CA2483361C - Icosahexahedron - Google Patents

Abstract

A Building Framework with 26 sides (Icosahexahedric) and 16 Vertices, known whimsically as a Silmaril. It is derived from the fusion of 2 Icosahedral Frameworks into it single Framework with 3 equidistant and parallel internal planes, one plane a perfectly symmetrical 12 sided elliptical equatorial, the other 2 planes being geometrically related 6 sided inversions, resulting in useful practical applications of the Building Framework. Having 20 identical equilateral triangles, 2 sides which are isosceles triangles, another 2 sides different but geometrically related isosceles triangles, and 2 quadrilaterals again geometrically related to the previous triangles. It is geodesic in many planes resulting in great structural strength, integrity and aesthetic appearance. External alignment points allow joining multiple similar frameworks into expanded building frameworks in many dimensions. Some dimensions are of the Fibonacci Ratio. It aligns mystically in 2 dimensions perfectly with the Star Constellation Orion depicting an ancient Universal Sign of Peace.

Description

ICOSAHEXAHEDRON
A space structure usable in maw) structures; building construction, housing, warehousing; and micro = or molecular space structures and applications.
BACKGROUND OF THE INVENTION
The project began as an exploration of various existing known space structures for possible use in building construction applications, along the lines of the work that Bucktninster Fn.tk:r achieved in his explorations of the 3-frequency Geodesic Dome which eventually unexpectedly led to the further discovery of applications in micro and molecular structures, specifically the discovery of the Carbon.60 molecule.
The project examined the feasibility of using the source structure of Fuller's work, the lcosahcciron, With various permutations applied to endeavour to discover new and novel uses of the structure, either through different mathematical transformations, or through fusion into newly synthemived structures.
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Research was done into the field of Geodesic Domes including Fuller's work and publications such as Domebook IT and Refried Domes. In Refried Domes is revealed many, many technical problems in using multi-frequency (high curvature) dome structures, summarized as:
1) high variety of dimensional sizes and angles requiring much increased labour.

2) as in item I whereby there is much material waste.

3) sound reflection problems due to the internal curved shape.

4) moisture problems in the roof due to no adequate ventilation strategy.

5) outward facing windows suffering from rain-shipping leakage problems.

6) difficulty in interface standard vertical internal walls to multi-angle outer walls.

7) difficulty in installing insulation into many diverse gaps.

8) all the above causing an inherent specialization in the field regarding Geodesic Structures.
In the world there have been many informal attempts to achieve an intangible unclear goal of using micro geodesic structures in a macro implementation that historically have often failed due to not completely and systematically meeting all of the above objectives.
There are some exceptions where various Geodesic Domes have been successfully implemented at World Fairs and various museums around the World, but albeit almost always at very greatly specialized expense and effort, hence failing several of the above objectives.
The current construction code is based on an orthogonal methodology that does not readily allow a generali7ation of geodesic structures, keeping them highly specialized because of the inherent problem of dealing with many diverse non-orthogonal angles.
Traditionally, orthogonal approaches to building structures have been dictated by the orthogonal nature of gravity. In taking a different approach to offsetthig loads under the force of gravity one has to employ more complex geometric and mathematical formulas to arrive at orthogonal equivalents that are only available through very specialized, and thus uncertain, means.
This also requires more specialized knowledge and resources that may or may not be available.
However the key uncertainty had lain in the fact that standard building materials and construction techniques are almost entirely oriented to the building methodologies currently in place. There was uncertainty in whether the Geometric Vectoring techniques needing to be employed would successfully arrive at values that match in efficient enough fractions, the standard dimensioning currently in use in the field of Building Construction.

A second uncertainty was in whether an efficient means of joining materials in non-orthogonal ways would require again, specialized joining mechanisms, defeating the purpose of the objectives, or whether a way of manipulating Geodesic Structures, perhaps through fusion, would lead to an efficient new way of joining elements with the required strength.
The project set out and successfully solved these technical problems, as well as making major new unexpected discoveries. Work resulted in employing the native Icosahedron, un-phased, leaving the large planar surfaces intact. Next analysis led to implementations whereby standard building dimensions, specifically the 4x8 foot standard sheathing/drywall panel, and the standard 16", or 24" dimensions were mapped into effective implementations of the Icosahedron's native large triangular panels, to solve problem 1, 2, 3, 6, and 7. Problem 4 was solved by employing a thick enough extrusion of the wall and roof system to allow thick enough insulation and air gap to meet building code for both. Problem 5 was solved by finding a window configuration that would see all windows slanted inward and elegantly configured doorways to be vertical.
Problem 8 was solved in solving problem 1 to 7 resulting in a successful de-specialization, or generalization, of a Geodesic Building Structure.
Further work led to discoveries a) of how to eliminate the roof-overhang leading to many advantages, b) an entirely new geometric shape known as the kosahexahedron with one variation, c) further repeatability in the design allowing high efficiency, and d) a way of incorporating the structural shape into macro, micro, and molecular applications which is the basis for this invention.
The project set out and successfully solved several diverse technical problems in the Geodesic Structures leading to a successful de-specialization, or generalization, as well as making major new unexpected discoveries. Work resulted in employing the native Icosahedron, un-phased, leaving the large planar surfaces intact. Next analysis led to implementations whereby standard building dimensions, specifically the 4x8 foot standard sheathing/drywall panel, and the standard 16", or 24" dimensions were mapped into effective implementations of the kosahedron's native large triangular panels, to solve problem 1, 2, 3, 6, and 7. Problem 4 was solved by employing a thick enough extrusion of the wall and roof system to allow thick enough insulation and air gap to meet building code for both. Problem 5 was solved by finding a window configuration that would see all windows slanted inward and elegantly configured doorways to be vertical.

õ _____________________________________________________________________ Problem 8 was solved in solving problem 1 to 7 resulting in a successful de-specialization, or generalization, of a Geodesic Building Structure.
Further work led to discoveries a) of how to eliminate the roof-overhang leading to many advantages, b) an entirely new geometric shape known as the Icosahexahedron with one variation, c) further repeatability in the design allowing high efficiency, and d) a way of incorporating the structural shape into macro, micro, and molecular applications which is the basis for this invention.
What began as an investigation of a micro structure, the kosahedron, for use as a macro structure leading to the discovery of a new synthesized structure which is the result of fusion between two Icosahedrons, came full-circle in being applicable to micro and molecular applications, in the pattern of Buckrninster Fuller.
SUMMARY OF THE INVENTION
The invention is a Geometric shape which is the result of fusing or merging two Icosahedrons into one, along vertices and panels native to the kosahedron, retaining all original vertices but introducing new interface planes that are completely native only to the new synthesized structure, an Icosahexahedron.
The new structure has 3 unique internal planes that are of specific use in the employment of the shape as a macro structure, which also explicitly contribute to the definition of the structure.
The original Icosahedron has 20 sides and 12 vertices, whereupon it is split into two 10 sided half-Icosahedrons, and fused together resulting in a shape that has the original 20 sides, but 6 new interface panels which are: 2 of which are the same, another 2 different, and 2 quadrilaterals different again, resulting in a structure that has 16 vertices.
There are several dimensional relationships in the structure that are based on the mathematical ratio PHI, otherwise known as the Fibonacci Sequence, and the Golden Ratio. Of note is that the ratio of the number of vertices of the Icosahexahedron to ifs number of panels, 16/26, is a mid increment ratio of PHI (between 13/21 and 21/34 in the sequence).

There is one variation of the shape where the 3 internal planes are not used resulting in a structure which is similar but less symmetrical, employing again the original 20 Icosahedron panels but interfaced by 6 identical triangles, creating a pure Icosahexahedron. In transforming the first form of the Icosahedron into the second, the 2 pairs of different triangles and the 2 quadrilaterals, become identical.
It is the first form of the Icosahexahedron having the 3 internal parallel planes, and external interface planes in 3 dimensions that is useful in macro building applications.
The Icosahexahedron also has a staggering exact alignment with the left side of the Star Constellation Orion, and a Transformational Correspondence to the right side.
In the drawings forming a part of this specification are:
Fig.1-6 Solid views of the structure front and off-angle Fig.7-12 Solid views of the structure side, top, and off-angle Fig.13-18 Wireframe views of the structure, front, side, top, off-angle.
Fig.19-30 Wireframe views of the structure, varying off-angle.
Fig.31-36 Wireframe views upside down, front, side, top, off-angle.
Fig.37-48 Wireframe views upside down, varying off-angle.
Fig.49-60 Development of base component equilateral triangles and angles.
Fig.61-72 Views of single base component Icosahedron,front,side, top, off-angle.
Fig.73-84 Views of interfaced base triangles to Icosahedron, varying.
Fig.85-87 Views of two base Icosahedrons aligned in proximity for fusing.
Fig.88-90 Views of base Icosahedron relevant alignment planes for fusing Fig.91-93 Views of base Icosahedron key panels to be removed for fusing.
Fig.94-96 Development of major fusion interface planes and points.
Fig.97-99 Major fusion interface planes aligned and points connected.
Fig.100 Exploded View off-angle of structure with non-regular panels.
Fig.101-103 Views of resultant non-regular panels off-angle, front, top Fig.104-115 Alternate views of structure derivation, front,top, off-angle.
Fig.116-126 Dimensional analysis of non-regular interface panels.
Fig.127-134 Development of structure major planes upper and lower.
Fig.135-149 Mathematical formulas for non-regular panels and upper plane.
Fig.150-158 Mathematical formulas for lower plane.of structure.
Fig.159-172 Mathematical formulas for vertical dimensions of structure.
- -Fig.174-197 Development of alternate 26 sided structure subcomponents Fig.198-209 Wireframe views of alternate structure with interface panels.
Fig.210-221 Solid views of alternate structure Fig.221-236 Component substructure placement within fused structure.
Fig.237-248 Molecular external alignments and interface planes.
Fig.249-273 Views of substructures resulting from fused structure.
Fig.274 Wireframe off-angle view of the 3 major internal usable planes.
Fig.275 Solid exploded off-angle view of the 3 major internal planes.
Fig.276-277 Solid exploded front & off-angle view of the major equatorial.
Fig.278-284 Mathematical analysis of the 3 major internal usable planes.
Fig. 285-289 Solid views of upside-down upper-plane dissected structure.
Fig.290-294 Solid views of rightside-up lower-plane dissected structure.
Fig.295-301 Solid views of rightside-up equatorial dissected structure.
Fig.302-308 Solid views of updside-down equatorial dissected structure.
Fig.309-313 Solid views of rightside-up lower-planed window configuration.
Fig.314-315 Wireframe views of lower-planed window/roof/wall config.
Fig.316-319 Wireframe views of lower-planed 3 major internal planes.
Fig.320-325 Views of two structures interfaced to each other at side plane.
Fig.326-331 Views of two structures stacked at lower-to-upper planes.
Fig.332-337 Views of double-phased strut support in main panels.
Fig.338-343 Views of orthogonal-phased strut support in main panels.
Fig.344-349 Views of double-ortho-phased strut support mixed derivative.
Fig.350-384 Views of substructures resulting from double-ortho mix.
Fig.385-392 Mathematical analysis of double and ortho phase strut-works.
Fig.393-397 Mathematical analysis of standard 16" or 24" stud-works Fig.398-399 Structural analysis of panel corner interface bolt-patterns.
Fig.400-404 Solid views of ventilated extruded wall and roof thickness.
Fig.405-410 Derivation of panel interface virtual beam structures. & bolts.
Fig.411-416 2.5 storey 2500 square foot free-standing house or warehouse.
Fig.417 Bottom view of structure showing bottom plane configuration.
Fig.418 Accurate sky view of Star Constellation Orion.
Fig.419 View of bottom view of structure aligning perfectly with Orion.
Fig.420 View of resulting interface matching Universal Symbol of Peace DETAILED DESCRIPTION OF THE INVENTION
The 26 sided semi-regular polyhedron can be a planar solid, as in Fig. 1, a wireframe, as in Fig. 13 or hollow solid with a shell thickness of varying depth defined by the definition vertices of the structure where all struts connect, extruded either inward toward inside the structure or outward away from the surface of the planes defined by the defining points of the structure, defined further in following discussion.
The structure is considered semi-regular for the reason that it is made up of 20 exact equilateral triangles, which are regular, but also a non-regular family of 3 pairs (another six) of isosceles triangles, which are all linked in their characteristics mathematically, to total 26 panels making up the structure.
Further, the structure has a unique resulting paradoxical semi-symmetry, or semi-asymmetry. From various views the structure is very symmetrical, as in Fig. 7 & 8 and 9 & 10. In other views it becomes very graphically asymmetrical, as in Fig. 1 and Fig. 5. Various views in between tend to show bizarre off-angle variations of the two, as in Fig 2,3,4,6,11 & 12. In other cases a strange kind of "hybrid-symmetry" can be viewed like in Fig. 22 & 43.
A tangible asymmetry in one plane can serve as a directional means of orientation of the structure.
In selecting one direction of the asymmetry as "rightside-up" it allows a tangible method of orienting the structure for identification and reference purposes. From this orientation the standard views of front, back, sides, top and underneath can be applied to the rightside-up orientation. A preferred orientation was selected based on a later developed view of a certain orientation identified as being most relevant to practical uses of the structure, resulting in the rightside-up orientation of Fig. 1.
This orientation was arrived at through the subsequent desire to remove the bottom cap of the structure shown in Fig. 257-260 in favour of using the top cap shown in Fig.
249-252 for the purpose of eliminating the only non-triangular panels in the structure which is desired for construction purposes, leaving only triangular panels making up the primarily useful structure as defined further below. All front, side, top and bottom views of the structure are based on this orientation.

Views that are employed are: front, back, top, underneath, left and right.
They are identified in the drawings with a cube present nearby the structure or other elements identified respectfully: F
(front), B (back), L (left), R (right), T (top) and U (underneath).
In vvireframe drawings solid lines represent struts that are on the viewer side of the structure, whereas dotted lines represent struts that are on the opposite side of the structure, as viewed transparently.
All 3D views of the structure are with Zero Perspective, i.e. non-isometric, so that the effect of symmetric can be seen in cases where perfect symmetry in any wireframe view is indicated by the absence of any dotted lines, where it can be inferred that other associated drawings that the dotted line is hidden immediately behind the solid line indicating perfect symmetry, as shown in Fig. 13, 15, & 31. Using any degree of perspective would introduce difficulty in properly understanding the degrees of symmetry throughout the structure.
Derivation of the structure begins in Fig. 49-60 starting with a simple Equilateral Triangle in Fig.
49. This is a very important basic building block in deriving the structure, and is given a unit dimension called "T" for each equal side of the triangle 0 as in Fig. 121, which can be any non-zero size, and is described further below. By mathematical definition the Angle U in Fig. 121 inside each corner of the triangle will always be 60 degrees The same triangle is shown in Fig. 2 from Right View slightly elevated. A
second identical triangle joined to the first along one edge is shown in Fig. 51 where the joining edge becomes an axis of rotation for the second triangle as in Fig. 52 identified as Axis A
forming Angle B between the two triangles in Fig. 53.
Note that the use of the the equilateral triangle as a building block was original derived from analysis of the Regular Polyhedron structure known as the kosahedron, a 20 sided structure. The Icosahexadron comes from the result of fusing two Icosahedrons together according to a certain protocol described further below. From that analysis it is established that the correct angle of rotation between the triangles in Fig. 53 to allow the pair to be used as a subcomponent making up an Icosahedron, is (rounded to one decimal place here and expanded further below) 138.2 degrees and can be viewed in Fig. 54-60.
This dual triangle component can be employed empirically facing in and joining only by its 3 vertices to adjacent identical components to arrive at the kosahedron structure. Following these rules will result in no other possible structure and will result in accuracy of the overall structure depending on the accuracy of size T and Angle B.
The characteristics of paradoxical-symmetry can be viewed in the Icosahedron as in Fig. 61-72, for which the invention amplifies into many new variations as though the two parent fused Icosahedrons result in a completely new unique variant characteristic offspring that is similar yet different.
The employment of the dual triangle component of Fig. 51 into the Icosahedron is shown in Fig.
73-84 whereby each component interfaces to each adjacent one by connected through the 3 vertices of each triangle at an angle of 138.2 degrees.
Note that there are other alternate mathematical methods of deriving the Icosahedron but this one is selected for it's simplicity and direct application in using 3D CAD tools to construct the base Icosahedron as a subcomponent to the invented Icosahexahedron.
Further, the invention can be created and modelled through various mathematical methods but in this case is presented in the same method as it was discovered, through manipulation of 3D
representations of the level of vertice structure of two adjacent kosahedrons.
In other words the derivation assumes two perfect Icosahedron base units with zero-width panel thickness that will be connected together perfectly at various vertices shown below.
To begin the process of fusing two kosahedrons together they need to be roughly oriented as in Fig. 85-87. Various experimental development was required to arrive at the final successful joining method that identified the various panels in Fig. 88 being removed as the result of slicing the two Icosahedron structures in Fig 85-87 to create the two new substructures STRI
and STR2 along the respective planes DI and D2 shown in Fig. 88-89, resulting in key vertices that can allow logical interface points at VTX1-1 to 5 and VTX2-1 to 5 along planes D1 and D2 Next, the panel families E, F, and G need to be removed from, the structures STRI and STR2 in Fig. 91-93 which results in the two new derived structures STR1b and STR2b in Fig. 94-96.
The two new structures then need to be aligned so that the plane PLN1 defined by vertices VI-1 to 4 matches the plane PLN2 defined by vertices V2-1 to 4. And also so that the plane PLN3 defined by vertices VI-5 to 7 and plane PLN4 defined by vertices V2-5-7 match each other.

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The two structures STR1b and STR2b are then joined together at the two interfaces il and i2 by connecting the vertices V1-2 and V2-1 together, as well as the vertices V1-3 and V2-4, all the while maintaining the planes PLN1 and PLN2 equal, as well as the planes PLN3 and PLN4 equal, which results in the new single synthesized partially complete structure in Fig. 97-99.
In Fig. 100 is an exploded view of all new panels created in fusing the two Icosahedrals resulting in panels J1-2, K1-2, and L1-2 making up the new panel types J, K, L, adding to existing panel type 0. In Figure 101 the struts Si to 4 are created by creating a) strut Si between the two top vertices V1-9 and V2-9 at the top of the fused structure creating the new panels J1 and J2, b) strut S3 between the bottom two vertices V1-8 and V2-8 creating the new panels Li and L2, c) strut S2 between the two front vertices V1-5 and V2-5 creating the new panel Kl, and d) strut S4 between the two back vertices V1-6 and V2-6 creating the new panel K2, as in the panel families J, L, K
shown at the bottom of Figure 100.
The connected vertices result in the completed invention, an Icosahexahedron, in Fig. 101-103, showing the regular panels in transparent (white) with the non-regular panels in grey.
The non-regular panels are interestingly related to each other and displayed viewed laterally from the front for clarity in Fig. 116.
In Fig. 117-119 the non-regular panels are showed in various two-dimensional special relationships which can be summarized as following.
a) In Fig. 101, strut S1=S3, amazingly, b) strut S2=S4, and c) 52=S1 *2 In Fig. 120 forward the variable R is assigned to the length of Si and the variable S is assigned to S2. The variable T is as mentioned the overall system structure base unit, which for practical purposes can be set to the value 1 to simplify derivations.
Of extreme significance are the following relationships also summarized in Fig. 125: a) R=S/2, b) S=2R, c) the short edge of panel J in Fig. 120, R, is identical to the long edge of Panel L in Fig.
123, as also shown graphically in Fig. 119, d) the long edge of Panel in Fig.
122, S, is identical to the long edge of Panel L in Fig. 123, as also shown in Fig. 119, e) as the overall structure is oriented, the vertical oriented struts all are size T, whereas the horizontal oriented struts Si to 4 all follow the above relationships, f) 3 Panel J's in Fig. 123 fit perfecdy dimensionally inside 1 Panel L
as in Fig. 124.

All internal angles are listed in Fig. 126 and can be calculated using standard mathematical geometry since all triangle side lengths are known.
Of structural significance is the orientation of the plane PLN5 in Fig. 97, hereby known as the Plane Top PT as shown in Fig. 274, and the plane PLN3/PLN4 (PLN3=PLN4) of Fig.

hereby known as Plane Bottom PB as shown in Fig. 274. In an analysis of these two planes there is a dual symmetry exists in that base components of each invert to create the other, as explained further in Pig. 127-134.
In an analysis of Plane Top PT first, in Fig. 127 & 131 looking down from above are two identical sided polygons known as pentagons, PN1 and PN2, which are the same as the two top and bottom planes of a regular kosahedron, as can be seen in Fig. 85 & 87.
Plane Top PT is derived as in Fig. 94 by connecting vertices V1-2 to V2-1 and vertices V1-3 to V2-4. This results in a geometric shape of Plane Top as in Fig. 129. The location of V1-9 (and V1-8) in STR1 is the same as vertex VTX1 in Fig. 128 which is also the intersection of lines DL1 and 2, and the location of V2-9 (and V2-9) in STR2 is the same as VTX2 in Fig.
132 which is also the intersection of the two lines DL3 and 4 resulting in the locations for the vertices VTX1 and 2 in Fig. 129 & 130 (& 133).
Plane Bottom PB is made up of the same two Pentagon components in an inverse way, as in Fig.
133 where the vertex in Fig. 94 V1-5 is connected to V2-5 by the distance S, similarly for the vertex V1-6 to V2-6, resulting in the pattern for Plane Bottom in Fig. 133 Looking down on the overall structure in Fig. 101-103 results in the same pattern as in Fig. 134, showing the overlapping views of Plane Top, and Plan Bottom, with the interaction of R in Fig.
129, and S in Fig. 134, which is extremely novel and intriguing.
A further analysis of relevant parameters of the invented fused structure is in Fig. 135-149, where in Fig. 135 is shown Plane Top where the previously derived values R and T are visible, and the new values P in Fig. 136 which is the perpendicular line from T looking in the downward plane across to the vertex VTX2 in Fig. 132. The value Z in Fig. 137 is the line segment from V1-2/V2-1 in Fig. 94 to the extension of the line segment T which also defmes 1/2 the length of the rectangle completely enclosing the shape of Plane Top.

Similarly the value ZZ in Fig. 138 is the other axis dimension which is half the difference between T and the width of the rectangle completely enclosing the shape of Plane Top.
The length of Plane Top PT is analyzed in Fig. 139 where it is equal to P + R
+ P = 2Z, which is also used to do an independent verification of R.
Fig. 143 reviews the relationships of the panels J, K, and L in the calculations in Fig. 140-149, for:
a) the Angle V in Fig. 140 & 144, b) the Angle X in Fig. 141 & 145, c) that Panel L in Fig. 142 & 143 is made up of exactly 3 Panel J's in Fig. 140, d) the Angle Q in Fig.
140 & 147, e) the Angle W in Fig. 141 & 148, f) the bottom corner angles of Panel L in Fig. 142 is equal to Q + V
in Fig. 149.
In Fig. 150 is a representation looking down on the overall structure with PT
and PB overlapping.
The dimension S at the bottom if the figure is mirrored in symmetry at the top of the figure with the apex of the triangle with base S touching the midpoint of the upper dimension S.
The dimension SW, Structure Width, is the width of the rectangle which fully encloses Plane Top PT (or similarly PB) with a large edge of both sides of the long side of the rectangle adjacent to the S dimension of PB for a portion of that length and the short edge of the rectangle adjacent to a length of T of the edge of PT.
The rectangle which fully encloses Plane Bottom PB has the same width as for PT except for the extension YY as seen at the extreme left center of the figure which is mirrored symmetrically on the right side.and is calculated in Fig. 151. The extension of these two dimensions is used to calculate the length of the enclosing rectangle by summing various previously calculated dimensions P and R as in Fig. 152.
The triangle in Fig. 157 is a bisected half of the isosceles triangle in Fig.
150 with it's base at the bottom dimension S and apex at the top of the figure. Fig. 153 is an alternate derivation of R by using other derived values as a verification of the value, using ZZ and T.
Fig. 155 is a review of the relationship between S and R.
The length PP in Fig. 150 and 156 is not relevant to the native Icosahedron or this structure, except in the context of looking down on either structure from the top, and measuring the distance laterally. In other words the true distance of the strut is T, but as viewed 2-dimensionally as in the figure is PP. This is a useful dimension for purposes of using the structure in practical . , space applications, for example if a support post were to be employed vertically at Val or VTX2, the distance to the nearest wall corner at the connection between ZZ and T.
In Fig. 158 the dimension UU is calculated in order to arrive at an alternate value of the total length of the rectangle enclosing PT by summing with the line segment S.
Finally, a major discovery in the invention is that the relationship between the enclosing structure width, SW in Fig. 154, and T, the unit length of the base sides of the subcomponent triangles making up 20 panels of the structure, is a perfect inverse of PHI, otherwise known as the "Fibonacci Number", or "Golden Ratio". Which means that viewed in various other ways is the same relation as PHI. I.e., an alternate equivalent relation is that T is equal to SW * PHI.
In Fig. 159-173 are analyses of horizontal front views of the overall structure summarized in Fig.
160 and developed as follows. Fig. 159 shows how intriguingly the intersection of Strut S3 in Fig.
101 with Plane Bottom PB as in Fig. 274, creates a perfect square as also shown in Fig. 172 element WW with side dimensions of R. A similar square is created in perfect symmetry in the front view in the vertical plane from this bottom square where the Strut Si in Fig. 101 intersects with Plane Top PT in Fig. 274. In aesthetic terms this is useful in applications as a macro space structure for habitation or warehousing in portraying balance ergonomically.
This is proven by the derivation of the value HH in Fig. 159 8c 166 as being equal to R
If the relation R to T, or other dimensions native to the invention can be found to match with atomic relations then new synthesized molecular substances will have been invented.
Further calculations for practical applications follow from Fig. 163 and 162 where Fig. 163 is an extraction of the right side of the overall structure in Fig. 159 with one of the base Equilateral triangles from the set of 20 making up the invention displayed in a plane perpendicular to the viewer in Fig. 162 (hence the Not Front indicator since the plane is not perpendicalar to front).
The practical dimension TI' is the vertical distance of the base component triangle calculated in Fig. 161 which would be relevant in calculations for ceiling height through the triangle if used as a passageway for human habitation or other functional uses. TT also gives the distance of a similar triangle viewed edge on in Fig. 163 making up what would be a roof panel in a housing application, another required dimension. Here it is also used to support the development of the various angles in Fig. 163 of BB, angle from ceiling to roof inclination, angle MM, angle from wall to vertical, angle QQ which is the 90 degree angle offset of the wall to the ground, and angle ¨ ,õ

B which is the angle of prime importance in the entire structure of which derivation and primary understanding is required in the process to allow the synthesis of the Icosahedron into the invention, as in Fig. 52-60.
The process to derive Angle B begins with derivation of Angle QQ in Fig. 163 and 165, to Angle 1VIIVI in Fig. 163 tic 167, proceeding to Angle BB in Fig. 163 8c, 164, finally to arrive at Angle B in Fig. 163 and Fig. 169.
Angle B can be rounded to 138.2 degrees as in Fig. 52-60.
Dimension GG in Fig. 163 is slightly different to 'IT in Fig. 162 because although the triangular panels are identical, in Fig. 163 there is a slight outward slope, which is indicated in the edge on triangle hidden in Figure 163 of the lower dimension panel denoted by U. So 11 is the panel height, GG is the vertical height, which are slightly different as shown in Fig. 171.
The total vertical height of the structure TH shown in Fig. 172 as the addition of the vertical dimensions of WW * 2 + AA is derived in Fig. 168 but summing previously known dimensions as related to T.
A further intriguing discovery in the invention is the relationship in Fig.
170 showing that the ratio between the vertical distance between the top (PT) and bottom plane (PB) as in Fig. 274, GG, (or the height of AA), to T, is PHI.
Further, it is found in Fig. 170 and summarized in Fig. 173 that the ratio of the height of the previously mentioned perfect square WW, to the height of the rectangle AA, is PHI.
The synthesized 26 sided polyhedron structure thus derived, is known as an ICOSAHEXABEDRON
from Icosa - 20 + Hexa 6 = 26. An alternate nomenclature for 6 is Hexa, resulting in ICOSASEXARBDRON
These are acceptable references in general terms, but to be more precise, the structure is symmetrical in 20 panels, then another two more J1 and J2, equal to themselves but not to the , other 20 panels, as in Fig. 100, then two more are different again K1 and K2 in Fig. 100, and finally another two more which are the Quadrilaterals Li and L2 in Fig. 100 as well.
To complete the definition may also be included the 3 internal planes PT, EQ, and PB as in Fig.
274 as they define the orientation of how to connect the two fused kosahedrons in Fig. 94-96.
So a more precise reference for the structure is:
20 + 2 + 2 + 2 which under nomenclature would be known as:
ICOSADUODUODUOHEDRON
or under a variation on the reference to two being "Do" rather than "Duo"
ICOSADODODOHEDRON
or two being "Di"
ICOSADIDIDIHEDRON
Finally, there is a whimsical identification which refers to the significance of a 26 sided structure identified in literature as being defined as: "A fictitious structure". Also which of note, having 26 sides coincides in a novel way with the number of letters in the alphabet.
A further connection is in an affinity with J.R.R. Tolkien's description of an ancient mythical structure of significance with power of influence due to many factors some of which were in the proportion of it's shape and the manner of it's grand making. For which after being the very root cause of endless war itself found its final resting place after proving too much a burden for the world, in the night sky as a lost star. To dramatically return only at a time when the world was deemed ready. In light of this it is considered within the bounds of apt novelty to further apply to the invention the name:
SILMARIL.

Of lesser note there is an alternate 26 sided polyhedron structure which will be briefly described, which is related to the invention in that it also has 26 sides, is similarly made from the fusion of two Icosahedrons, has 6 extra sides, and is of novelty interest as a parallel invention but is not identified as having the same practical applications in macro building structures to the degree of great utility of the primary invention previously described above.
It is shown in Fig. 108-209 in wireframe view and Fig. 210-221 in solid view.
Following is a description of the synthesis of this second structure whereupon the description will revert back to the primary invention.
This second 26 sided polygon structure has the characteristics of similarly being two fused Icosahedrons, but without the alignment of the 3 internal Planes PT, EQ, and PB in Fig. 274, which are completely non existent in this structure The structure has symmetry in two planes, and the six extra interface panels which are different again from the base 20 panels, are in this case, identical to each other resulting in the detailed nomenclature ICOSAHEXAHEDRON (and/or ICOSASEWEEDRON) is completely accurate since the 6 extra panels are all identical.
To construct this structure is similar to the ICOSADUODUODUOHEDRON in that two base ICOSAHEDRONS have various panels removed and then joined at various logically convenient vertices.
The difference between the two synthesized structures is in that this one does not align with the internal planes intact, but rather depends completely on the joining at 3 co-planar vertices instead.
Meaning also, that after joining the two together, no extra connection struts are required.
In Fig. 174 is a slight off-angle front view of an Icosahedron. In Fig. 175 is a Front view.
Note that an alternate method of making both this structure and the Icosaduoduoduohedron are presented in Fig. 176, in that rather than follow the process outline previously shown in Fig. 91-93, & 104-115, an alternate method is to simply take ONE Icosahedron, and split it along a natural equatorial that follows it's strut connections.
This results in the two half-kosahedrons in Fig. 176 and also creates the same resultant structures in Fig. 94-96, in an alternate method.
In this method, now one of the half-Icosahedrons has to be rotated 180 degrees as in Fig. 177 &
178 to arrive at the orientation in Fig. 179, which is identical to the result of the process in Fig.
94.
From this point it would be possible to align the internal planes PT, EQ, and PB as described and arrive at the synthesis of the kosaduoduoduohedron.
But to create this alternate structure the two structures STR1 and STR2 are joined differently.
First are some views of the nature of each half-kosahedron shown in Fig. 180-185, including an axis of rotation which is perpendicular to the central panel of the structure, identified as AX4 in Fig. 186.
To align the two half-Icosahedrons to make this alternate structure requires aligning them so that the Axis AX4 for both halfs, is equal, as in Fig. 186-189, next one half must be rotated about Axis AX4 180 degrees as in Fig. 190 & 191, whereupon the two halves can be joined at 3 native vertices as in Fig. 192. To finish, the Struts SU1, SU2 and SU3 in Fig. 193 &
194 must be connected, resulting in the complete structure in Fig. 194, a pure Icosahexadron.
Further wireframe views are in Fig. 195-197. In Fig. 198-209 are shown the new resultant interface panels in grey, which total 6 and which are all identical isosceles triangles, where the other 20 native Icosahedron Equilateral triangular panels are in transparent (white), with solid views in Fig. 210-221.
It is a novel structure that has geodesic strength, is symmetrical in 2 planes (facing front, left to right, and top to bottom, as in Fig. 213, 220).
The structure also exhibits the similar trait in the Icosaduoduoduohedron of "hybrid-symmetry", as in Fig. 214, and semi-asymmetry in Fig. 210, 211, 212, 215, 216, 217, 218, 219, and 221.

¨ _ Continuing on with the primary invention, the Icosaduoduoduohedron, which from this point forth will be referred to again with it's general reference the Icosahexahedron, in Fig. 222-236 is an analysis of substructures which result from the synthesis of the structure.
There are 6 substructures shown in Fig. 249-273 further described below, whereas in Fig. 222-236 the actual placements of these 6 subcomponents are shown.
The overlapping interfaces of the interlocking subcomponents contribute to great geodesic strength at the same time due to the diversity of slightly different shapes contribute to an aesthetic effect in the structure.
In Fig. 237-248 are addressed external interface interactions between several structures due to several innate external planes resulting in the resultant fused structure.
In Fig. 237 two Icosahexahedrons can be stacked and joined at Strut S3 of Fig.
101 of Fig. 237 STR-U, and Strut S1 of Fig. 101 of Fig. 237 STR-L.
This results in the substructure DCUL in Fig. 237, a novel structure which defines physically the nature of the interface between two Icosahexahedrons in this plane further shown as Plane MPT in Fig. 247 and Fig. 246 with 3 units stacked vertically, and also in Fig. 238, 239, 240, & 241.
Next, there is a natural vertical surface which is Panel K1 and K2 in Fig. 100 which allows the sharing of the Plane MPF in Fig. 247 among several interfaced Icosahexahedrons in that plane, also shown in Fig. 240, 241, 242 & 243.
Finally, in the third orthogonal dimension of 3D space, the common vertices of joined kosahedrons create the Plane MPR in Fig. 247, where each corner of the Plane MPR is the interface point for two adjacent Icosahexahedrons in that plane, also shown in Fig. 238, 239,242 8c 243.
A variation on interfaces that flips various units around in various planes is explored in Fig. 244, 245, and 248.
Such orientations and interfaces will have use in micro structures allowing a large base unit, the kosahexahedron, but with many planar connection points, which will tend to create a novel material with properties of strength.

woesswaw...1110.1.% .11.11" vants-va.

At a molecular level the synthesis of two Icosahedrons into one fused one will create a new synthesized material that similarly will have benefit of low mass due to large molecules but many connection points in 3 dimensions.
The native substructures that result from the synthesis of the Icosahexahedron are listed in Fig.
249-273.
Fig. 249-252 are views of the HEXADUOCAP (6 + 2), named for being a 6 common plus 2 extra sided pyramid that is defined as the slice created by Plane Top PT in Fig. 274 resulting in the top component in Fig. 275..
Fig. 253-256 are views of the DECADUOSECf (10 + 2), the middle section created by slicing the Icosahexahedron at both PT and PB in Fig. 274, which has 10 identical sides, with two K
panels as in Fig. 100, which appears as an elliptical (or oval) shaped, 12 sided drum-kit also shown as the middle section in Fig. 275.
Fig. 257-260 are views of the 1ETRADUOCAP (4 + 2), the bottom section defined by slicing the Icosahexahedron at PB in Fig. 274 resulting in the 4 common sides and 2 L
Panels in Fig. 100 6 sided pyramid, also shown as the bottom section in Fig. 275.
Fig. 261-264 are views of the PENTACAP (5) sided pyramid which is native to the Icosahedron and represents the unchanged parts of the Fused Icosahedrons making up the invention.
Fig. 265-268 are views of the TETRAUNIUNICAP (4 + 1 + 1) sided pyramid which are the substructures resulting from the interface of 4 Equilateral triangles, a J
Panel, and a K Panel from Fig. 100.
Fig. 269-273 are views of the INTEHEXAPENTACAP (integrated 6 and 5 sided) pyramid, which is the result of part of a Pentacap with one L Panel in Fig. 100, the L
panel being a Quadrilateral can be thought to be be one panel as in the view Fig. 269, making a non-regular Pentacap, or, inherently can also simultaneously be considered to be dissected into 2 triangles as in Fig. 270, meaning that it is also a non-regular Hexacap (6 sided pyramid) Note that the 6 identified substructures described above, 5 are completely unique to the invention, i.e. the Icosahexahedron. This means that do they do not knowingly exist in any other currently existing polyhedron structures. They result inherently because of the planar and connection characteristics in joining two Icosahedrons together along PT and PB at the vertices described. As such they are unique substructures of the invention.
However, the Pentacap substructure is inherent to the Icosahedron polyhedron and as such is not unique to the invention and is well known as a pentagonal structure.
The kosahexadron has a natural equatorial, EQ, as in Fig. 276,277 and defined in Fig. 274. EQ
is also one of 3 planes in Fig. 274 that are unique to the invention and are the result of merging planes in the two fused Icosashedrons into one on each of the 3 levels.
Each level makes for a convenient and useful application as a floor or ceiling in various further configurations of the invention.
Hence follows a further analysis of PT, EQ, and PB in Fig. 278-283. Fig. 278 &
279 show plane Top PT and some characteristics. First, it is 6 sided, a stretched Hexagon.
Also every dimension is T, the system base unit. The rectangle enclosing PT is described previously and reviewed in Fig.
279 where the calculation of the Angles Al and A.2 are in Fig. 282.
In a practical application as a building structure, in a preferred configuration (rightside-up) PT
serves as a very convenient and useful roof-line, as described further below.
The Plane Bottom PB in Fig. 280 & 281 is also a stretched Hexagon but is different in shape and has 4 T dimensions and 2 S dimensions. The rectangle enclosing PB also completely encloses the entire structure which is useful for building construction purposes, and SL
and SW are defined in more detail previously in Fig. 152 & 154. The Angles B1 and B2 are further described in Fig.
282.
Again in a practical application, PB serves as a very convenient and useful ground floor-line as described further below.
The Equatorial EQ in Fig. 283 & 284 is a hybrid of Plane Top PT and Plane Bottom PB, since struts interfacing the upper and lower plane pass though the equatorial definition points. This results in a plane that is 12 sided, is elliptical (or oval) shaped, and has dimensions that at half of T
or half of S ( R ). The length of EQ is SL YY as in Fig. 150 and has the same width as both PT
and PB.
oeve.ueveakmaximorm* =

Again EQ serves as a very convenient and useful floor or ceiling line as described further below.
Note that intriguingly, because PT, EQ, and PR all have the same width, this means that the panels K in Fig. 100, are perfectly completely vertical planes.
This is directly useful in building construction applications because a) it allows a natural door placement allowing a vertical door and not the slight slope out or in that occurs in the native Equilateral panels and b) it allows applications where one or more Icosahexahedrons can be joined along this vertical interface side by side as described below, and c) ergonomically it allows for some standard wall orientations in a structure that otherwise may be too overwhelming in it's non-orthodoxy.
Various combinations of slicing and removing substructures from the kosahexahedron result in various novel configurations, some of which are well suited to building construction applications.
The first variation, considered the least desirable, is in by turning the kosahexahedron upside-down (bottom-up), slicing it at PT and removing the HEXADUOCAP in Fig. 249 resulting from the slice, which results in the structure in Fig. 285-289. This is undesirable simply because it uses the TETRADUOCAP in Fig. 257 for a roof, which includes the Quadrilateral panel L in Fig.
100, whereas for purposes of geodesic strength and simplicity of structure, it is preferable to use a configuration made up only of triangular panels, which is what results when using the preferred "rightside-up" configuration which puts the triangular-only BEXADUOCAP in the roof position, as in Fig. 290-294 This preferred configuration also allows for other features like upward pointing K Panels which better allows a doorway or window structure whereas the upside-down triangle in the inverted structure would make passage through impossible.
The preferred configuration slices the rightside-up structure along Plane Bottom PB and removes the TETRADUOCAP in Fig. 257 resulting in Fig. 290-294.
In this configuration, by sizing T to a practical size that would allow the structure in Fig. 290-294 to be a two storey structure, would as a benefit allow the EQ plane to serve as a logical second floor/first floor ceiling, as described further below.

However, the same structure can be sliced at the Equatorial EQ to arrive at the convenient one storey resulting structure ideal as a bungalow, cottage, garage, or shop, as in Fig. 295-301 for which again has a perfectly vertical plan on either side in this case being a dissected K Panel in Fig.
100, that if sized properly, would allow a standard doorway.
For comparison purposes the upside-down version is shown in Fig. 302-308 which again is not so ideal for building construction at least, for the reasons described above.
Developing the preferred configuration further in Fig. 309-313 shows a preferred window configuration, which is convenient and useful in building construction applications because a) it allows window structures to slope slightly outward eliminating the problem of windows shipping water, b) any of these openings could also server as doors allowed to be vertical with a small amount of vertical support blocking making a vertical doorway., c) a novel aesthetic effect results, d) a continuous roof-line is possible from the roof all the way to the ground in a triangular wall configuration, albeit requiring specialized eaves-troughing and roof ventilation in the design used.
Further views of this preferred configuration shows PT, EQ, and PB, the window configuration, floor and ceiling configuraitions in Fig. 314-319.
Note that inherent in using the Icosahexahedron in such applications lend well to Post-and-Beam consrucdon techniques that would allow using geometrical mathematical techniques applied to the strut configuration to be directly applicable, as opposed to a Frame type approach.
This allows further applications of open type structures like Pavilions or Salt Domes where only the roof needs to be covered.
In Fig. 320-325 is shown a further development where one or more units can be constructed connected together sideways with the K Panel in Fig. 100 as the logical interface where a large opening between units would be allowed, allowing two or more units to be connected into one functional dwelling or other functional application.
This sideways connection is allowable in either a two or one storey configuration where either PB
or EQ are the floorline as in the figures.
In Fig. 326-331 is shown a further development where one or more units can be constructed stacked vertically resulting in a new novel application that as in Fig. 326 would create a 4 storey _ structure (with attic space) that is its own unique shape. In this scenario all panels are repeatable across levels.
With proper sizing and joining methodologies, this configuration can be utilized to create a stacked Highrise building that uses standard, repeatable strut components that would result in a very novel oscillating floor type effect where the window effects would allow for a repeating diamond effect, as in Fig. 330 on the front and back side views of the structure, and a contrasting divergent diffractive window effect from the side views.
This structure has inherent geodesic strength due to the oscillating fold-effect between adjacent floors that allows efficient use of materials and contruction labour.
To focus back to aspects of the base invention, to develop the Post-and-Beam approach, there is a natural support-strut configuration allowed by the fact the majority of panels making up the structure, can be dissected at their mid-points and have support struts connecte there, adding strength and also allowing weaker materials to be used since the T struts have support at all their midpoints, as in panel QP1,2, &3 in Fig. 332.
The elements TP1, 2, & 3 are known as "Tri-panels", being triangular building panels. The elements QP1, 2 &3 are known as "Quadpanels" since they are made up of 4 joined Tr-panels.
Hence a method of building up Quad-panels from Tr-panels allows repeatability, strength, and efficiency in building construction, shown in Fig. 332-337.
A further development is a more relevant orthogonal arrangement as in Fig. 338-343, where the Element QP1b, 2b, &3b represent a migration toward more efficiency in that standard building construction practice tends to be naturally oriented to orthogonal structures, due to the force of gravity which is vertical.
This configuration has the beneficial side-effect of allowing a larger, rectangular entrance way through the K Panel in Fig. 100, or the QP2b panel in Fig. 338, as further shown in Fig. 339-343.
A further development upon analysis shows that a logical mixture of both panel types is beneficial since the roof and wail panels do not require openings, hence could make use of the advanges of the QP1, 2, &3 configuration, and then the lower floor window and doorway panels could make use of the QP1b, 2b, and 3b configuration, as illustrated in Fig. 344-349 In employing the above method various substructure components arise which are summarized in Fig. 350-384, which are similar to the already described components in Fig.
249-273 except for some additions will arise due to the use of the QP method employed above.
The structures Fig. 350-364 are already described. However in Fig. 365-369 is the result of a lower corner in the QPb strategy.
Fig. 370-374 is the similar resulting corner adjacent to a K panel in Fig.
100.
Fig. 375-379 is the substructure resulting from a QPb employment between a wall and roof interface at the front Ka' panel Interface in Fig. 100.
Fig. 380-384 is the substructure resulting in the corner QPb employment between a wall and roff interace at the Pentacap interface in Fig.! 264.
In Fig. 385-392 are various dimensional values for the different panel configurations. Fig. 384 shows the dimensions for a QP1a (same as QP1).
Fig. 386 shows the equivalent QPb configuration.
Fig. 387 shows the QP2a panel, which is the same as a K Panel in Fig. 100, with the equivalent QPb configuration in Fig. 388.
Fig. 389 Shows the J Panel in Fig. 100 as a QP3a configuration, with the equivalent QPb configuration in Fig.390.
Fig. 391 shows the QP configuration for a L Panel from Fig.100, and the equivalent QPb configuration in Fig. 392.
Thus the Post-and-Beam strategy is set now the internal stud-works to it has to be established as in Fig. 393-397, where all standard building construction configurations are applied.

WI IMPRIAMMIAIIM110109.-This means a way of fitting standard 16" or 24" studs-on-center in between the strut works developed in the QP and QPb types decribed above.
In doing so 5 different configurations were arrived at that allow for convenient and useful sizing of struts, as shown in Fig. 393-397.
Next the strategy for connecting struts in the QP configuration are shown in Fig. 398. This method uniquely allows using standard bolts (or screws as shown) to connect beams (or struts) together into Tii-panels as standard, repeatable building units that are easy to fabricate, are efficient, strong and sized to allow transport in standard trucks.
These panels are then builtup at the construction site into Quad-panels as in Fig. 399 by applying the bolt patterns at the interface connections where a virtual dimension point (VDP) occurs.
A Virtual Dimension Point is defined as: any point on the inside of the structural shape of the Icosahexahedron allowing for a reference that is independent of beam or stud thickness. In other words all dimensioning and measurements are relative to all the vertices in the invention as previously defined in a zero-thickness structure, that attains thickness in walls, roof, etc, by extruding beam and stud thickness OUT from the VDP's, which are simply the vertice co-ordinates as identified by the Icosahexahedron shape definition.
In this way co-ordinates of the outer dimensions of building components do not have to be mapped but are kept track off at the subcomponent level.
Hence the structure is defined independent of wall and roof thickness. A
structure with a roof and wall of thickness 12" has all the exact same Virtual Dimension Points as a same sized structure with a 18" thick roof and wall.
This is demonstrated in Fig.400-404 an application of a two storey building construction which is a residential dwelling where the wall and roof thicknesses are extruded out in the plane of each building panel.
This results in a triangular valley between each plane extruding outward, as in View I, all of which are identical in size and triangular shape at the interfaces between the 20 Equilateral Triangles as in View2, but where various different panels interface as in Viewl, 3, & 4, the valley width (not depth) is different, usually smaller as at the interface between an 0 Panel and a J Panel, although at the very top of the structure as in View4 central where two J Panels meet the valley is bigger.
The nature of this valley is utilized, uniquely, by synthesizing it implicitly into a substructure known as a "Virtual-Beam", or V-Beam, which is essentially a hollow triangular beam.
With some additional support this structure becomes what is known in the Building Construction field as probably THE strongest building element possible.
This is born out by the fact that ALL large-capacity construction cranes of the kind that can be seen constructing Highrise Buildings, where huge weights have to be maneuvered about at large fulcrum swing, utilize exactly this structure of a hollow triangular beam.
In this invention, the effect of the triangular valleys forming at the panel interface points, by extruding them out the thickness of the wall and roof, is completely usefully and elegantly utilized by simply reinforcing the outward gap of the valley so that a triangular beam by definition results, as in Fig. 405-410. Fig 407 shows two possible modes of filling in the gaps between panels resulting in a very strong interface between building panels.
In Fig. 406 the method employs a lateral placement of support blocking to create the V-Beam, whereas in Fig. 407 a series of inserted blocks achieve a slightly different version.
Bolt (or screw) patterns for either approach are shown in Fig. 408-410.
This has the elegant side-effect of effectively emplacing strong hollow triangular beams from each major vertice in the Icosahexadron structure, implementing a very effectively strong Post-and-Beam strategy that also has the extra benefit of being completely geodesic adding even more strength.
Added to these two effects is a powerful third: the Shell Effect. It results from the fact there are essentially two shapes one inside the other connected by each inner Virtual Dimension Point to the corresponding out point at the thickness of the beams or studs, making a shell of that thickness that has great strength of integrity.
Which when added to the geodesic nature and triangular beam employment makes the Icosahexahedron building construction structure strong enough to be free-standing without internal load-bearing walls or beam span structures, although there is nothing to prevent an application that would use these structures anyway for various purposes like being a basis for walls or other useful structures in a dwelling.
But this free-standing capability allows the structure to be used judiciously as an open-concept structure, where one application is to buildup an internal room system entirely out of a very flexible free-standing mezzanine structure which rests entirely on the first floor (or even feasibly the basement floor), which itself can be designed out of completely unrelated thematic modes like steel tube beam or any other architecturally sound method that would contract very aesthetically with the non-orthodoxy of the Icosahexahedron structure.
Also, in free-standing warehouse applications where large objects need to be stored, or for example like in salt or other chemical storage domes.
In Fig. 411-416 are shown examples of a two-story free-standing structure with a door entry at front and window ways at the 4 corner locations that allow outward sloping windows eliminating any moisture entry problems that may occur on inward sloping windows.
A roof ventilation strategy that allows air to flow from the edges of the walls up into the roof and out the top allows for the elimination of the necessary for eaves-troughs at the wall-roof boundary.
This also eliminates the need for down-spouts since the equivalent to an eaves-trough can be run along the edge of each triangular wall panel to be exhausted at ground level by default.
The elimination of the roof-overhang and eaves-trough, geodesic structure, triangulated beam system, and shell effect, also all contribute to the structure being effectively a wind-resilient structure having applications in hurricane-prone locations where the preferred doorway/window configuration provides a natural pre-prepared plywood placement strategy for quickly and easily preparing a structure for severe oncoming weather, and is probably very effective without any such added measures in that the structure itself is aerodynamic.
Wrmd-flow is very forgiving of shapes that flare away, but destruction to flat surfaces, as used in most conventional building structures.
The most aerodynamic shape is the head of a whale, or a sphere, because it flares away, even though a fair portion of the front surface can be reasonable considered to be fairly flat to the wind.

The Icosahexahedron is similar where from any view angle, all walls flare away back from the viewer in an aerodynamic way, allowing high wind to flow around the house easily rather than getting caught up in destructive vortices underneath eaves-troughs, roof overhangs, and flat surfaces with square flaring back effects which itself causes vortices. And in most cases, the very way that conventional roves are fastened to wall structures is often not taken very seriously as builders consider the immense weight of a trussed roof structure, the mistake in not realizing that once the wind gets underneath the leading overhang of a roof that is not fastened with extreme integrity, it becomes a perfect wing, with the expected resultant outcome of flying away suddenly.
Further, in a frame construction building, the building strategy at play is that it is the placement of the plywood sheathing that gives triangulated strength to what is actually a very week frame structure.
In any frame structure that does not have the sheathing applied, it can very easily be knocked over just by leaning agsinst it, even one that hss all it's own primary fasteners in place. This is why they have to be very securely braced until the sheathing is applied.
But the problem is, during extreme weather one of the two things that happens is first, there I a sudden drop in barometric pressure as the weather system arrives, second, high-wind. Both have the effect of applying hostile forces directly to the sheathing, whereupon all it takes is for the first few sheets be torn away, and the forces inside the house then contribute in a chain reaction to tear the the rest of them away. The more this happens, the more that skew forces in the now weakened frame, actually contribute to pushing off the remaining sheathing mechanistically.
At this point it is very easy for wind forces to get underneath the overhang of the roof and carry it away the wing-shape actually contributing to lift in the structure.
None of this is at issue in the Icosahexadron design, in that it is Pm-and-Beam: it has great strength of integrity with NO sheathing in place; there is no roof overhang, the connection between the roof and the wall is of high integrity and is the same technique as every where else in the structure; the lack of eaves-I:mud-ling eliminates destructive vortex formation, and the overall shape is very aerodynamic from any angle of oncoming wind.
In employing the structure as a macro structure, eliminating the eaves-trough/roof-overhang has certain advantages and disadvantages. The advantages are: a) no eaves-troughs to clean out, b) no down-spouts required, c) better aerodynamics at the roof-line contributing to wind-resilience, and d) improved aesthetic appearance when taken in conjunction with other necessary design factors.
But the following problems are introduced a) the interface between the roof and wall becomes non-standard, b) without an overhang there is no convenient shelter for walkways, c) the roof must be extended directly to the ground which makes traditional attic ventilation through the underneath of the roof overhang impossible.
Building structures according to municipal building-code requirements must have adequate ventilation in the roof. To accomplish this and address the other factors several design elements were employed: a) making the roof thick enough to have adequate code insulation and air gap, b) making the roof continuous with the walls, c) employ ventilation openings at the interface between the roof and wall, resulting in a wall the same thickness as the roof, but not requiring the same depth of insulation, resulting d) in the advantage of repeatability in design and manufacture of wall panels because they are identical to roof panels, e) the roof is ventilated through air openings in the downward angle struts of the walls as in Fig. 400 in the sample wall panel with vertices included in Viewl, View2, and View3, where the roof is ventilated through a series of internal air openings builtin to the traditional over-hange location at the interface between roof and wall, flowing up through similar air gaps inside the wall into the roof, where the air inlets are lower down along the downward angles from Viewl to View3, and View2 to View3.
Hence airflow is up through the bottom triangular panels edges through air openings, through the wall up into the roof and out traditional roof vents at the top of the roof.
The advantages of all this are a) cathedral ceiling inherent in the design allowing use of attic space, b) rain-troughs run at an angle downard and meet at 'View3 where a simple drain removes rain, resulting in self-cleaning rain-troughs, c) the rain-troughs double as down-spouts. d) since there are two troughs per wall, they can be smaller and less visible, e) the overall structure becomes very aerodynamic and hence wind-resilient, f) since the interface between roof and wall is continuous, last, and not least, the problem of ice-damming is completely eliminated in this design, a major achievement in macro building structure design.
Moving on to one final extraordinary feature of the extremely versatile kosahexahedron is evident in viewing it from the bottom, i.e. upside-down, i.e. in viewing the TETRADUOCAP in Fig.
258, where evident is the hour-glass shape made up from the interface of the L
Panels in Fig. 100.

...01,4254 MPOWVISKGE.

In looking down upon the TETRADTJOCAP hourglass, the angle between the native Equilateral triangles is 72 degrees, i.e. 360/5, as derived in Fig. 135. This means that the angle of the lines flaring out from the hourglass in Fig. 419 defined by the angle between the two line segments iX1-iX2, and iX2-iX3, is exactly 144 degrees (72 * 2).
This is is also empirically known to be the exact same angle in one side of the hourglass shape in the Star Constellation Orion, specifically the left side, where upon observation the two shapes are strikingly similar, as in Fig. 417 and Fig. 418.
But when the two are overlayed graphically, the novelty of the alignment of this angle between the two structures is astoundingly perfectly identical, as in Fig. 419. Resulting further in the 3 interface points iX1, iX2, and iX3.
Further, in the novel views of points iX4 to iX8 the following is observed: a) the quadrilateral defined by the points iX4, iX5, iX6, and iX7, as viewed off-angle, under transformation is the same as the L Panel of Fig. 100 native to the Icosahexaduoduoduohedron, and b) the triangle defined by points iX4, iX7, and iX8 which is connected along one edge to the quadrilateral previously described, similarly as to that occurring in the Icosahexaduoduoduohedron, represents the transformation of the two points of the adjoining quadrilateral line segment into one single point thus showing the transformation of the L Panel into an interface panel in transforming the Icosahexaduoduoduohedron of Fig. 274 into the pure Icosahexahedron of Fig. 198-221, in transforming the structure in Fig. 102 into the structure of Fig. 198 whereby the j1 panel is mapped to the a panel, the K1 panel to the KX, and the L1 to the LX
respectively. As in the process of rotating the two structures STR1 and STR2 in Fig. 95 about the axis defined by V1-2 and V1-3 in Fig. 94 such that in Fig. 102 the short edge of J1 increases, the short edge of K1 decreases, and the short edge of Li decreases to zero as the points V1-8 and V2-8 of Fig. 94 are transformed together as in the described rotation inFig. 98, resulting in the pure Icosahexadron.
A further study of the interface at iX1 as in Fig. 420, shows a depiction of the ancient Universal Symbol of Peace.
______________ e. __ "WM. _____________________ votanparmas. __

Claims (15)

1. A Building Framework which is modular with 26 sides (Icosahexahedral) and 16 vertices which is a synthesis of two half Icosahedral frameworks which have been dissected longitudinally along their equatorial struts, fused together by inverting one of the halves and then connecting the 2 halves at the 2 upper extending vertices of each half, retaining all the original vertices of both structures but also introducing on the surface 6 new planes and 4 new struts, and internally 3 new planes which are parallel and equidistant, 2 of the internal planes of which upper and lower are 6-sided but alternatingly, with the plane in the middle being 2-sided, where on the surface of the structure there are 20 sides which are all identical equilateral triangles, 2 sides are isosceles triangles, another 2 sides are isosceles triangles with base equal to twice the base of the previous 2 triangles, and another 2 sides are quadrilaterals allowing a perfect inclusion of 3 of the first 2 triangles alternatingly.

2. The Double-Layer Shell Building Framework resulting from extruding outward of the sides of the Building Framework of Claim 1 giving thickness to the sides of the Building Framework by introducing struts of the length of the depth of the desired thickness perpendicular to the side planes of the Building Framework of Claim 1, connected at the side vertices extruded out and subsequently connecting the resulting new strut ends to each closest new adjacent strut end resulting in essentially two Building Frameworks of Claim 1 of different dimensions one inside the other connected by interconnecting struts as defined by the inner Building and introducing gaps between sides in the outer layer shell which are defined in width by the depth of the shell.

3. A Building Floor/Ceiling defined by either of the lower, upper, or central internal planes made up of either the central equatorial plane perfectly dissecting the Building Framework of Claim 1 about its natural vertex connections resulting in a 12 sided, elliptical floor/ceiling, or by either of the two different upper or lower planes defined by natural vertex connections of the Building Framework of Claim 1 which are equidistant to the central equatorial plane in opposite directions, both of which are different but are both 6 sided in inverse relation, and transform geometrically into each other through relationship with the equatorial plane of the Building Framework of Claim 1.

4. The 12 Sided, 6 End-Point Sided Tubular Sub-Building Framework created by removing the top and bottom caps from the Building Framework of Claim 1 which is a 6-sided ellipse at both ends but contains an alternating symmetrical transformational-correspondence from one end of the tube to the other, allowing adjacent tube frameworks to be connected along common interface points or edges by alternating each subsequent unit, extending the Tubular Framework into an indefinitely extendable articulated pipe with an alternating hexagonal cross-section, or extended vertically in a similar Building Framework to implement a building tower or stack.

5. The Triangular (3-Sided) Tubular Longitudinal Sub-Building Framework of the Double-Layer Shell Framework of Claim 2 which are hollow or semi-hollow triangular beams resulting from the integration of struts connecting the inner and outer Building Frameworks of Claim 2 and all their inner and outer side edges.

6. The Small Triangular Sub-Building Panel Units, which are sized to allow conventional orthogonal sized building materials to be placed within standard orthogonal stud dimensions but within the constraints of triangle shaped panels, 4 of which comprise larger Triangular Sub-building Panel Units and retain the orthogonal-to-triangular material mapping feature by extension, 4 of which joined correspond directly to the sides of the Building Framework of Claim 1, are joined at the 3 corners without any special joining mechanism but rather by cutting one adjacent strut end at each corner in a 60 degree angle and connecting two struts at the corner together with a simple bolt or screw pattern that utilizes the 60 degree angle in making the joint.

7. The Large Triangular Building Panel Units which correspond directly to the sides of the Building Framework of Claim 1, which are made up of 4 of the Small Triangular Sub-Building Panel Units of Claim 6 joined together flush and parallel with a simple bolt or screw pattern, which are thenjoined at their inner edges to implement the Double-Layer Shell Building Framework of Claim 2 , by placing standard bolt or screw fasteners at the inner joint then an accompanying strut-work or blocking at the outer gap, the integration of which creates the Triangular Tubular Longitudinal Sub-Building Framework of Claim 5.

8. The Process of Constructing the Small Triangular Sub-Building Panel Units of Claim 6 without any special fasteners by using a standard bolt or screw method with the panel end points cut at a 60 degree angle to facilitate joining the end points together.

9. The Process of Constructing the Large Triangular Sub-Building Units of Claim 7 where the Small Triangular Sub-building Panel Units of Claim 6 are joined together through a simple bolt or screw and plate pattern with the small units flush and adjacent together creating integrated beam structures internal to the Large Triangular Sub-Building Unit of Claim 7.

10. The Process of Constructing the Double-Layer Shell Building Framework of Claim 2 where the Large Triangular Sub-Building Panel Units of Claim 7 are joined first along one edge through a hinge method, where two adjacent panels are hinged at their inner edges together to facilitate rough placement of the two panels into their position in the Building Framework of Claim 2 as defined by the inner edges of the panels, where all subsequent Large Triangular Panels are hinged on one edge onto the placed units in the framework one edge at a time, in sequence, until the entire Building Framework is connected all along all inner edges by hinges allowing flexible adjustments to panel placements until all panels are finally properly placed comprising the Shell Building Framework of Claim 2, whereupon final bolt or screw placements are done at the inner panel interfaces, then further strut-work or blocking is done at the outer interfaces with further bolt or screw placements to make the final unit connections, whereupon the hinges are then removed resulting in the Double-Layer Shell Framework of Claim 6 which is 2 Building Frameworks of Claim 1 one contained within the other but without final permanent bracing in the outer interfaces and a dimension gap between sides in the outer-layer Building Framework of Claim 1.

11. The Process of Construction where the gaps between adjacent joined panels of the sides of the Shell Building Framework of Claim 2 making up the Triangular Tubular Longitudinal hollow or semi-hollow beams of Claim 5 created by the depth between the inner and outer layers in the Shell Building Framework of Claim 2, are filled with triangular supports (blocking) which fit the insertion angle of the triangular gap such that the base side of the isosceles triangular support faces in the direction of the outward extrusion making a semi-hollow beam, or instead, a single longitudinal brace is used to enclose the open end of the triangular hollow beam at the gap, into an enclosed triangular completely hollow beam which encloses the outer layer gaps between sides of the outer-layer of the Shell Building Framework of Claim 2, making it a complete implementation of 2 Building Frameworks of Claim 1 contained within the other with the outer one expanded at the layer surface sides by the enclosed gap.

12. The set of Concave Sub-Building Frameworks (trusses) with a minimum of 3 sides which result from dissecting the Shell Building Framework of Claim 2 across all possible adjacent edges, but excluding the Concave Building Sub-Framework of the set which is a 5 Sided Icosacap which is native to the Prior Art Icosahedron.

13. The Building Framework resulting from removing the inner Building Framework of the Shell Building Framework of Claim 2 resulting in only the single outer layer framework retaining the extra connecting struts where their placement was defined by the original placement of the vertices of the inner Building Framework.

14. The Building Framework of Claim 1 where strut sizes may deviate freely from the constraints of Claim 1 but still retain the same number of vertices in each side: 3 for any triangular side and 4 for any quadrilateral side, resulting in a freely non-symmetrical 26 Sided 16 Vertex Building Framework.

15. The 26 Sided 15 Vertex Building Framework resulting from a transformation of the Building Framework of Claim 1 where the first 2 isosceles triangles are increased in size of the their base, then the second 2 isosceles triangles are decreased in size of their base, so that the first and second sets now are equal in base size, subsequently making the 2 quadrilaterals transform into the same equal triangles as they, as well such that the 20 identical equilateral triangles retain configuration as two fused Icosahedrons joined at interface points by the 6 newly transformed triangles, but no longer with the 3 internal parallel planes, whereby the two vertices previously connected by two edges of the connected quadrilaterals merge into one vertex, thereby reducing the vertex count of this Building Framework in still retaining the original 26 sides, but now has one less vertex to total 15 vertices.