IL47517A - Kaleidoscope - Google Patents

Description

ŋšleidoscopes The present Invention relates to kaleidoscopes. More ^ particularly the present nvention relates to spherical and spatial kaleidoscopes.

The mathematical concept of a kaleidoscope and Its various applications based on multiple reflections from an arrangement of multiple reflecting surfaces have been known since ancient times. Today the term kaleidoscope evokes the familiar mage of the child's toy 1n the form of an optical nstrument n which bits of colored glass etc. 1n a rotating tube are shown by reflection 1n continually changing symmetrical forms. In fact the term kaleidoscope Is derived from combinations of the Greek word kalos which means beautiful, the Greek word e1do(s) which means form and the word scope.

Thus kaleidoscopes have been used to generate and view beautiful forms for almost as long as the polished reflecting surface has been known.

The basic kaleidoscopic function is the generation of a periodic regular geometric phenomenon which results from the multiplication of an elementary unit through application of symmetry operations of reflection.

Neretofor there have been known and described kaleidoscopes which cause one dimensional linear expansion as s the case of two parallel mirrors reflecting an elementary cell unit placed therebetween and kaleidoscopes which cause two-dimensional, plane expansions according to plane reflecting regular and semi-regular networks such as found 1n the child's toy kaleidoscope.

The present invention 1s directed to two additional classes of kaleidoscopes hereinafter referred to as spherical kaleidoscopes and spatial kaleidoscopes.

- Spherical kaleidoscopes are those which cause two-dimensional spherical expansions In reference to a symmetry center according to patterns of convex regular and semi-regular polyhedra.

Similarly spatial kaleidoscopes cause three dimensional spatial expansion and result n spatial networks and lattices.

More specifically the present Invention provides novel spherical kaleidoscopes In the general form of triangular pyramids comprising three plane reflecting surfaces Interfaclngly arranged and attached together to form three corner edges wherein each corner edge Is common to a pair of adjacent planes, and lines extending along each of said corner edges meet at a common convergence point which 1s the center of symmetry of said kaleidoscope and further characterized 1n that the dihedral angles between each set of two adjacent plane surfaces are selected from the groupings consisting of 90°, 60° and 60°; 90°. 60° and 45°: 90e, 60° and 36°j and 90°, 90° and o , wherein a 1s any angle greater than 0° and less than 90° satisfying the equation vertices of said tetrahedron constitutes the convergence point of a spherical kaleidoscope as defined above wherein the dihedral angles between each of the three sets of two adjacent plane surfaces defining each of said vertices are selected from the groupings consisting of 90°. 90° and 45°; 90°. 60° and 60°; and 90°, 60° and 45°.

The spherical and spatial kaleidoscopes of the present Invention have many useful applications Including: - - 1. Teaching device and llustration for all geometrical s periodic configurations whose periodicity 1s a function of ^ reflections. 2. Aid in planning structures based on the above mentioned periodic geometric configurations. The aid In planning consists In Instant realization by producing a model of the periodical elementary unit of the structure and putting It Into the suitable kaleidoscope. Planning and research In the field of regular constructions such as geodesic domes, regular space trusses, regular modular grids, and even research on crystallography which demands drawing of elaborate space networks, will be made easier by this method of Illustration. 3. Toys. This Is the traditional use of kaleidoscopes, but the kaleidoscopes described here greatly surpass present achievements. 4. For exhibitions, decorations and advertising.

The medium of mirrors 1s an Inexhaustible source of fantastic optical effects, but the Impact of the proposed kaleidoscope has Its source 1n the unique relationship between the spli se means and the overwhelming variety of results.

While the Invention will now be described 1n connection with certain preferred embodiments, 1t will be understood that It 1s not Intended to limit the Invention to these particular embodiments. On the contrary, It Is Intended to cover all alternatives , modifications and equivalent arrangements as may be Included within the scope of the Invention as defined by the appended claims. Nevertheless 1t 1s believed that the embodiments of the Invention will be more fully understood from a consideration of the following Illustrative description.

Referring first to the spherical kaleidoscopes, said " kaleidoscopes break up Into two categories: ^ a) Kaleidoscopes wherein the dihedral angles are selected from the grouping 90°, 90° and a wherein a 1s any angle greater than 0° and less than 90° satisfying the equation a* ]j~ end m 1s a whole number; and b) kaleidoscopes herein the dihedral angles are selected from the groupings consisting of 90°, 60° and 60*; 90°, 60° and 45°; and 90°, 60° and 36°.

The kaleidoscopes of the first type are prepared by placing one plane reflecting surface, e.g. a mirror on a horizontal plane and then placing two additional plane reflecting surfaces on a vertical plane at right angles to the first sur&ce whereby there 1s formed 90° angles between the first and second and the first and third mirrors. The second and thtrd mirrors are attached together along a common edge wherein the angle a between said second and third mirrors fulfills the definition set forth above. Thus o can be 1°, 2°. 3°, 3*6°, 4°. 4.5°, 5% 6β , etc.

The construction of the spherical kaleidoscopes of the second type 1s more difficult since the creation of a generally triangulai* pyramidal form having dihedral angles between adjacent surfaces of other than 90e cannot be readily done as with the first type o spherical kaleidoscopes.

The generation of such kaleidoscopes however 1s carried out with the aid of existing regular solid geometric figures Including the tetrahedron, the cube or octahedron and the lcosahedron or dodecahedron* In each of said figures the respective desired spherical kaleidoscope 1s generated and defined by the area enclosed when the points constituting the center of the figure, the center of one face of t e figure, the center of one edge of said face and the vertex point of said edge are Interconnected; the point constituting the center of the figure also constituting the center of symmetry of the kaleidoscope. .

The spatial kaleidoscopes of the present Invention are in the general form of a convex closed tetrahedron wherein each of the vertices of said tetrahedron constitutes the convergence point of a spherical kaleidoscope as defined.

Thus for example spatial kaleidoscopes according to the present Invention can be manufactured from tetrahedrons generated within the cubic symmetry group. Starting, e.g.* with a cube with an edge length of 2 1t Is possible. to generate a tetrahedron representing 1/48 the volume of said cube and sub-groups thereof wherein the four triangular surfaces of said tetrahedrons have from the following groupings: a) 2, and si 2, {Ϊ and T; 2, b) 2, and ϊ/ϊ.

The region close to said convergence point can be truncated along a plane triangular section or said open triangular section can br~ composed of three concentric arcs having equal radii and a common center which coincides with said convergence point formed by cutting curved arcs in each of the three surfaces converging on sa d center of symmetry so that each point on each and all of said respective arcs Is equidistant from said convergence point.

In order that the Invention may be understood more fully, reference should be made to the following Illustrative description read in conjunction with the accompanying drawings 1n which: Fig, 1 Is a schematic «1ew of a truncated spherical kaleidoscope according to the present invention and an Image generated thereby and viewable therethrough.

With specific reference now to the figure 1n detail t Is stressed that the particulars shown are by way of example and for purposes of Illustrative discussion only and are presented 1n the cause of providing what Is believed to be the most useful and readily understood description of the principles and conceptual aspects of the Invention. In this regard no attempt Is made to show structural details of the system and Its apparatus 1n more detail than 1s necessary for a fundamental understanding of the Invention the description taken with the drawing making apparent to those skilled In the art how the several forms of the Invention may be embodied 1n practice.

Referring now to F1g. 1 there Is llustrated spherical kaleidoscope O LM derived from the icosahedral -dodecahedral symmetry group and having dihedral angles of 90°, 60° and 36°, OKLM 1s a kaleidoscope built of three mirrors, meeting along the lines of OK, OL, OM and having a center of symmetry at pint 0.

Said kaleidoscope 1s truncated close to convergence point 0 to form a spherical triangular section ABC composed of three circular concentric arcs AB, BC and AC having equal rad11 and a common The triangle ABC Is an elementairy„ periodic surface unit representing 1/120 of the spherical envelope. ~ In case the truncation Is along straight lines (some plane truncation)* we get within the kaleidoscope a polyhedron with 120 plane facets.

Point P within the space of ABC will be multiplied 120 times through reflections and 1f radiating from point P there are, drawn three simple diverging lines have angles of 90, 150 and 120 therebetween (e.g. lines pointing to 3, 6 and 11 o'clock on a clock face) there will be created a pattern of quadrangles, hexagons and decagons, as shown, viewable within said kaleidoscope when looking from the direction of the arrow.

In case point P 1s made to move within the space of ABC and connected with straight lines perpendicularly to the plane mirrors of the caleldoscope one can view all the transformations of convex polyhedra (spherical) belonging to the lcosahedral-dodecahedral symmetry group, each spherical polyhedron generated as a function of the location of point P within the space of the kaleidoscope.

In order to study the subject of transformations of polyhedra one dynamic model will be more efficient and direct than an Infinite number of static Illustrations, and this Is but one application of the kaleidoscopes of the present Invention.

It will be evident to those skilled 1n the art that the nvention 1s not limited to the details of the foregoing Illustrative embodiments and that the present Invention ma be embodied In other specific forms without departing from the spirit or essential attributes thereof. It 1s therefore desired that the present embodiments be considered In all respects as Illustrative and not restrictive, reference being made to the appended claims, rather than to the foregoing description, 1n which 1t 1s Intended to claim , all modifications coming within the scope and spl\p1t

Claims (9)

WHAT IS CLAIMED IS:

1. A spherical kaleidoscope 1n the general form of a triangular -pyramid comprising three plane reflecting surfaces Interfaclngly^ arranged and attached together to form three corner edges, wherein each corner edge 1s common to a pair of adjacent planes, and lines extending along each of said corner edges meet at a common convergence point which 1s the center of symmetry of said kaleidoscope and further characterized in that the dihedral angles between each set of two adjacent plane surfaces are selected from the groupings consisting of 90°, 60° and 60°j 90°, 60° and 45° ; 90°, 60° and 36°; and 90°, 90° and g wherein a 1s any angle greater than 0° and less than 90® satisfying the equation o * J and m 1s a whole number.

2. A spherical kaleidoscope according to claim 1 wherein said dihedral angles are selected from the groupings consisting of 90°, 60° and 60° ; 90°, 60° and 45°; and 90°, 60° and 36°.

3. A spherical akal e1 doscope according to claim 1 wherein the region close to said convergence point Is truncated to form an open substantially triangular section at one end thereof.

4. A spherical kaleidoscope according to claim 3 wherein the region close to said convergence point 1s truncated along a plane triangular section.

5. A spherical kaleidoscope according to claim 3 wherein said open triangular section 1s composed of three concentric arcs having equal radii, and a common center which coincides w th said convergence point.

6. A spherical kaleidoscope according to claim 1 further comprising a viewing aperture positioned at the diverging end of said reflecting surface arrangement.

7. A spatial kaleidoscope 1n the general form of a convex closed tetrahedron comprising four inwardly facing plane reflecting surfaces wherein each of the vertices of said tetrahedron constitutes the convergence point of a spherical kaleidoscope as defined in claim 1 wherein the dihedral angles between each of the three sets of two adjacent plane surfaces defining each of said vertices are selected from the groupings consisting of 90°, 90° and 45°; 90°, 60° and 60° ; and 90°, 60° a d 45°.

8. A spatial kaleidoscope according to claim 7 wherein the four triangular surfaces of said tetrahedron have relative edge lengths selected from the following groupings: a) 2, and Vzt- 2, ^ and 2, \TS and ^3 and 2, fz and ^T; b) Z fz and ^3; 2, f? and i 1 , Ή> and and 1 , fi and ; c) 1 , |3 and f2; 1. fi and fi j 1 , 1 and {¥; and 1 * 1 and ^2; and multiples thereof.

9. , A spherical kaleidoscope substantially as herein described and with reference to the accompanying drawings. A spatial kaleidoscope substantially as herein described. For the Applicant Wolff, Bregman and Goller