3855	−0.6979
  269	0.7073	−0.3312	−0.5963
  270	0.7540	−0.4267	−0.4540
  271	−0.9937	0.0774	−0.0813
  272	−0.9660	0.1253	−0.2259
  273	−0.9490	0.0000	−0.3154
  274	−0.9660	−0.1253	−0.2259
  275	−0.9937	−0.0774	−0.0813
  276	−0.6405	0.4470	−0.5963
  277	−0.6211	0.3047	−0.6979
  278	−0.7078	0.1550	−0.6571
  279	−0.8203	0.1503	−0.5196
  280	−0.8397	0.2926	−0.4181
  281	−0.7466	0.4396	−0.4540
  282	−0.8397	−0.2926	−0.4181
  283	−0.8203	−0.1503	−0.5196
  284	−0.7078	−0.1550	−0.6571
  285	−0.6211	−0.3047	−0.6979
  286	−0.6405	−0.4470	−0.5963
  287	−0.7466	−0.4396	−0.4540
  288	0.6865	0.0774	−0.7230
  289	0.6865	−0.0774	−0.7230
  290	0.5694	−0.1253	−0.8124
  291	0.4971	0.0000	−0.8677
  292	0.5694	0.1253	−0.8124
  293	0.0798	−0.4470	−0.8714
  294	−0.0023	−0.3047	−0.9342
  295	0.0895	−0.1550	−0.9616
  296	0.2650	−0.1503	−0.9342
  297	0.3472	−0.2926	−0.8714
  298	0.2538	−0.4396	−0.8361
  299	0.3472	0.2926	−0.8714
  300	0.2650	0.1503	−0.9342
  301	0.0895	0.1550	−0.9616
  302	−0.0023	0.3047	−0.9342
  303	0.0798	0.4470	−0.8714
  304	0.2538	0.4396	−0.8361
  305	−0.4103	0.5558	−0.7230
  306	−0.2762	0.6332	−0.7230
  307	−0.1762	0.5558	−0.8124
  308	−0.2485	0.4305	−0.8677
  309	−0.3932	0.4305	−0.8124
  310	−0.4270	0.1544	−0.8714
  311	−0.2627	0.1544	−0.9342
  312	−0.1790	0.0000	−0.9616
  313	−0.2627	−0.1544	−0.9342
  314	−0.4270	−0.1544	−0.8714
  315	−0.5077	0.0000	−0.8361
  316	−0.2762	−0.6332	−0.7230
  317	−0.4103	−0.5558	−0.7230
  318	−0.3932	−0.4305	−0.8124
  319	−0.2485	−0.4305	−0.8677
  320	−0.1762	−0.5558	−0.8124

• TABLE 2
  Vertex	x/D	y/D	z/D	Face	Group of vertices

  1	0.0166	0.0382	0.4983	1	96	168	240	216	144
  2	0.0166	−0.0382	−0.4983	2	97	169	241	217	145
  3	−0.0166	−0.0382	0.4983	3	98	170	242	218	146
  4	−0.0166	0.0382	−0.4983	4	99	171	243	219	147
  5	0.4983	0.0166	0.0382	5	100	172	244	221	149
  6	0.4983	−0.0166	−0.0382	6	101	173	245	220	148
  7	−0.4983	−0.0166	0.0382	7	102	174	246	223	151
  8	−0.4983	0.0166	−0.0382	8	103	175	247	222	150
  9	0.0382	0.4983	0.0166	9	104	176	248	226	154
  10	0.0382	−0.4983	−0.0166	10	105	177	249	227	155
  11	−0.0382	−0.4983	0.0166	11	106	178	250	224	152
  12	−0.0382	0.4983	−0.0166	12	107	179	251	225	153
  13	0.0979	0.0465	0.4881	13	72	26	0	12	60	84
  14	0.0979	−0.0465	−0.4881	14	72	84	192	264	230	134
  15	−0.0979	−0.0465	0.4881	15	72	134	158	110	50	26
  16	−0.0979	0.0465	−0.4881	16	73	25	3	15	63	87
  17	0.4881	0.0979	0.0465	17	73	87	195	267	229	133
  18	0.4881	−0.0979	−0.0465	18	73	133	157	109	49	25
  19	−0.4881	−0.0979	0.0465	19	74	24	2	14	62	86
  20	−0.4881	0.0979	−0.0465	20	74	86	194	266	228	132
  21	0.0465	0.4881	0.0979	21	74	132	156	108	48	24
  22	0.0465	−0.4881	−0.0979	22	75	27	1	13	61	85
  23	−0.0465	−0.4881	0.0979	23	75	85	193	265	231	135
  24	−0.0465	0.4881	−0.0979	24	75	135	159	111	51	27
  25	0.0333	−0.1043	0.4879	25	76	28	4	16	64	88
  26	0.0333	0.1043	−0.4879	26	76	88	196	268	232	136
  27	−0.0333	0.1043	0.4879	27	76	136	160	112	52	28
  28	−0.0333	−0.1043	−0.4879	28	77	29	5	17	65	89
  29	0.4879	−0.0333	0.1043	29	77	89	197	269	233	137
  30	0.4879	0.0333	−0.1043	30	77	137	161	113	53	29
  31	−0.4879	0.0333	0.1043	31	78	30	6	18	66	90
  32	−0.4879	−0.0333	−0.1043	32	78	90	198	270	234	138
  33	0.1043	−0.4879	0.0333	33	78	138	162	114	54	30
  34	0.1043	0.4879	−0.0333	34	79	31	7	19	67	91
  35	−0.1043	0.4879	0.0333	35	79	91	199	271	235	139
  36	−0.1043	−0.4879	−0.0333	36	79	139	163	115	55	31
  37	0.1443	−0.0179	0.4784	37	80	33	8	20	68	92
  38	0.1443	0.0179	−0.4784	38	80	92	200	272	237	141
  39	−0.1443	0.0179	0.4784	39	80	141	165	117	57	33
  40	−0.1443	−0.0179	−0.4784	40	81	32	9	21	69	93
  41	0.4784	−0.1443	0.0179	41	81	93	201	273	236	140
  42	0.4784	0.1443	−0.0179	42	81	140	164	116	56	32
  43	−0.4784	0.1443	0.0179	43	82	34	11	23	71	95
  44	−0.4784	−0.1443	−0.0179	44	82	95	203	275	238	142
  45	0.0179	−0.4784	0.1443	45	82	142	166	118	58	34
  46	0.0179	0.4784	−0.1443	46	83	35	10	22	70	94
  47	−0.0179	0.4784	0.1443	47	83	94	202	274	239	143
  48	−0.0179	−0.4784	−0.1443	48	83	143	167	119	59	35
  49	0.1142	−0.0920	0.4780	49	372	360	252	204	312	364
  50	0.1142	0.0920	−0.4780	50	372	364	256	208	316	368
  51	−0.1142	0.0920	0.4780	51	372	368	260	212	320	360
  52	−0.1142	−0.0920	−0.4780	52	373	325	277	349	297	333
  53	0.4780	−0.1142	0.0920	53	373	333	285	357	293	329
  54	0.4780	0.1142	−0.0920	54	373	329	281	353	289	325
  55	−0.4780	0.1142	0.0920	55	374	324	276	348	296	332
  56	−0.4780	−0.1142	−0.0920	56	374	332	284	356	292	328
  57	0.0920	−0.4780	0.1142	57	374	328	280	352	288	324
  58	0.0920	0.4780	−0.1142	58	375	361	253	205	313	365
  59	−0.0920	0.4780	0.1142	59	375	365	257	209	317	369
  60	−0.0920	−0.4780	−0.1142	60	375	369	261	213	321	361
  61	0.1304	0.1181	0.4680	61	376	326	278	350	298	334
  62	0.1304	−0.1181	−0.4680	62	376	334	286	358	294	330
  63	−0.1304	−0.1181	0.4680	63	376	330	282	354	290	326
  64	−0.1304	0.1181	−0.4680	64	377	363	255	207	315	367
  65	0.4680	0.1304	0.1181	65	377	367	259	211	319	371
  66	0.4680	−0.1304	−0.1181	66	377	371	263	215	323	363
  67	−0.4680	−0.1304	0.1181	67	378	362	254	206	314	366
  68	−0.4680	0.1304	−0.1181	68	378	366	258	210	318	370
  69	0.1181	0.4680	0.1304	69	378	370	262	214	322	362
  70	0.1181	−0.4680	−0.1304	70	379	327	279	351	299	335
  71	−0.1181	−0.4680	0.1304	71	379	335	287	359	295	331
  72	−0.1181	0.4680	−0.1304	72	379	331	283	355	291	327
  73	0.0000	0.1784	0.4671	73	48	108	180	168	96	36
  74	0.0000	0.1784	−0.4671	74	49	109	181	169	97	37
  75	0.0000	−0.1784	0.4671	75	50	110	182	170	98	38
  76	0.0000	−0.1784	−0.4671	76	51	111	183	171	99	39
  77	0.4671	0.0000	0.1784	77	52	112	184	172	100	40
  78	0.4671	0.0000	−0.1784	78	53	113	185	173	101	41
  79	−0.4671	0.0000	0.1784	79	54	114	186	174	102	42
  80	−0.4671	0.0000	−0.1784	80	55	115	187	175	103	43
  81	0.1784	0.4671	0.0000	81	56	116	188	176	104	44
  82	0.1784	−0.4671	0.0000	82	57	117	189	177	105	45
  83	−0.1784	0.4671	0.0000	83	58	118	190	178	106	46
  84	−0.1784	−0.4671	0.0000	84	59	119	191	179	107	47
  85	0.0830	0.1847	0.4571	85	60	12	36	96	144	120
  86	0.0830	−0.1847	−0.4571	86	61	13	37	97	145	121
  87	−0.0830	−0.1847	0.4571	87	62	14	38	98	146	122
  88	−0.0830	0.1847	−0.4571	88	63	15	39	99	147	123
  89	0.4571	0.0830	0.1847	89	64	16	41	101	148	124
  90	0.4571	−0.0830	−0.1847	90	65	17	40	100	149	125
  91	−0.4571	−0.0830	0.1847	91	66	18	43	103	150	126
  92	−0.4571	0.0830	−0.1847	92	67	19	42	102	151	127
  93	0.1847	0.4571	0.0830	93	68	20	46	106	152	128
  94	0.1847	−0.4571	−0.0830	94	69	21	47	107	153	129
  95	−0.1847	−0.4571	0.0830	95	70	22	44	104	154	130
  96	−0.1847	0.4571	−0.0830	96	71	23	45	105	155	131
  97	0.2123	−0.0080	0.4526	97	228	266	310	226	248	336
  98	0.2123	0.0080	−0.4526	98	229	267	311	227	249	337
  99	−0.2123	0.0080	0.4526	99	230	264	308	224	250	338
  100	−0.2123	−0.0080	−0.4526	100	231	265	309	225	251	339
  101	0.4526	−0.2123	0.0080	101	232	268	300	216	240	340
  102	0.4526	0.2123	−0.0080	102	233	269	301	217	241	341
  103	−0.4526	0.2123	0.0080	103	234	270	302	218	242	342
  104	−0.4526	−0.2123	−0.0080	104	235	271	303	219	243	343
  105	0.0080	−0.4526	0.2123	105	236	273	305	221	244	344
  106	0.0080	0.4526	−0.2123	106	237	272	304	220	245	345
  107	−0.0080	0.4526	0.2123	107	238	275	307	223	246	346
  108	−0.0080	−0.4526	−0.2123	108	239	274	306	222	247	347
  109	0.1621	−0.1505	0.4484	109	288	352	340	240	168	180
  110	0.1621	0.1505	−0.4484	110	289	353	341	241	169	181
  111	−0.1621	0.1505	0.4484	111	290	354	342	242	170	182
  112	−0.1621	−0.1505	−0.4484	112	291	355	343	243	171	183
  113	0.4484	−0.1621	0.1505	113	292	356	344	244	172	184
  114	0.4484	0.1621	−0.1505	114	293	357	345	245	173	185
  115	−0.4484	0.1621	0.1505	115	294	358	346	246	174	186
  116	−0.4484	−0.1621	−0.1505	116	295	359	347	247	175	187
  117	0.1505	−0.4484	0.1621	117	296	348	336	248	176	188
  118	0.1505	0.4484	−0.1621	118	297	349	337	249	177	189
  119	−0.1505	0.4484	0.1621	119	298	350	338	250	178	190
  120	−0.1505	−0.4484	−0.1621	120	299	351	339	251	179	191
  121	0.2055	0.1169	0.4406	121	312	204	120	144	216	300
  122	0.2055	−0.1169	−0.4406	122	313	205	121	145	217	301
  123	−0.2055	−0.1169	0.4406	123	314	206	122	146	218	302
  124	−0.2055	0.1169	−0.4406	124	315	207	123	147	219	303
  125	0.4406	0.2055	0.1169	125	316	208	124	148	220	304
  126	0.4406	−0.2055	−0.1169	126	317	209	125	149	221	305
  127	−0.4406	−0.2055	0.1169	127	318	210	126	150	222	306
  128	−0.4406	0.2055	−0.1169	128	319	211	127	151	223	307
  129	0.1169	0.4406	0.2055	129	320	212	128	152	224	308
  130	0.1169	−0.4406	−0.2055	130	321	213	129	153	225	309
  131	−0.1169	−0.4406	0.2055	131	322	214	130	154	226	310
  132	−0.1169	0.4406	−0.2055	132	323	215	131	155	227	311
  133	0.0497	−0.2387	0.4365	133	48	36	12	0	2	24
  134	0.0497	0.2387	−0.4365	134	49	37	13	1	3	25
  135	−0.0497	0.2387	0.4365	135	50	38	14	2	0	26
  136	−0.0497	−0.2387	−0.4365	136	51	39	15	3	1	27
  137	0.4365	−0.0497	0.2387	137	52	40	17	5	4	28
  138	0.4365	0.0497	−0.2387	138	53	41	16	4	5	29
  139	−0.4365	0.0497	0.2387	139	54	42	19	7	6	30
  140	−0.4365	−0.0497	−0.2387	140	55	43	18	6	7	31
  141	0.2387	−0.4365	0.0497	141	56	44	22	10	9	32
  142	0.2387	0.4365	−0.0497	142	57	45	23	11	8	33
  143	−0.2387	0.4365	0.0497	143	58	46	20	8	11	34
  144	−0.2387	−0.4365	−0.0497	144	59	47	21	9	10	35
  145	0.2396	0.0521	0.4358	145	60	120	204	252	192	84
  146	0.2396	−0.0521	−0.4358	146	61	121	205	253	193	85
  147	−0.2396	−0.0521	0.4358	147	62	122	206	254	194	86
  148	−0.2396	0.0521	−0.4358	148	63	123	207	255	195	87
  149	0.4358	0.2396	0.0521	149	64	124	208	256	196	88
  150	0.4358	−0.2396	−0.0521	150	65	125	209	257	197	89
  151	−0.4358	−0.2396	0.0521	151	66	126	210	258	198	90
  152	−0.4358	0.2396	−0.0521	152	67	127	211	259	199	91
  153	0.0521	0.4358	0.2396	153	68	128	212	260	200	92
  154	0.0521	−0.4358	−0.2396	154	69	129	213	261	201	93
  155	−0.0521	−0.4358	0.2396	155	70	130	214	262	202	94
  156	−0.0521	0.4358	−0.2396	156	71	131	215	263	203	95
  157	0.1314	−0.2239	0.4274	157	132	228	336	348	276	156
  158	0.1314	0.2239	−0.4274	158	133	229	337	349	277	157
  159	−0.1314	0.2239	0.4274	159	134	230	338	350	278	158
  160	−0.1314	−0.2239	−0.4274	160	135	231	339	351	279	159
  161	0.4274	−0.1314	0.2239	161	136	232	340	352	280	160
  162	0.4274	0.1314	−0.2239	162	137	233	341	353	281	161
  163	−0.4274	0.1314	0.2239	163	138	234	342	354	282	162
  164	−0.4274	−0.1314	−0.2239	164	139	235	343	355	283	163
  165	0.2239	−0.4274	0.1314	165	140	236	344	356	284	164
  166	0.2239	0.4274	−0.1314	166	141	237	345	357	285	165
  167	−0.2239	0.4274	0.1314	167	142	238	346	358	286	166
  168	−0.2239	−0.4274	−0.1314	168	143	239	347	359	287	167
  169	0.2525	−0.0570	0.4278	169	288	180	108	156	276	324
  170	0.2525	0.0570	−0.4278	170	289	181	109	157	277	325
  171	−0.2525	0.0570	0.4278	171	290	182	110	158	278	326
  172	−0.2525	−0.0570	−0.4278	172	291	183	111	159	279	327
  173	0.4278	−0.2525	0.0570	173	292	184	112	160	280	328
  174	0.4278	0.2525	−0.0570	174	293	185	113	161	281	329
  175	−0.4278	0.2525	0.0570	175	294	186	114	162	282	330
  176	−0.4278	−0.2525	−0.0570	176	295	187	115	163	283	331
  177	0.0570	−0.4278	0.2525	177	296	188	116	164	284	332
  178	0.0570	0.4278	−0.2525	178	297	189	117	165	285	333
  179	−0.0570	0.4278	0.2525	179	298	190	118	166	286	334
  180	−0.0570	−0.4278	−0.2525	180	299	191	119	167	287	335
  181	0.2345	−0.1280	0.4227	181	308	264	192	252	360	320
  182	0.2345	0.1280	−0.4227	182	309	265	193	253	361	321
  183	−0.2345	0.1280	0.4227	183	310	266	194	254	362	322
  184	−0.2345	−0.1280	−0.4227	184	311	267	195	255	363	323
  185	0.4227	−0.2345	0.1280	185	300	268	196	256	364	312
  186	0.4227	0.2345	−0.1280	186	301	269	197	257	365	313
  187	−0.4227	0.2345	0.1280	187	302	270	198	258	366	314
  188	−0.4227	−0.2345	−0.1280	188	303	271	199	259	367	315
  189	0.1280	−0.4227	0.2345	189	304	272	200	260	368	316
  190	0.1280	0.4227	−0.2345	190	305	273	201	261	369	317
  191	−0.1280	0.4227	0.2345	191	306	274	202	262	370	318
  192	−0.1280	−0.4227	−0.2345	192	307	275	203	263	371	319
  193	0.1147	0.2508	0.4171
  194	0.1147	−0.2508	−0.4171
  195	−0.1147	−0.2508	0.4171
  196	−0.1147	0.2508	−0.4171
  197	0.4171	0.1147	0.2508
  198	0.4171	−0.1147	−0.2508
  199	−0.4171	−0.1147	0.2508
  200	−0.4171	0.1147	−0.2508
  201	0.2508	0.4171	0.1147
  202	0.2508	−0.4171	−0.1147
  203	−0.2508	−0.4171	0.1147
  204	−0.2508	0.4171	−0.1147
  205	0.2374	0.1793	0.4019
  206	0.2374	−0.1793	−0.4019
  207	−0.2374	−0.1793	0.4019
  208	−0.2374	0.1793	−0.4019
  209	0.4019	0.2374	0.1793
  210	0.4019	−0.2374	−0.1793
  211	−0.4019	−0.2374	0.1793
  212	−0.4019	0.2374	−0.1793
  213	0.1793	0.4019	0.2374
  214	0.1793	−0.4019	−0.2374
  215	−0.1793	−0.4019	0.2374
  216	−0.1793	0.4019	−0.2374
  217	0.2966	0.0401	0.4005
  218	0.2966	−0.0401	−0.4005
  219	−0.2966	−0.0401	0.4005
  220	−0.2966	0.0401	−0.4005
  221	0.4005	0.2966	0.0401
  222	0.4005	−0.2966	−0.0401
  223	−0.4005	−0.2966	0.0401
  224	−0.4005	0.2966	−0.0401
  225	0.0401	0.4005	0.2966
  226	0.0401	−0.4005	−0.2966
  227	−0.0401	−0.4005	0.2966
  228	−0.0401	0.4005	−0.2966
  229	0.0162	−0.3030	0.3974
  230	0.0162	0.3030	−0.3974
  231	−0.0162	0.3030	0.3974
  232	−0.0162	−0.3030	−0.3974
  233	0.3974	−0.0162	0.3030
  234	0.3974	0.0162	−0.3030
  235	−0.3974	0.0162	0.3030
  236	−0.3974	−0.0162	−0.3030
  237	0.3030	−0.3974	0.0162
  238	0.3030	0.3974	−0.0162
  239	−0.3030	0.3974	0.0162
  240	−0.3030	−0.3974	−0.0162
  241	0.3045	−0.0273	0.3956
  242	0.3045	0.0273	−0.3956
  243	−0.3045	0.0273	0.3956
  244	−0.3045	−0.0273	−0.3956
  245	0.3956	−0.3045	0.0273
  246	0.3956	0.3045	−0.0273
  247	−0.3956	0.3045	0.0273
  248	−0.3956	−0.3045	−0.0273
  249	0.0273	−0.3956	0.3045
  250	0.0273	0.3956	−0.3045
  251	−0.0273	0.3956	0.3045
  252	−0.0273	−0.3956	−0.3045
  253	0.1932	0.2475	0.3891
  254	0.1932	−0.2475	−0.3891
  255	−0.1932	−0.2475	0.3891
  256	−0.1932	0.2475	−0.3891
  257	0.3891	0.1932	0.2475
  258	0.3891	−0.1932	−0.2475
  259	−0.3891	−0.1932	0.2475
  260	−0.3891	0.1932	−0.2475
  261	0.2475	0.3891	0.1932
  262	0.2475	−0.3891	−0.1932
  263	−0.2475	−0.3891	0.1932
  264	−0.2475	0.3891	−0.1932
  265	0.0643	0.3089	0.3879
  266	0.0643	−0.3089	−0.3879
  267	−0.0643	−0.3089	0.3879
  268	−0.0643	0.3089	−0.3879
  269	0.3879	0.0643	0.3089
  270	0.3879	−0.0643	−0.3089
  271	−0.3879	−0.0643	0.3089
  272	−0.3879	0.0643	−0.3089
  273	0.3089	0.3879	0.0643
  274	0.3089	−0.3879	−0.0643
  275	−0.3089	−0.3879	0.0643
  276	−0.3089	0.3879	−0.0643
  277	0.1766	−0.2744	0.3788
  278	0.1766	0.2744	−0.3788
  279	−0.1766	0.2744	0.3788
  280	−0.1766	−0.2744	−0.3788
  281	0.3788	−0.1766	0.2744
  282	0.3788	0.1766	−0.2744
  283	−0.3788	0.1766	0.2744
  284	−0.3788	−0.1766	−0.2744
  285	0.2744	−0.3788	0.1766
  286	0.2744	0.3788	−0.1766
  287	−0.2744	0.3788	0.1766
  288	−0.2744	−0.3788	−0.1766
  289	0.2793	−0.1750	0.3760
  290	0.2793	0.1750	−0.3760
  291	−0.2793	0.1750	0.3760
  292	−0.2793	−0.1750	−0.3760
  293	0.3760	−0.2793	0.1750
  294	0.3760	0.2793	−0.1750
  295	−0.3760	0.2793	0.1750
  296	−0.3760	−0.2793	−0.1750
  297	0.1750	−0.3760	0.2793
  298	0.1750	0.3760	−0.2793
  299	−0.1750	0.3760	0.2793
  300	−0.1750	−0.3760	−0.2793
  301	0.3335	0.0901	0.3615
  302	0.3335	−0.0901	−0.3615
  303	−0.3335	−0.0901	0.3615
  304	−0.3335	0.0901	−0.3615
  305	0.3615	0.3335	0.0901
  306	0.3615	−0.3335	−0.0901
  307	−0.3615	−0.3335	0.0901
  308	−0.3615	0.3335	−0.0901
  309	0.0901	0.3615	0.3335
  310	0.0901	−0.3615	−0.3335
  311	−0.0901	−0.3615	0.3335
  312	−0.0901	0.3615	−0.3335
  313	0.3054	0.1650	0.3599
  314	0.3054	−0.1650	−0.3599
  315	−0.3054	−0.1650	0.3599
  316	−0.3054	0.1650	−0.3599
  317	0.3599	0.3054	0.1650
  318	0.3599	−0.3054	−0.1650
  319	−0.3599	−0.3054	0.1650
  320	−0.3599	0.3054	−0.1650
  321	0.1650	0.3599	0.3054
  322	0.1650	−0.3599	−0.3054
  323	−0.1650	−0.3599	0.3054
  324	−0.1650	0.3599	−0.3054
  325	0.2518	−0.2492	0.3528
  326	0.2518	0.2492	−0.3528
  327	−0.2518	0.2492	0.3528
  328	−0.2518	−0.2492	−0.3528
  329	0.3528	−0.2518	0.2492
  330	0.3528	0.2518	−0.2492
  331	−0.3528	0.2518	0.2492
  332	−0.3528	−0.2518	−0.2492
  333	0.2492	−0.3528	0.2518
  334	0.2492	0.3528	−0.2518
  335	−0.2492	0.3528	0.2518
  336	−0.2492	−0.3528	−0.2518
  337	0.0612	−0.3504	0.3514
  338	0.0612	0.3504	−0.3514
  339	−0.0612	0.3504	0.3514
  340	−0.0612	−0.3504	−0.3514
  341	0.3514	−0.0612	0.3504
  342	0.3514	0.0612	−0.3504
  343	−0.3514	0.0612	0.3504
  344	−0.3514	−0.0612	−0.3504
  345	0.3504	−0.3514	0.0612
  346	0.3504	0.3514	−0.0612
  347	−0.3504	0.3514	0.0612
  348	−0.3504	−0.3514	−0.0612
  349	0.1395	−0.3376	0.3414
  350	0.1395	0.3376	−0.3414
  351	−0.1395	0.3376	0.3414
  352	−0.1395	−0.3376	−0.3414
  353	0.3414	−0.1395	0.3376
  354	0.3414	0.1395	−0.3376
  355	−0.3414	0.1395	0.3376
  356	−0.3414	−0.1395	−0.3376
  357	0.3376	−0.3414	0.1395
  358	0.3376	0.3414	−0.1395
  359	−0.3376	0.3414	0.1395
  360	−0.3376	−0.3414	−0.1395
  361	0.2185	0.3031	0.3322
  362	0.2185	−0.3031	−0.3322
  363	−0.2185	−0.3031	0.3322
  364	−0.2185	0.3031	−0.3322
  365	0.3322	0.2185	0.3031
  366	0.3322	−0.2185	−0.3031
  367	−0.3322	−0.2185	0.3031
  368	−0.3322	0.2185	−0.3031
  369	0.3031	0.3322	0.2185
  370	0.3031	−0.3322	−0.2185
  371	−0.3031	−0.3322	0.2185
  372	−0.3031	0.3322	−0.2185
  373	0.2887	0.2887	0.2887
  374	0.2887	0.2887	−0.2887
  375	0.2887	−0.2887	0.2887
  376	0.2887	−0.2887	−0.2887
  377	−0.2887	0.2887	0.2887
  378	−0.2887	0.2887	−0.2887
  379	−0.2887	−0.2887	0.2887
  380	−0.2887	−0.2887	−0.2887

• TABLE 3
  Sphere	x/D	y/D	z/D	d/D

  1	0.1437	0.0000	−0.7810	0.6114
  2	−0.0718	0.1244	−0.7811	0.6115
  3	−0.0718	−0.1244	−0.7811	0.6115
  4	0.1418	0.2456	−0.7424	0.6127
  5	0.1418	−0.2456	−0.7424	0.6127
  6	−0.2836	0.0000	−0.7424	0.6127
  7	0.3189	0.1091	−0.6648	0.5125
  8	−0.0649	0.3307	−0.6648	0.5125
  9	0.3189	−0.1091	−0.6648	0.5125
  10	−0.2539	0.2216	−0.6648	0.5125
  11	−0.0649	−0.3307	−0.6648	0.5125
  12	−0.2539	−0.2216	−0.6648	0.5125
  13	0.3860	0.0000	−0.5053	0.2902
  14	−0.1930	0.3343	−0.5053	0.2902
  15	−0.1930	−0.3343	−0.5053	0.2902
  16	0.3413	0.3423	−0.6301	0.6115
  17	0.1258	0.4668	−0.6301	0.6115
  18	0.3413	−0.3423	−0.6301	0.6115
  19	0.1258	−0.4668	−0.6301	0.6115
  20	−0.4672	0.1244	−0.6301	0.6115
  21	−0.4672	−0.1244	−0.6301	0.6115
  22	0.4838	0.1766	−0.5388	0.5125
  23	−0.0890	0.5073	−0.5388	0.5125
  24	0.4838	−0.1766	−0.5388	0.5125
  25	−0.3948	0.3307	−0.5388	0.5125
  26	−0.0890	−0.5073	−0.5388	0.5125
  27	−0.3948	−0.3307	−0.5388	0.5125
  28	0.5858	0.0000	−0.4610	0.5125
  29	−0.2929	0.5073	−0.4610	0.5125
  30	−0.2929	−0.5073	−0.4610	0.5125
  31	0.3139	0.5437	−0.4864	0.6115
  32	0.3139	−0.5437	−0.4864	0.6115
  33	−0.6278	0.0000	−0.4864	0.6114
  34	0.5130	0.3974	−0.4589	0.6127
  35	0.0876	0.6430	−0.4589	0.6127
  36	0.5130	−0.3974	−0.4589	0.6127
  37	−0.6006	0.2456	−0.4589	0.6127
  38	0.0876	−0.6430	−0.4589	0.6127
  39	−0.6006	−0.2456	−0.4589	0.6127
  40	0.6611	0.2116	−0.3857	0.6114
  41	−0.1473	0.6784	−0.3858	0.6115
  42	0.6612	−0.2116	−0.3857	0.6115
  43	−0.5138	0.4668	−0.3857	0.6115
  44	−0.1473	−0.6784	−0.3857	0.6115
  45	−0.5138	−0.4668	−0.3857	0.6115
  46	0.4350	0.5351	−0.2829	0.5125
  47	0.2460	0.6442	−0.2829	0.5125
  48	0.4350	−0.5351	−0.2829	0.5125
  49	0.2460	−0.6442	−0.2829	0.5125
  50	−0.6809	0.1091	−0.2829	0.5125
  51	−0.6809	−0.1091	−0.2829	0.5125
  52	0.7424	0.0000	−0.2836	0.6127
  53	−0.3712	0.6430	−0.2836	0.6127
  54	−0.3712	−0.6430	−0.2836	0.6127
  55	0.6337	0.4129	−0.2421	0.6115
  56	0.0407	0.7553	−0.2421	0.6115
  57	0.6337	−0.4129	−0.2421	0.6115
  58	−0.6745	0.3423	−0.2421	0.6115
  59	0.0408	−0.7553	−0.2421	0.6115
  60	−0.6745	−0.3423	−0.2421	0.6115
  61	0.3123	0.5409	−0.1193	0.2902
  62	0.3123	−0.5409	−0.1193	0.2902
  63	−0.6246	0.0000	−0.1193	0.2902
  64	0.7500	0.2116	−0.1533	0.6115
  65	0.7500	−0.2116	−0.1533	0.6115
  66	−0.1917	0.7553	−0.1533	0.6115
  67	−0.5582	0.5437	−0.1533	0.6115
  68	−0.1917	−0.7553	−0.1533	0.6115
  69	−0.5582	−0.5437	−0.1533	0.6115
  70	0.5128	0.5351	−0.0791	0.5125
  71	0.2070	0.7117	−0.0791	0.5125
  72	0.5128	−0.5351	−0.0791	0.5125
  73	0.2070	−0.7117	−0.0791	0.5125
  74	−0.7198	0.1766	−0.0791	0.5125
  75	−0.7198	−0.1766	−0.0791	0.5125
  76	0.7439	0.0000	−0.0469	0.5125
  77	−0.3720	0.6442	−0.0469	0.5125
  78	−0.3720	−0.6442	−0.0469	0.5125
  79	0.6883	0.3974	0.0000	0.6127
  80	0.0000	0.7948	0.0000	0.6127
  81	0.6883	−0.3974	0.0000	0.6127
  82	−0.6883	0.3974	0.0000	0.6127
  83	0.0000	−0.7948	0.0000	0.6127
  84	−0.6883	−0.3974	0.0000	0.6127
  85	0.3720	0.6442	0.0469	0.5125
  86	0.3720	−0.6442	0.0469	0.5125
  87	−0.7439	0.0000	0.0469	0.5125
  88	0.7198	0.1766	0.0791	0.5125
  89	0.7198	−0.1766	0.0791	0.5125
  90	−0.2070	0.7117	0.0791	0.5125
  91	−0.5128	0.5351	0.0791	0.5125
  92	−0.2070	−0.7117	0.0791	0.5125
  93	−0.5128	−0.5351	0.0791	0.5125
  94	0.5582	0.5437	0.1533	0.6115
  95	0.1917	0.7553	0.1533	0.6115
  96	0.5582	−0.5437	0.1533	0.6115
  97	0.1917	−0.7553	0.1533	0.6115
  98	−0.7500	0.2116	0.1533	0.6115
  99	−0.7500	−0.2116	0.1533	0.6115
  100	0.6246	0.0000	0.1193	0.2902
  101	−0.3123	0.5409	0.1193	0.2902
  102	−0.3123	−0.5409	0.1193	0.2902
  103	0.6745	0.3423	0.2421	0.6115
  104	−0.0408	0.7553	0.2421	0.6115
  105	0.6745	−0.3423	0.2421	0.6115
  106	−0.6337	0.4129	0.2421	0.6115
  107	−0.0407	−0.7553	0.2421	0.6115
  108	−0.6337	−0.4129	0.2421	0.6115
  109	0.3712	0.6430	0.2836	0.6127
  110	0.3712	−0.6430	0.2836	0.6127
  111	−0.7424	0.0000	0.2836	0.6127
  112	0.6809	0.1091	0.2829	0.5125
  113	0.6809	−0.1091	0.2829	0.5125
  114	−0.2460	0.6442	0.2829	0.5125
  115	−0.4350	0.5351	0.2829	0.5125
  116	−0.2460	−0.6442	0.2829	0.5125
  117	−0.4350	−0.5351	0.2829	0.5125
  118	0.5138	0.4668	0.3857	0.6115
  119	0.1473	0.6784	0.3857	0.6115
  120	0.5138	−0.4668	0.3857	0.6115
  121	−0.6612	0.2116	0.3857	0.6115
  122	0.1473	−0.6784	0.3858	0.6115
  123	−0.6611	−0.2116	0.3857	0.6114
  124	0.6006	0.2456	0.4589	0.6127
  125	−0.0876	0.6430	0.4589	0.6127
  126	0.6006	−0.2456	0.4589	0.6127
  127	−0.5130	0.3974	0.4589	0.6127
  128	−0.0876	−0.6430	0.4589	0.6127
  129	−0.5130	−0.3974	0.4589	0.6127
  130	0.6278	0.0000	0.4864	0.6114
  131	−0.3139	0.5437	0.4864	0.6115
  132	−0.3139	−0.5437	0.4864	0.6115
  133	0.2929	0.5073	0.4610	0.5125
  134	0.2929	−0.5073	0.4610	0.5125
  135	−0.5858	0.0000	0.4610	0.5125
  136	0.3948	0.3307	0.5388	0.5125
  137	0.0890	0.5073	0.5388	0.5125
  138	0.3948	−0.3307	0.5388	0.5125
  139	−0.4838	0.1766	0.5388	0.5125
  140	0.0890	−0.5073	0.5388	0.5125
  141	−0.4838	−0.1766	0.5388	0.5125
  142	0.4672	0.1244	0.6301	0.6115
  143	0.4672	−0.1244	0.6301	0.6115
  144	−0.1258	0.4668	0.6301	0.6115
  145	−0.3413	0.3423	0.6301	0.6115
  146	−0.1258	−0.4668	0.6301	0.6115
  147	−0.3413	−0.3423	0.6301	0.6115
  148	0.1930	0.3343	0.5053	0.2902
  149	0.1930	−0.3343	0.5053	0.2902
  150	−0.3860	0.0000	0.5053	0.2902
  151	0.2539	0.2216	0.6648	0.5125
  152	0.0649	0.3307	0.6648	0.5125
  153	0.2539	−0.2216	0.6648	0.5125
  154	−0.3189	0.1091	0.6648	0.5125
  155	0.0649	−0.3307	0.6648	0.5125
  156	−0.3189	−0.1091	0.6648	0.5125
  157	0.2836	0.0000	0.7424	0.6127
  158	−0.1418	0.2456	0.7424	0.6127
  159	−0.1418	−0.2456	0.7424	0.6127
  160	0.0718	0.1244	0.7811	0.6115
  161	0.0718	−0.1244	0.7811	0.6115
  162	−0.1437	0.0000	0.7810	0.6114

• TABLE 4
  Vertex	x/D	y/D	z/D	Face	Group of vertices

  1	0.0304	0.0117	0.4989	1	204	276	384	312	216
  2	0.0304	−0.0117	−0.4989	2	205	277	385	313	217
  3	−0.0304	−0.0117	0.4989	3	206	278	386	314	218
  4	−0.0304	0.0117	−0.4989	4	207	279	387	315	219
  5	0.4989	0.0304	0.0117	5	208	280	388	317	221
  6	0.4989	−0.0304	−0.0117	6	209	281	389	316	220
  7	−0.4989	−0.0304	0.0117	7	210	282	390	319	223
  8	−0.4989	0.0304	−0.0117	8	211	283	391	318	222
  9	0.0117	0.4989	0.0304	9	212	284	392	322	226
  10	0.0117	−0.4989	−0.0304	10	213	285	393	323	227
  11	−0.0117	−0.4989	0.0304	11	214	286	394	320	224
  12	−0.0117	0.4989	−0.0304	12	215	287	395	321	225
  13	0.0804	−0.0287	0.4926	13	108	50	24	72	132	156
  14	0.0804	0.0287	−0.4926	14	108	156	264	360	302	194
  15	−0.0804	0.0287	0.4926	15	108	194	182	98	38	50
  16	−0.0804	−0.0287	−0.4926	16	109	49	27	75	135	159
  17	0.4926	−0.0804	0.0287	17	109	159	267	363	301	193
  18	0.4926	0.0804	−0.0287	18	109	193	181	97	37	49
  19	−0.4926	0.0804	0.0287	19	110	48	26	74	134	158
  20	−0.4926	−0.0804	−0.0287	20	110	158	266	362	300	192
  21	0.0287	−0.4926	0.0804	21	110	192	180	96	36	48
  22	0.0287	0.4926	−0.0804	22	111	51	25	73	133	157
  23	−0.0287	0.4926	0.0804	23	111	157	265	361	303	195
  24	−0.0287	−0.4926	−0.0804	24	111	195	183	99	39	51
  25	0.0406	0.0756	0.4926	25	112	52	28	76	136	160
  26	0.0406	−0.0756	−0.4926	26	112	160	268	364	304	196
  27	−0.0406	−0.0756	0.4926	27	112	196	184	100	40	52
  28	−0.0406	0.0756	−0.4926	28	113	53	29	77	137	161
  29	0.4926	0.0406	0.0756	29	113	161	269	365	305	197
  30	0.4926	−0.0406	−0.0756	30	113	197	185	101	41	53
  31	−0.4926	−0.0406	0.0756	31	114	54	30	78	138	162
  32	−0.4926	0.0406	−0.0756	32	114	162	270	366	306	198
  33	0.0756	0.4926	0.0406	33	114	198	186	102	42	54
  34	0.0756	−0.4926	−0.0406	34	115	55	31	79	139	163
  35	−0.0756	−0.4926	0.0406	35	115	163	271	367	307	199
  36	−0.0756	0.4926	−0.0406	36	115	199	187	103	43	55
  37	0.0708	−0.0922	0.4863	37	116	57	32	80	140	164
  38	0.0708	0.0922	−0.4863	38	116	164	272	368	309	201
  39	−0.0708	0.0922	0.4863	39	116	201	189	105	45	57
  40	−0.0708	−0.0922	−0.4863	40	117	56	33	81	141	165
  41	0.4863	−0.0708	0.0922	41	117	165	273	369	308	200
  42	0.4863	0.0708	−0.0922	42	117	200	188	104	44	56
  43	−0.4863	0.0708	0.0922	43	118	58	35	83	143	167
  44	−0.4863	−0.0708	−0.0922	44	118	167	275	371	310	202
  45	0.0922	−0.4863	0.0708	45	118	202	190	106	46	58
  46	0.0922	0.4863	−0.0708	46	119	59	34	82	142	166
  47	−0.0922	0.4863	0.0708	47	119	166	274	370	311	203
  48	−0.0922	−0.4863	−0.0708	48	119	203	191	107	47	59
  49	0.0102	−0.1163	0.4862	49	612	588	456	468	600	592
  50	0.0102	0.1163	−0.4862	50	612	592	460	472	604	596
  51	−0.0102	0.1163	0.4862	51	612	596	464	476	608	588
  52	−0.0102	−0.1163	−0.4862	52	613	577	565	549	489	585
  53	0.4862	−0.0102	0.1163	53	613	585	573	545	485	581
  54	0.4862	0.0102	−0.1163	54	613	581	569	541	481	577
  55	−0.4862	0.0102	0.1163	55	614	576	564	548	488	584
  56	−0.4862	−0.0102	−0.1163	56	614	584	572	544	484	580
  57	0.1163	−0.4862	0.0102	57	614	580	568	540	480	576
  58	0.1163	0.4862	−0.0102	58	615	589	457	469	601	593
  59	−0.1163	0.4862	0.0102	59	615	593	461	473	605	597
  60	−0.1163	−0.4862	−0.0102	60	615	597	465	477	609	589
  61	0.1376	−0.0055	0.4807	61	616	578	566	550	490	586
  62	0.1376	0.0055	−0.4807	62	616	586	574	546	486	582
  63	−0.1376	0.0055	0.4807	63	616	582	570	542	482	578
  64	−0.1376	−0.0055	−0.4807	64	617	591	459	471	603	595
  65	0.4807	−0.1376	0.0055	65	617	595	463	475	607	599
  66	0.4807	0.1376	−0.0055	66	617	599	467	479	611	591
  67	−0.4807	0.1376	0.0055	67	618	590	458	470	602	594
  68	−0.4807	−0.1376	−0.0055	68	618	594	462	474	606	598
  69	0.0055	−0.4807	0.1376	69	618	598	466	478	610	590
  70	0.0055	0.4807	−0.1376	70	619	579	567	551	491	587
  71	−0.0055	0.4807	0.1376	71	619	587	575	547	487	583
  72	−0.0055	−0.4807	−0.1376	72	619	583	571	543	483	579
  73	0.1006	0.0965	0.4802	73	12	0	2	26	48	36
  74	0.1006	−0.0965	−0.4802	74	13	1	3	27	49	37
  75	−0.1006	−0.0965	0.4802	75	14	2	0	24	50	38
  76	−0.1006	0.0965	−0.4802	76	15	3	1	25	51	39
  77	0.4802	0.1006	0.0965	77	16	5	4	28	52	40
  78	0.4802	−0.1006	−0.0965	78	17	4	5	29	53	41
  79	−0.4802	−0.1006	0.0965	79	18	7	6	30	54	42
  80	−0.4802	0.1006	−0.0965	80	19	6	7	31	55	43
  81	0.0965	0.4802	0.1006	81	20	10	9	33	56	44
  82	0.0965	−0.4802	−0.1006	82	21	11	8	32	57	45
  83	−0.0965	−0.4802	0.1006	83	22	8	11	35	58	46
  84	−0.0965	0.4802	−0.1006	84	23	9	10	34	59	47
  85	0.1473	0.0548	0.4747	85	132	228	348	396	264	156
  86	0.1473	−0.0548	−0.4747	86	133	229	349	397	265	157
  87	−0.1473	−0.0548	0.4747	87	134	230	350	398	266	158
  88	−0.1473	0.0548	−0.4747	88	135	231	351	399	267	159
  89	0.4747	0.1473	0.0548	89	136	232	352	400	268	160
  90	0.4747	−0.1473	−0.0548	90	137	233	353	401	269	161
  91	−0.4747	−0.1473	0.0548	91	138	234	354	402	270	162
  92	−0.4747	0.1473	−0.0548	92	139	235	355	403	271	163
  93	0.0548	0.4747	0.1473	93	140	236	356	404	272	164
  94	0.0548	−0.4747	−0.1473	94	141	237	357	405	273	165
  95	−0.0548	−0.4747	0.1473	95	142	238	358	406	274	166
  96	−0.0548	0.4747	−0.1473	96	143	239	359	407	275	167
  97	0.1206	−0.1289	0.4678	97	180	192	300	432	420	288
  98	0.1206	0.1289	−0.4678	98	181	193	301	433	421	289
  99	−0.1206	0.1289	0.4678	99	182	194	302	434	422	290
  100	−0.1206	−0.1289	−0.4678	100	183	195	303	435	423	291
  101	0.4678	−0.1206	0.1289	101	184	196	304	436	424	292
  102	0.4678	0.1206	−0.1289	102	185	197	305	437	425	293
  103	−0.4678	0.1206	0.1289	103	186	198	306	438	426	294
  104	−0.4678	−0.1206	−0.1289	104	187	199	307	439	427	295
  105	0.1289	−0.4678	0.1206	105	188	200	308	440	428	296
  106	0.1289	0.4678	−0.1206	106	189	201	309	441	429	297
  107	−0.1289	0.4678	0.1206	107	190	202	310	442	430	298
  108	−0.1289	−0.4678	−0.1206	108	191	203	311	443	431	299
  109	0.0000	0.1784	0.4671	109	336	288	420	564	576	480
  110	0.0000	0.1784	−0.4671	110	337	289	421	565	577	481
  111	0.0000	−0.1784	0.4671	111	338	290	422	566	578	482
  112	0.0000	−0.1784	−0.4671	112	339	291	423	567	579	483
  113	0.4671	0.0000	0.1784	113	340	292	424	568	580	484
  114	0.4671	0.0000	−0.1784	114	341	293	425	569	581	485
  115	−0.4671	0.0000	0.1784	115	342	294	426	570	582	486
  116	−0.4671	0.0000	−0.1784	116	343	295	427	571	583	487
  117	0.1784	0.4671	0.0000	117	344	296	428	572	584	488
  118	0.1784	−0.4671	0.0000	118	345	297	429	573	585	489
  119	−0.1784	0.4671	0.0000	119	346	298	430	574	586	490
  120	−0.1784	−0.4671	0.0000	120	347	299	431	575	587	491
  121	0.1827	−0.0417	0.4636	121	456	588	608	516	396	348
  122	0.1827	0.0417	−0.4636	122	457	589	609	517	397	349
  123	−0.1827	0.0417	0.4636	123	458	590	610	518	398	350
  124	−0.1827	−0.0417	−0.4636	124	459	591	611	519	399	351
  125	0.4636	−0.1827	0.0417	125	460	592	600	520	400	352
  126	0.4636	0.1827	−0.0417	126	461	593	601	521	401	353
  127	−0.4636	0.1827	0.0417	127	462	594	602	522	402	354
  128	−0.4636	−0.1827	−0.0417	128	463	595	603	523	403	355
  129	0.0417	−0.4636	0.1827	129	464	596	604	524	404	356
  130	0.0417	0.4636	−0.1827	130	465	597	605	525	405	357
  131	−0.0417	0.4636	0.1827	131	466	598	606	526	406	358
  132	−0.0417	−0.4636	−0.1827	132	467	599	607	527	407	359
  133	0.1111	0.1573	0.4614	133	12	60	84	72	24	0
  134	0.1111	−0.1573	−0.4614	134	13	61	85	73	25	1
  135	−0.1111	−0.1573	0.4614	135	14	62	86	74	26	2
  136	−0.1111	0.1573	−0.4614	136	15	63	87	75	27	3
  137	0.4614	0.1111	0.1573	137	16	64	89	77	29	5
  138	0.4614	−0.1111	−0.1573	138	17	65	88	76	28	4
  139	−0.4614	−0.1111	0.1573	139	18	66	91	79	31	7
  140	−0.4614	0.1111	−0.1573	140	19	67	90	78	30	6
  141	0.1573	0.4614	0.1111	141	20	68	94	82	34	10
  142	0.1573	−0.4614	−0.1111	142	21	69	95	83	35	11
  143	−0.1573	−0.4614	0.1111	143	22	70	92	80	32	8
  144	−0.1573	0.4614	−0.1111	144	23	71	93	81	33	9
  145	0.1760	−0.1012	0.4569	145	96	180	288	336	252	144
  146	0.1760	0.1012	−0.4569	146	97	181	289	337	253	145
  147	−0.1760	0.1012	0.4569	147	98	182	290	338	254	146
  148	−0.1760	−0.1012	−0.4569	148	99	183	291	339	255	147
  149	0.4569	−0.1760	0.1012	149	100	184	292	340	256	148
  150	0.4569	0.1760	−0.1012	150	101	185	293	341	257	149
  151	−0.4569	0.1760	0.1012	151	102	186	294	342	258	150
  152	−0.4569	−0.1760	−0.1012	152	103	187	295	343	259	151
  153	0.1012	−0.4569	0.1760	153	104	188	296	344	260	152
  154	0.1012	0.4569	−0.1760	154	105	189	297	345	261	153
  155	−0.1012	0.4569	0.1760	155	106	190	298	346	262	154
  156	−0.1012	−0.4569	−0.1760	156	107	191	299	347	263	155
  157	0.0612	0.1989	0.4546	157	228	240	372	468	456	348
  158	0.0612	−0.1989	−0.4546	158	229	241	373	469	457	349
  159	−0.0612	−0.1989	0.4546	159	230	242	374	470	458	350
  160	−0.0612	0.1989	−0.4546	160	231	243	375	471	459	351
  161	0.4546	0.0612	0.1989	161	232	244	376	472	460	352
  162	0.4546	−0.0612	−0.1989	162	233	245	377	473	461	353
  163	−0.4546	−0.0612	0.1989	163	234	246	378	474	462	354
  164	−0.4546	0.0612	−0.1989	164	235	247	379	475	463	355
  165	0.1989	0.4546	0.0612	165	236	248	380	476	464	356
  166	0.1989	−0.4546	−0.0612	166	237	249	381	477	465	357
  167	−0.1989	−0.4546	0.0612	167	238	250	382	478	466	358
  168	−0.1989	0.4546	−0.0612	168	239	251	383	479	467	359
  169	0.2009	0.0711	0.4523	169	264	396	516	536	504	360
  170	0.2009	−0.0711	−0.4523	170	265	397	517	537	505	361
  171	−0.2009	−0.0711	0.4523	171	266	398	518	538	506	362
  172	−0.2009	0.0711	−0.4523	172	267	399	519	539	507	363
  173	0.4523	0.2009	0.0711	173	268	400	520	528	508	364
  174	0.4523	−0.2009	−0.0711	174	269	401	521	529	509	365
  175	−0.4523	−0.2009	0.0711	175	270	402	522	530	510	366
  176	−0.4523	0.2009	−0.0711	176	271	403	523	531	511	367
  177	0.0711	0.4523	0.2009	177	272	404	524	532	512	368
  178	0.0711	−0.4523	−0.2009	178	273	405	525	533	513	369
  179	−0.0711	−0.4523	0.2009	179	274	406	526	534	514	370
  180	−0.0711	0.4523	−0.2009	180	275	407	527	535	515	371
  181	0.1113	−0.1901	0.4489	181	432	560	452	548	564	420
  182	0.1113	0.1901	−0.4489	182	433	561	453	549	565	421
  183	−0.1113	0.1901	0.4489	183	434	562	454	550	566	422
  184	−0.1113	−0.1901	−0.4489	184	435	563	455	551	567	423
  185	0.4489	−0.1113	0.1901	185	436	552	444	540	568	424
  186	0.4489	0.1113	−0.1901	186	437	553	445	541	569	425
  187	−0.4489	0.1113	0.1901	187	438	554	446	542	570	426
  188	−0.4489	−0.1113	−0.1901	188	439	555	447	543	571	427
  189	0.1901	−0.4489	0.1113	189	440	556	448	544	572	428
  190	0.1901	0.4489	−0.1113	190	441	557	449	545	573	429
  191	−0.1901	0.4489	0.1113	191	442	558	450	546	574	430
  192	−0.1901	−0.4489	−0.1113	192	443	559	451	547	575	431
  193	0.0510	−0.2154	0.4483	193	12	36	96	144	120	60
  194	0.0510	0.2154	−0.4483	194	13	37	97	145	121	61
  195	−0.0510	0.2154	0.4483	195	14	38	98	146	122	62
  196	−0.0510	−0.2154	−0.4483	196	15	39	99	147	123	63
  197	0.4483	−0.0510	0.2154	197	16	40	100	148	124	64
  198	0.4483	0.0510	−0.2154	198	17	41	101	149	125	65
  199	−0.4483	0.0510	0.2154	199	18	42	102	150	126	66
  200	−0.4483	−0.0510	−0.2154	200	19	43	103	151	127	67
  201	0.2154	−0.4483	0.0510	201	20	44	104	152	128	68
  202	0.2154	0.4483	−0.0510	202	21	45	105	153	129	69
  203	−0.2154	0.4483	0.0510	203	22	46	106	154	130	70
  204	−0.2154	−0.4483	−0.0510	204	23	47	107	155	131	71
  205	0.2287	−0.0184	0.4443	205	72	84	168	240	228	132
  206	0.2287	0.0184	−0.4443	206	73	85	169	241	229	133
  207	−0.2287	0.0184	0.4443	207	74	86	170	242	230	134
  208	−0.2287	−0.0184	−0.4443	208	75	87	171	243	231	135
  209	0.4443	−0.2287	0.0184	209	76	88	172	244	232	136
  210	0.4443	0.2287	−0.0184	210	77	89	173	245	233	137
  211	−0.4443	0.2287	0.0184	211	78	90	174	246	234	138
  212	−0.4443	−0.2287	−0.0184	212	79	91	175	247	235	139
  213	0.0184	−0.4443	0.2287	213	80	92	176	248	236	140
  214	0.0184	0.4443	−0.2287	214	81	93	177	249	237	141
  215	−0.0184	0.4443	0.2287	215	82	94	178	250	238	142
  216	−0.0184	−0.4443	−0.2287	216	83	95	179	251	239	143
  217	0.2367	0.0315	0.4393	217	252	336	480	540	444	324
  218	0.2367	−0.0315	−0.4393	218	253	337	481	541	445	325
  219	−0.2367	−0.0315	0.4393	219	254	338	482	542	446	326
  220	−0.2367	0.0315	−0.4393	220	255	339	483	543	447	327
  221	0.4393	0.2367	0.0315	221	256	340	484	544	448	328
  222	0.4393	−0.2367	−0.0315	222	257	341	485	545	449	329
  223	−0.4393	−0.2367	0.0315	223	258	342	486	546	450	330
  224	−0.4393	0.2367	−0.0315	224	259	343	487	547	451	331
  225	0.0315	0.4393	0.2367	225	260	344	488	548	452	332
  226	0.0315	−0.4393	−0.2367	226	261	345	489	549	453	333
  227	−0.0315	−0.4393	0.2367	227	262	346	490	550	454	334
  228	−0.0315	0.4393	−0.2367	228	263	347	491	551	455	335
  229	0.1692	0.1724	0.4378	229	300	362	506	500	560	432
  230	0.1692	−0.1724	−0.4378	230	301	363	507	501	561	433
  231	−0.1692	−0.1724	0.4378	231	302	360	504	502	562	434
  232	−0.1692	0.1724	−0.4378	232	303	361	505	503	563	435
  233	0.4378	0.1692	0.1724	233	304	364	508	492	552	436
  234	0.4378	−0.1692	−0.1724	234	305	365	509	493	553	437
  235	−0.4378	−0.1692	0.1724	235	306	366	510	494	554	438
  236	−0.4378	0.1692	−0.1724	236	307	367	511	495	555	439
  237	0.1724	0.4378	0.1692	237	308	369	513	496	556	440
  238	0.1724	−0.4378	−0.1692	238	309	368	512	497	557	441
  239	−0.1724	−0.4378	0.1692	239	310	371	515	498	558	442
  240	−0.1724	0.4378	−0.1692	240	311	370	514	499	559	443
  241	0.2129	0.1273	0.4341	241	372	408	528	520	600	468
  242	0.2129	−0.1273	−0.4341	242	373	409	529	521	601	469
  243	−0.2129	−0.1273	0.4341	243	374	410	530	522	602	470
  244	−0.2129	0.1273	−0.4341	244	375	411	531	523	603	471
  245	0.4341	0.2129	0.1273	245	376	412	532	524	604	472
  246	0.4341	−0.2129	−0.1273	246	377	413	533	525	605	473
  247	−0.4341	−0.2129	0.1273	247	378	414	534	526	606	474
  248	−0.4341	0.2129	−0.1273	248	379	415	535	527	607	475
  249	0.1273	0.4341	0.2129	249	380	416	536	516	608	476
  250	0.1273	−0.4341	−0.2129	250	381	417	537	517	609	477
  251	−0.1273	−0.4341	0.2129	251	382	418	538	518	610	478
  252	−0.1273	0.4341	−0.2129	252	383	419	539	519	611	479
  253	0.2218	−0.1307	0.4286	253	60	120	204	216	168	84
  254	0.2218	0.1307	−0.4286	254	61	121	205	217	169	85
  255	−0.2218	0.1307	0.4286	255	62	122	206	218	170	86
  256	−0.2218	−0.1307	−0.4286	256	63	123	207	219	171	87
  257	0.4286	−0.2218	0.1307	257	64	124	208	221	173	89
  258	0.4286	0.2218	−0.1307	258	65	125	209	220	172	88
  259	−0.4286	0.2218	0.1307	259	66	126	210	223	175	91
  260	−0.4286	−0.2218	−0.1307	260	67	127	211	222	174	90
  261	0.1307	−0.4286	0.2218	261	68	128	212	226	178	94
  262	0.1307	0.4286	−0.2218	262	69	129	213	227	179	95
  263	−0.1307	0.4286	0.2218	263	70	130	214	224	176	92
  264	−0.1307	−0.4286	−0.2218	264	71	131	215	225	177	93
  265	0.0707	0.2558	0.4238	265	144	252	324	276	204	120
  266	0.0707	−0.2558	−0.4238	266	145	253	325	277	205	121
  267	−0.0707	−0.2558	0.4238	267	146	254	326	278	206	122
  268	−0.0707	0.2558	−0.4238	268	147	255	327	279	207	123
  269	0.4238	0.0707	0.2558	269	148	256	328	280	208	124
  270	0.4238	−0.0707	−0.2558	270	149	257	329	281	209	125
  271	−0.4238	−0.0707	0.2558	271	150	258	330	282	210	126
  272	−0.4238	0.0707	−0.2558	272	151	259	331	283	211	127
  273	0.2558	0.4238	0.0707	273	152	260	332	284	212	128
  274	0.2558	−0.4238	−0.0707	274	153	261	333	285	213	129
  275	−0.2558	−0.4238	0.0707	275	154	262	334	286	214	130
  276	−0.2558	0.4238	−0.0707	276	155	263	335	287	215	131
  277	0.2665	−0.0428	0.4209	277	240	168	216	312	408	372
  278	0.2665	0.0428	−0.4209	278	241	169	217	313	409	373
  279	−0.2665	0.0428	0.4209	279	242	170	218	314	410	374
  280	−0.2665	−0.0428	−0.4209	280	243	171	219	315	411	375
  281	0.4209	−0.2665	0.0428	281	244	172	220	316	412	376
  282	0.4209	0.2665	−0.0428	282	245	173	221	317	413	377
  283	−0.4209	0.2665	0.0428	283	246	174	222	318	414	378
  284	−0.4209	−0.2665	−0.0428	284	247	175	223	319	415	379
  285	0.0428	−0.4209	0.2665	285	248	176	224	320	416	380
  286	0.0428	0.4209	−0.2665	286	249	177	225	321	417	381
  287	−0.0428	0.4209	0.2665	287	250	178	226	322	418	382
  288	−0.0428	−0.4209	−0.2665	288	251	179	227	323	419	383
  289	0.1599	−0.2213	0.4189	289	444	552	492	384	276	324
  290	0.1599	0.2213	−0.4189	290	445	553	493	385	277	325
  291	−0.1599	0.2213	0.4189	291	446	554	494	386	278	326
  292	−0.1599	−0.2213	−0.4189	292	447	555	495	387	279	327
  293	0.4189	−0.1599	0.2213	293	448	556	496	388	280	328
  294	0.4189	0.1599	−0.2213	294	449	557	497	389	281	329
  295	−0.4189	0.1599	0.2213	295	450	558	498	390	282	330
  296	−0.4189	−0.1599	−0.2213	296	451	559	499	391	283	331
  297	0.2213	−0.4189	0.1599	297	452	560	500	392	284	332
  298	0.2213	0.4189	−0.1599	298	453	561	501	393	285	333
  299	−0.2213	0.4189	0.1599	299	454	562	502	394	286	334
  300	−0.2213	−0.4189	−0.1599	300	455	563	503	395	287	335
  301	0.0403	−0.2720	0.4176	301	504	536	416	320	394	502
  302	0.0403	0.2720	−0.4176	302	505	537	417	321	395	503
  303	−0.0403	0.2720	0.4176	303	506	538	418	322	392	500
  304	−0.0403	−0.2720	−0.4176	304	507	539	419	323	393	501
  305	0.4176	−0.0403	0.2720	305	508	528	408	312	384	492
  306	0.4176	0.0403	−0.2720	306	509	529	409	313	385	493
  307	−0.4176	0.0403	0.2720	307	510	530	410	314	386	494
  308	−0.4176	−0.0403	−0.2720	308	511	531	411	315	387	495
  309	0.2720	−0.4176	0.0403	309	512	532	412	316	389	497
  310	0.2720	0.4176	−0.0403	310	513	533	413	317	388	496
  311	−0.2720	0.4176	0.0403	311	514	534	414	318	391	499
  312	−0.2720	−0.4176	−0.0403	312	515	535	415	319	390	498
  313	0.2795	0.0379	0.4128
  314	0.2795	−0.0379	−0.4128
  315	−0.2795	−0.0379	0.4128
  316	−0.2795	0.0379	−0.4128
  317	0.4128	0.2795	0.0379
  318	0.4128	−0.2795	−0.0379
  319	−0.4128	−0.2795	0.0379
  320	−0.4128	0.2795	−0.0379
  321	0.0379	0.4128	0.2795
  322	0.0379	−0.4128	−0.2795
  323	−0.0379	−0.4128	0.2795
  324	−0.0379	0.4128	−0.2795
  325	0.2683	−0.0968	0.4106
  326	0.2683	0.0968	−0.4106
  327	−0.2683	0.0968	0.4106
  328	−0.2683	−0.0968	−0.4106
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  332	−0.4106	−0.2683	−0.0968
  333	0.0968	−0.4106	0.2683
  334	0.0968	0.4106	−0.2683
  335	−0.0968	0.4106	0.2683
  336	−0.0968	−0.4106	−0.2683
  337	0.2157	−0.1901	0.4091
  338	0.2157	0.1901	−0.4091
  339	−0.2157	0.1901	0.4091
  340	−0.2157	−0.1901	−0.4091
  341	0.4091	−0.2157	0.1901
  342	0.4091	0.2157	−0.1901
  343	−0.4091	0.2157	0.1901
  344	−0.4091	−0.2157	−0.1901
  345	0.1901	−0.4091	0.2157
  346	0.1901	0.4091	−0.2157
  347	−0.1901	0.4091	0.2157
  348	−0.1901	−0.4091	−0.2157
  349	0.1788	0.2285	0.4072
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  351	−0.1788	−0.2285	0.4072
  352	−0.1788	0.2285	−0.4072
  353	0.4072	0.1788	0.2285
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  355	−0.4072	−0.1788	0.2285
  356	−0.4072	0.1788	−0.2285
  357	0.2285	0.4072	0.1788
  358	0.2285	−0.4072	−0.1788
  359	−0.2285	−0.4072	0.1788
  360	−0.2285	0.4072	−0.1788
  361	0.0200	0.2917	0.4056
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  363	−0.0200	−0.2917	0.4056
  364	−0.0200	0.2917	−0.4056
  365	0.4056	0.0200	0.2917
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  367	−0.4056	−0.0200	0.2917
  368	−0.4056	0.0200	−0.2917
  369	0.2917	0.4056	0.0200
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  371	−0.2917	−0.4056	0.0200
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  373	0.2647	0.1351	0.4021
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  375	−0.2647	−0.1351	0.4021
  376	−0.2647	0.1351	−0.4021
  377	0.4021	0.2647	0.1351
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  379	−0.4021	−0.2647	0.1351
  380	−0.4021	0.2647	−0.1351
  381	0.1351	0.4021	0.2647
  382	0.1351	−0.4021	−0.2647
  383	−0.1351	−0.4021	0.2647
  384	−0.1351	0.4021	−0.2647
  385	0.2980	−0.0080	0.4014
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  387	−0.2980	0.0080	0.4014
  388	−0.2980	−0.0080	−0.4014
  389	0.4014	−0.2980	0.0080
  390	0.4014	0.2980	−0.0080
  391	−0.4014	0.2980	0.0080
  392	−0.4014	−0.2980	−0.0080
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  395	−0.0080	0.4014	0.2980
  396	−0.0080	−0.4014	−0.2980
  397	0.1296	0.2704	0.4001
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  399	−0.1296	−0.2704	0.4001
  400	−0.1296	0.2704	−0.4001
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  403	−0.4001	−0.1296	0.2704
  404	−0.4001	0.1296	−0.2704
  405	0.2704	0.4001	0.1296
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  407	−0.2704	−0.4001	0.1296
  408	−0.2704	0.4001	−0.1296
  409	0.2977	0.0856	0.3925
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  411	−0.2977	−0.0856	0.3925
  412	−0.2977	0.0856	−0.3925
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  415	−0.3925	−0.2977	0.0856
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  419	−0.0856	−0.3925	0.2977
  420	−0.0856	0.3925	−0.2977
  421	0.1484	−0.2776	0.3884
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  424	−0.1484	−0.2776	−0.3884
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  429	0.2776	−0.3884	0.1484
  430	0.2776	0.3884	−0.1484
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  432	−0.2776	−0.3884	−0.1484
  433	0.0888	−0.3025	0.3881
  434	0.0888	0.3025	−0.3881
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  439	−0.3881	0.0888	0.3025
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  442	0.3025	0.3881	−0.0888
  443	−0.3025	0.3881	0.0888
  444	−0.3025	−0.3881	−0.0888
  445	0.3111	−0.1174	0.3734
  446	0.3111	0.1174	−0.3734
  447	−0.3111	0.1174	0.3734
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  449	0.3734	−0.3111	0.1174
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  451	−0.3734	0.3111	0.1174
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  455	−0.1174	0.3734	0.3111
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  459	−0.2337	−0.2368	0.3732
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  481	0.2603	−0.2143	0.3692
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  483	−0.2603	0.2143	0.3692
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  490	0.2143	0.3692	−0.2603
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  538	0.0842	−0.3533	−0.3436
  539	−0.0842	−0.3533	0.3436
  540	−0.0842	0.3533	−0.3436
  541	0.3091	−0.1762	0.3513
  542	0.3091	0.1762	−0.3513
  543	−0.3091	0.1762	0.3513
  544	−0.3091	−0.1762	−0.3513
  545	0.3513	−0.3091	0.1762
  546	0.3513	0.3091	−0.1762
  547	−0.3513	0.3091	0.1762
  548	−0.3513	−0.3091	−0.1762
  549	0.1762	−0.3513	0.3091
  550	0.1762	0.3513	−0.3091
  551	−0.1762	0.3513	0.3091
  552	−0.1762	−0.3513	−0.3091
  553	0.3491	−0.0753	0.3499
  554	0.3491	0.0753	−0.3499
  555	−0.3491	0.0753	0.3499
  556	−0.3491	−0.0753	−0.3499
  557	0.3499	−0.3491	0.0753
  558	0.3499	0.3491	−0.0753
  559	−0.3499	0.3491	0.0753
  560	−0.3499	−0.3491	−0.0753
  561	0.0753	−0.3499	0.3491
  562	0.0753	0.3499	−0.3491
  563	−0.0753	0.3499	0.3491
  564	−0.0753	−0.3499	−0.3491
  565	0.1931	−0.3025	0.3481
  566	0.1931	0.3025	−0.3481
  567	−0.1931	0.3025	0.3481
  568	−0.1931	−0.3025	−0.3481
  569	0.3481	−0.1931	0.3025
  570	0.3481	0.1931	−0.3025
  571	−0.3481	0.1931	0.3025
  572	−0.3481	−0.1931	−0.3025
  573	0.3025	−0.3481	0.1931
  574	0.3025	0.3481	−0.1931
  575	−0.3025	0.3481	0.1931
  576	−0.3025	−0.3481	−0.1931
  577	0.2495	−0.2708	0.3383
  578	0.2495	0.2708	−0.3383
  579	−0.2495	0.2708	0.3383
  580	−0.2495	−0.2708	−0.3383
  581	0.3383	−0.2495	0.2708
  582	0.3383	0.2495	−0.2708
  583	−0.3383	0.2495	0.2708
  584	−0.3383	−0.2495	−0.2708
  585	0.2708	−0.3383	0.2495
  586	0.2708	0.3383	−0.2495
  587	−0.2708	0.3383	0.2495
  588	−0.2708	−0.3383	−0.2495
  589	0.2392	0.2873	0.3320
  590	0.2392	−0.2873	−0.3320
  591	−0.2392	−0.2873	0.3320
  592	−0.2392	0.2873	−0.3320
  593	0.3320	0.2392	0.2873
  594	0.3320	−0.2392	−0.2873
  595	−0.3320	−0.2392	0.2873
  596	−0.3320	0.2392	−0.2873
  597	0.2873	0.3320	0.2392
  598	0.2873	−0.3320	−0.2392
  599	−0.2873	−0.3320	0.2392
  600	−0.2873	0.3320	−0.2392
  601	0.3254	0.1894	0.3289
  602	0.3254	−0.1894	−0.3289
  603	−0.3254	−0.1894	0.3289
  604	−0.3254	0.1894	−0.3289
  605	0.3289	0.3254	0.1894
  606	0.3289	−0.3254	−0.1894
  607	−0.3289	−0.3254	0.1894
  608	−0.3289	0.3254	−0.1894
  609	0.1894	0.3289	0.3254
  610	0.1894	−0.3289	−0.3254
  611	−0.1894	−0.3289	0.3254
  612	−0.1894	0.3289	−0.3254
  613	0.2887	0.2887	0.2887
  614	0.2887	0.2887	−0.2887
  615	0.2887	−0.2887	0.2887
  616	0.2887	−0.2887	−0.2887
  617	−0.2887	0.2887	0.2887
  618	−0.2887	0.2887	−0.2887
  619	−0.2887	−0.2887	0.2887
  620	−0.2887	−0.2887	−0.2887

• TABLE 5
  Sphere	x/D	y/D	z/D	d/D

  1	0.3189	0.0000	0.5160	0.2254
  2	0.3189	0.0000	−0.5160	0.2254
  3	−0.3189	0.0000	0.5160	0.2254
  4	−0.3189	0.0000	−0.5160	0.2254
  5	0.5160	−0.3189	0.0000	0.2254
  6	0.5160	0.3189	0.0000	0.2254
  7	−0.5160	0.3189	0.0000	0.2254
  8	−0.5160	−0.3189	0.0000	0.2254
  9	0.0000	−0.5160	0.3189	0.2254
  10	0.0000	0.5160	−0.3189	0.2254
  11	0.0000	0.5160	0.3189	0.2254
  12	0.0000	−0.5160	−0.3189	0.2254
  13	0.0794	0.2166	0.7473	0.5800
  14	0.0159	0.3710	0.6884	0.5800
  15	−0.0954	0.2425	0.7375	0.5800
  16	−0.0794	0.2166	−0.7473	0.5800
  17	−0.0159	0.3710	−0.6884	0.5800
  18	0.0954	0.2425	−0.7375	0.5800
  19	−0.0794	−0.2166	0.7473	0.5800
  20	−0.0159	−0.3710	0.6884	0.5800
  21	0.0954	−0.2425	0.7375	0.5800
  22	0.0794	−0.2166	−0.7473	0.5800
  23	0.0159	−0.3710	−0.6884	0.5800
  24	−0.0954	−0.2425	−0.7375	0.5800
  25	0.7473	0.0794	0.2166	0.5800
  26	0.6884	0.0159	0.3710	0.5800
  27	0.7375	−0.0954	0.2425	0.5800
  28	0.7473	−0.0794	−0.2166	0.5800
  29	0.6884	−0.0159	−0.3710	0.5800
  30	0.7375	0.0954	−0.2425	0.5800
  31	−0.7473	−0.0794	0.2166	0.5800
  32	−0.6884	−0.0159	0.3710	0.5800
  33	−0.7375	0.0954	0.2425	0.5800
  34	−0.7473	0.0794	−0.2166	0.5800
  35	−0.6884	0.0159	−0.3710	0.5800
  36	−0.7375	−0.0954	−0.2425	0.5800
  37	0.2166	0.7473	0.0794	0.5800
  38	0.3710	0.6884	0.0159	0.5800
  39	0.2425	0.7375	−0.0954	0.5800
  40	0.2166	−0.7473	−0.0794	0.5800
  41	0.3710	−0.6884	−0.0159	0.5800
  42	0.2425	−0.7375	0.0954	0.5800
  43	−0.2166	0.7473	−0.0794	0.5800
  44	−0.3710	0.6884	−0.0159	0.5800
  45	−0.2425	0.7375	0.0954	0.5800
  46	−0.2166	−0.7473	0.0794	0.5800
  47	−0.3710	−0.6884	0.0159	0.5800
  48	−0.2425	−0.7375	−0.0954	0.5800
  49	0.4459	0.3763	0.5208	0.5800
  50	0.5208	0.4459	0.3763	0.5800
  51	0.3763	0.5208	0.4459	0.5800
  52	0.3665	0.5049	−0.4717	0.5800
  53	0.5049	0.4717	−0.3665	0.5800
  54	0.4717	0.3665	−0.5049	0.5800
  55	0.3665	−0.5049	0.4717	0.5800
  56	0.5049	−0.4717	0.3665	0.5800
  57	0.4717	−0.3665	0.5049	0.5800
  58	0.4459	−0.3763	−0.5208	0.5800
  59	0.5208	−0.4459	−0.3763	0.5800
  60	0.3763	−0.5208	−0.4459	0.5800
  61	−0.3665	0.5049	0.4717	0.5800
  62	−0.5049	0.4717	0.3665	0.5800
  63	−0.4717	0.3665	0.5049	0.5800
  64	−0.4459	0.3763	−0.5208	0.5800
  65	−0.5208	0.4459	−0.3763	0.5800
  66	−0.3763	0.5208	−0.4459	0.5800
  67	−0.4459	−0.3763	0.5208	0.5800
  68	−0.5208	−0.4459	0.3763	0.5800
  69	−0.3763	−0.5208	0.4459	0.5800
  70	−0.3665	−0.5049	−0.4717	0.5800
  71	−0.5049	−0.4717	−0.3665	0.5800
  72	−0.4717	−0.3665	−0.5049	0.5800
  73	0.0315	−0.0822	0.7752	0.5761
  74	0.0315	0.0822	−0.7752	0.5761
  75	−0.0315	0.0822	0.7752	0.5761
  76	−0.0315	−0.0822	−0.7752	0.5761
  77	0.7752	−0.0315	0.0822	0.5761
  78	0.7752	0.0315	−0.0822	0.5761
  79	−0.7752	0.0315	0.0822	0.5761
  80	−0.7752	−0.0315	−0.0822	0.5761
  81	0.0822	−0.7752	0.0315	0.5761
  82	0.0822	0.7752	−0.0315	0.5761
  83	−0.0822	0.7752	0.0315	0.5761
  84	−0.0822	−0.7752	−0.0315	0.5761
  85	0.1888	0.3367	0.6780	0.5761
  86	0.1888	−0.3367	−0.6780	0.5761
  87	−0.1888	−0.3367	0.6780	0.5761
  88	−0.1888	0.3367	−0.6780	0.5761
  89	0.6780	0.1888	0.3367	0.5761
  90	0.6780	−0.1888	−0.3367	0.5761
  91	−0.6780	−0.1888	0.3367	0.5761
  92	−0.6780	0.1888	−0.3367	0.5761
  93	0.3367	0.6780	0.1888	0.5761
  94	0.3367	−0.6780	−0.1888	0.5761
  95	−0.3367	−0.6780	0.1888	0.5761
  96	−0.3367	0.6780	−0.1888	0.5761
  97	0.1573	−0.3877	0.6585	0.5761
  98	0.1573	0.3877	−0.6585	0.5761
  99	−0.1573	0.3877	0.6585	0.5761
  100	−0.1573	−0.3877	−0.6585	0.5761
  101	0.6585	−0.1573	0.3877	0.5761
  102	0.6585	0.1573	−0.3877	0.5761
  103	−0.6585	0.1573	0.3877	0.5761
  104	−0.6585	−0.1573	−0.3877	0.5761
  105	0.3877	−0.6585	0.1573	0.5761
  106	0.3877	0.6585	−0.1573	0.5761
  107	−0.3877	0.6585	0.1573	0.5761
  108	−0.3877	−0.6585	−0.1573	0.5761
  109	0.3218	−0.3875	0.5958	0.5761
  110	0.3218	0.3875	−0.5958	0.5761
  111	−0.3218	0.3875	0.5958	0.5761
  112	−0.3218	−0.3875	−0.5958	0.5761
  113	0.5958	−0.3218	0.3875	0.5761
  114	0.5958	0.3218	−0.3875	0.5761
  115	−0.5958	0.3218	0.3875	0.5761
  116	−0.5958	−0.3218	−0.3875	0.5761
  117	0.3875	−0.5958	0.3218	0.5761
  118	0.3875	0.5958	−0.3218	0.5761
  119	−0.3875	0.5958	0.3218	0.5761
  120	−0.3875	−0.5958	−0.3218	0.5761
  121	0.2903	0.4385	0.5763	0.5761
  122	0.2903	−0.4385	−0.5763	0.5761
  123	−0.2903	−0.4385	0.5763	0.5761
  124	−0.2903	0.4385	−0.5763	0.5761
  125	0.5763	0.2903	0.4385	0.5761
  126	0.5763	−0.2903	−0.4385	0.5761
  127	−0.5763	−0.2903	0.4385	0.5761
  128	−0.5763	0.2903	−0.4385	0.5761
  129	0.4385	0.5763	0.2903	0.5761
  130	0.4385	−0.5763	−0.2903	0.5761
  131	−0.4385	−0.5763	0.2903	0.5761
  132	−0.4385	0.5763	−0.2903	0.5761
  133	0.1373	0.0529	0.7524	0.5487
  134	0.1373	−0.0529	−0.7524	0.5487
  135	−0.1373	−0.0529	0.7524	0.5487
  136	−0.1373	0.0529	−0.7524	0.5487
  137	0.7524	−0.1373	−0.0529	0.5487
  138	0.7524	0.1373	0.0529	0.5487
  139	−0.7524	0.1373	−0.0529	0.5487
  140	−0.7524	−0.1373	0.0529	0.5487
  141	−0.0529	−0.7524	0.1373	0.5487
  142	−0.0529	0.7524	−0.1373	0.5487
  143	0.0529	0.7524	0.1373	0.5487
  144	0.0529	−0.7524	−0.1373	0.5487
  145	0.2584	−0.2488	0.6775	0.5487
  146	0.2584	0.2488	−0.6775	0.5487
  147	−0.2584	0.2488	0.6775	0.5487
  148	−0.2584	−0.2488	−0.6775	0.5487
  149	0.6775	−0.2584	0.2488	0.5487
  150	0.6775	0.2584	−0.2488	0.5487
  151	−0.6775	0.2584	0.2488	0.5487
  152	−0.6775	−0.2584	−0.2488	0.5487
  153	0.2488	−0.6775	0.2584	0.5487
  154	0.2488	0.6775	−0.2584	0.5487
  155	−0.2488	0.6775	0.2584	0.5487
  156	−0.2488	−0.6775	−0.2584	0.5487
  157	0.3439	0.2815	0.6247	0.5487
  158	0.3439	−0.2815	−0.6247	0.5487
  159	−0.3439	−0.2815	0.6247	0.5487
  160	−0.3439	0.2815	−0.6247	0.5487
  161	0.6247	0.3439	0.2815	0.5487
  162	0.6247	−0.3439	−0.2815	0.5487
  163	−0.6247	−0.3439	0.2815	0.5487
  164	−0.6247	0.3439	−0.2815	0.5487
  165	0.2815	0.6247	0.3439	0.5487
  166	0.2815	−0.6247	−0.3439	0.5487
  167	−0.2815	−0.6247	0.3439	0.5487
  168	−0.2815	0.6247	−0.3439	0.5487
  169	0.1211	0.4709	0.5927	0.5487
  170	0.1211	−0.4709	−0.5927	0.5487
  171	−0.1211	−0.4709	0.5927	0.5487
  172	−0.1211	0.4709	−0.5927	0.5487
  173	0.5927	0.1211	0.4709	0.5487
  174	0.5927	−0.1211	−0.4709	0.5487
  175	−0.5927	−0.1211	0.4709	0.5487
  176	−0.5927	0.1211	−0.4709	0.5487
  177	0.4709	0.5927	0.1211	0.5487
  178	0.4709	−0.5927	−0.1211	0.5487
  179	−0.4709	−0.5927	0.1211	0.5487
  180	−0.4709	0.5927	−0.1211	0.5487
  181	0.2066	−0.5036	0.5399	0.5487
  182	0.2066	0.5036	−0.5399	0.5487
  183	−0.2066	0.5036	0.5399	0.5487
  184	−0.2066	−0.5036	−0.5399	0.5487
  185	0.5399	−0.2066	0.5036	0.5487
  186	0.5399	0.2066	−0.5036	0.5487
  187	−0.5399	0.2066	0.5036	0.5487
  188	−0.5399	−0.2066	−0.5036	0.5487
  189	0.5036	−0.5399	0.2066	0.5487
  190	0.5036	0.5399	−0.2066	0.5487
  191	−0.5036	0.5399	0.2066	0.5487
  192	−0.5036	−0.5399	−0.2066	0.5487
  193	0.1942	−0.1019	0.7251	0.5304
  194	0.1942	0.1019	−0.7251	0.5304
  195	−0.1942	0.1019	0.7251	0.5304
  196	−0.1942	−0.1019	−0.7251	0.5304
  197	0.7251	−0.1942	0.1019	0.5304
  198	0.7251	0.1942	−0.1019	0.5304
  199	−0.7251	0.1942	0.1019	0.5304
  200	−0.7251	−0.1942	−0.1019	0.5304
  201	0.1019	−0.7251	0.1942	0.5304
  202	0.1019	0.7251	−0.1942	0.5304
  203	−0.1019	0.7251	0.1942	0.5304
  204	−0.1019	−0.7251	−0.1942	0.5304
  205	0.2387	0.1739	0.6976	0.5304
  206	0.2387	−0.1739	−0.6976	0.5304
  207	−0.2387	−0.1739	0.6976	0.5304
  208	−0.2387	0.1739	−0.6976	0.5304
  209	0.6976	0.2387	0.1739	0.5304
  210	0.6976	−0.2387	−0.1739	0.5304
  211	−0.6976	−0.2387	0.1739	0.5304
  212	−0.6976	0.2387	−0.1739	0.5304
  213	0.1739	0.6976	0.2387	0.5304
  214	0.1739	−0.6976	−0.2387	0.5304
  215	−0.1739	−0.6976	0.2387	0.5304
  216	−0.1739	0.6976	−0.2387	0.5304
  217	0.4037	−0.2369	0.5957	0.5304
  218	0.4037	0.2369	−0.5957	0.5304
  219	−0.4037	0.2369	0.5957	0.5304
  220	−0.4037	−0.2369	−0.5957	0.5304
  221	0.5957	−0.4037	0.2369	0.5304
  222	0.5957	0.4037	−0.2369	0.5304
  223	−0.5957	0.4037	0.2369	0.5304
  224	−0.5957	−0.4037	−0.2369	0.5304
  225	0.2369	−0.5957	0.4037	0.5304
  226	0.2369	0.5957	−0.4037	0.5304
  227	−0.2369	0.5957	0.4037	0.5304
  228	−0.2369	−0.5957	−0.4037	0.5304
  229	0.0445	−0.4882	0.5776	0.5304
  230	0.0445	0.4882	−0.5776	0.5304
  231	−0.0445	0.4882	0.5776	0.5304
  232	−0.0445	−0.4882	−0.5776	0.5304
  233	0.5776	−0.0445	0.4882	0.5304
  234	0.5776	0.0445	−0.4882	0.5304
  235	−0.5776	0.0445	0.4882	0.5304
  236	−0.5776	−0.0445	−0.4882	0.5304
  237	0.4882	−0.5776	0.0445	0.5304
  238	0.4882	0.5776	−0.0445	0.5304
  239	−0.4882	0.5776	0.0445	0.5304
  240	−0.4882	−0.5776	−0.0445	0.5304
  241	0.4757	0.2094	0.5512	0.5304
  242	0.4757	−0.2094	−0.5512	0.5304
  243	−0.4757	−0.2094	0.5512	0.5304
  244	−0.4757	0.2094	−0.5512	0.5304
  245	0.5512	0.4757	0.2094	0.5304
  246	0.5512	−0.4757	−0.2094	0.5304
  247	−0.5512	−0.4757	0.2094	0.5304
  248	−0.5512	0.4757	−0.2094	0.5304
  249	0.2094	0.5512	0.4757	0.5304
  250	0.2094	−0.5512	−0.4757	0.5304
  251	−0.2094	−0.5512	0.4757	0.5304
  252	−0.2094	0.5512	−0.4757	0.5304
  253	0.2679	0.0224	0.6585	0.4368
  254	0.2679	−0.0224	−0.6585	0.4368
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• The foregoing description and drawings should be considered as illustrative only of the principles of the invention. The invention may be configured in a variety of shapes and sizes and is not intended to be limited by the embodiment shown. In addition, the statements made with respect to one embodiment apply to the other embodiments, unless otherwise specifically noted. For example, the statements regarding FIG. 2(a) with respect to size, shape and geometry apply equally to the embodiments of FIGS. 3, 7-9, 12. It is further understood that the description and scope of invention apply equally (though the descriptions have not been repeated) for each structure that is the same or similar between each of the various embodiment, and whether or not those structures have been assigned a similar reference numeral.
• Numerous applications of the invention will readily occur to those skilled in the art. Therefore, it is not desired to limit the invention to the specific examples disclosed or the exact construction and operation shown and described. Rather, all suitable modifications and equivalents may be resorted to, falling within the scope of the invention.

Claims (17)

1. A golf ball comprising:

a body having an outer shell with an outersurface; and

a pattern formed in the outer surface of said body, the pattern comprising a polyhedron having a plurality of flat faces, each of said plurality of flat faces having one or more sharp edges.

2. The golf ball of claim 1, wherein said plurality of faces are circumscribed in a sphere, wherein only the sharp corners forming vertices of the polyhedron lie on the sphere.

3. The golf ball of claim 2, said sphere having a diameter of at least 1·68 in.

4. The golf ball of claim 1, wherein at least one of the plurality of faces of the polyhedron contains one or more dimples.

5. The golf ball of claim 1, wherein said plurality of faces are each in a plane.

6. The golf ball of claim 1, wherein said plurality of faces are contiguous to touch one another and form a single continuous outer surface of said body.

7. The golf ball of claim 1, wherein said plurality of faces are at an angle with respect to one another to define said one or more sharp edges and said one or more sharp corners.

8. The golf ball of claim 1, wherein said pattern comprises a Goldberg polyhedron.

9. The golf ball of claim 1, wherein said plurality of faces comprise a plurality of first faces having a first shape and a plurality of second faces having a second shape.

10. The golf ball of claim 9, wherein said first shape comprises a pentagon and said second shape comprises a hexagon.

11. The golf ball of claim 9, wherein said plurality of first faces comprise twelve and said plurality of second faces comprise 150.

12. The golf ball of claim 9, wherein a ration of said plurality of first faces to said plurality of second faces comprises 12.5:1.

13. The golf ball of claim 1, wherein said plurality of flat faces having one or more sharp corners.

14. The golf ball of claim 1, wherein said edges are linear.

15. The golf ball of claim 1, wherein two neighboring flat faces form an angle substantially less than 180 degrees.

16. The golf ball of claim 1, wherein said sharp edges have a radius of curvature that is less than 0.001 D, where D is the diameter of a circumscribed sphere of said golf ball.

17. A method of forming a golf ball, comprising:

forming an outer surface; and,

forming a pattern in the outer surface, the pattern having a plurality of flat surfaces defining sharp edges and points therebetween.

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