CA2483361A1 - Icosahexahedron - Google Patents

Abstract

A 26 sided semi-regular polyhedron structure known generally as an ICOSAHEXAHEDRON, or ICOSASEXAHEDRON, more specifically an ICOSADUODUODUOHEDRON, whimsically a SILMARIL. Having 20 identical equilateral triangles, and 3 pairs of different but mathematically related triangles/quadrilaterals. It is composed of 2 ICOSAHEDRON structures fused together to make a single synthesized structure which results in 3 parallel internal planes, one which is a perfectly symmetrical elliptical 12 sided equatorial; which have useful applications in macro use of the structure, and has external alignment points and planes in 3 dimensions useful as macro, micro or molecular structural interfaces. It is geodesic in many planes resulting in great structural strength, integrity and aesthetic appearance. It has some dimensions which are of the mathematical ratio PHI, and it aligns in one dimension perfectly with the Star Constellation Orion resulting in a further alignment which is a depiction of an ancient Universal Sign of Peace.

Description

26-SIDED 16-VERTEX ICOSA~i' ~'XA'~EI)RC~N SFA.(~E
S'I'RUCI'~JRE
A. space structure usable is macro structures; building constiv.~ttion, housing, warehousing; and micro or molecular space structures and applications.
BACI~~ROII'ND aF TIH~ IN'VL~'7CION
The project began as an exploration of various existing known space structures for possible use im building construction applications, along the lines of the work that Buckminster Fuller achieved in his explorations of the 3-frequency CeOdesic 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 Icosahedron, with various permuations applied to endeavour to discovery new and novel uses of the structure, either through different mathematical transformations, or through fizsion into newly synthesized str uctures.

Research was done into the field of Geodesic Domes including Fuller's work and publications such as Domebaok II 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 1 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.
) outward facing windows suffering from rain-shipping leakaige problems.
6) difficulty in interface standard vertical internal walls to mufti-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 gaa~l 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 l:he above objectives.
The current construction code is based an an orthogonal methodology that does not readily allow a generalization 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 hams been dictated by the orthogonal nature of gravity. In taking a different approach to offsetting 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 f'~eld of Building Construction.

A second uncertainty was in whether an efficient means of joining materials in non-arthogonal 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 welt 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 4x$ foot standard sheathing jdrywall panel, and the standard 16", or 24'" dimensions were mapped into effective implementations of the Icosahedron's native Large triangular panels, to solve problem l, 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 coxifiguration 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 elimuzate the rooF overhang leading to many advantages, b) an entirely new geometric shape known as the Icosahexahedron with one variation, c) fixrther repeatability in the design allowing high efficiency, and d) a way of incorporating ti~.e structural shape into macro, micro, and molecular applications which is the basis far this invention.
The project set aut and successfully solved several diverse techuc~ical 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 Icosahedran, 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 ?,4" dimensions were mapped into effective unplementations of the Ieosahedron"s native Iaxge triangular panels, to solve problem l, 2, 3, l, 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 salved 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 Icosahexahedron with one variation, c) fiirrher repeatability in the design allowing high efficiency, and d) a way of incorporating this 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 Icosahedron, 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 Bucklninster Fuller.
su,~~r.~RY aF ~~t°~tor~
The invention is a Geometric shape which is the result of fusilig or merging two Icosahedrons into one, along vertices and panels native to the Icosahedron, retaining all original vertices but introducing new interface planes that are completely native only to the new synthesized structure, an lcosahexahedron.
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 24 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 ~, 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 Icosahe~cahedron to it's number of panels, 16/26, is a mid increment ratio of PHI (between X3/21 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 Ieosahexahedron 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.l-6 Solid views of the structure front and off angle Fig.7-12 Solid views of the structure side, top, and oflrangie Fig.l3-18 Wireframe views of the structure, front, side, top, off angle.
Fig.l9-30 Wireframe views of the structure, varying ofl-angle.
Fig.31-36 Wireframe views upside down, front, side, top, oflF 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.I00 Exploded View off angle of structure with non-regular panels.
Fig.I01-103 Views of resultant non-regular panels ofI=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.I27-I34 Development of structure major planes upper and lower.
Fig.135-I49 Mathematical formulas for non-regular panels and upper plane.
Fig.150-158 Mathematical formulas for lower plane.of structure.
Fig.159-I72 Mathematical formulas for vertical dimensions of structure.

Fig.174-197Development of alternate 26 sided structure subcomponents Fig.l98-209Wirefiame views of alternate structure with interface panels.

Fig.2i0-221Solid views of alternate structure Fig.221-236Component substructure placement within fused structure.

Fig.237-248Molecular external alignments and interface planes.

Fig.249-273Views 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.276-277Solid exploded front & ofl=angle view of the major equatorial.

Fig.278-Z84Mathematical analysis of the 3 major internal usable planes.

Fig.285-289Solid views of upside-down upper-plane dissecr_ed structure.

Fig.290-294Solid views of rightside-up lower-plane dissected structure.

Fig.295-301Solid views of rightside-up equatorial dissected smzcture.

Fig.302-308Solid views of updside-down equatorial dissected structure.

Fig.309-313Solid views of riglztside-up lower-planed window configuration.

Fig.314-315Wireframe views of lower-planed window/roof/wall config.

Fig.3i6-319Wireframe views of lower-planed 3 major internal planes.

Fig.320-325Views of two structures interfaced to each other at side plane.

Fig.326-331Views of two structures stacked at lower-to-upper planes.

Fig.332-337Views of double-phased strut support in maul panels.

Fig.338-343Views of orthogonal-phased strut support in main panels.

Fig.344-349Views of double-artho-phased strut support mixed derivative.

Fig.350-384Views of substructures resulting from double-ortho mix.

Fig.385-392Mathematical analysis of double and ortho pha:>e strut-works.

Fig.393-397Mathematical analysis of standard i6" or 24"
stud-works Fig.398-399Structural analysis of panel corner interface boht-patterns.

Fig.400-404Solid views of ventilated extruded wall and roa:~thickxaess.

Fig.405-4i0Derivation of panel interface virtual beam structures. & bolts.

Fig.411-4162.5 storey 2500 square foot free-standing houss~ 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 I~ETATLED DESCR.~1''I"~t~N t~F THE IN'V'F~ON
The 26 sided semi-regular polyhedron can be a planar solid, a;s 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 fixrther 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 mathema~:ically, 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 & l~. ~n other views it becomes very graphically asymmetrical, as in Fig. 1 and Fig. 5. Various views in ~~etween tend to show bizarre off angle variations of the two, as in Fig 2,3,4,6,11 & 12. Yn 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 idcntif~ed as being mast relevant to practical uses c>f the structure, resulting in the rightside-up orientation of Fig. I.
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 far construction purposes, leaving only triangular panels making up the primarily usefixt 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 ui the drawings with a cube present nearby the structure or other elements identified respectfully: F
(front), B (back), I. (left), R (right), T (top) and U (underneath).
In wireframe 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 oiF
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. I3, I S, & 3I . 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 JFig.
49. This is a very important basic building block in deriving tl~e structure, and is given a unit dimension called "T" far each equal side of the triangle O as in Fig. I2I, which can be any non-zero size, and is described further below. By mathematical definition the Angle U in Fig. ICI
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. 5I 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. 5 3.
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 Icosahedron, a 20 sided structure. ~Che Icasahexadron 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) I38.2 degrees and can be viewed in Fig. ~4-60.
This dual triangle component can be employed empirically facing in and joining only by it°s 3 vertices to adjacent identical components to arrive at the Icosahedron structure. Following the;>e 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 Icosahedeon as irz Fig. bI-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 Icasahedron 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~ SAD 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 Ievel of vertice structure of two adjacent Icosahedrons.
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 Icosahedrons 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 sulsstructures STRI
and STR2 along the respective planes DI and D2 shown in Fig. 88-8~, resulting in key vertices that can allow logical interface points at VTXI-I to 5 and VT~~2-I to 5 along planes D1 and Next, the panel families E, F, and G need to be removed from. the structures STRI and STR2 in Fig. 9I-93 which results in the two new derived structures ST'Rlb and STR2b in Fig. 94-96.
The two new structures then need to be aligned so that the plane PLNl defined by vertices VI-1.
to 4 matches the plane PLN2 defined by vertices V2-1 to 4. A,nd also so that the plane PLN3 defined by vertices VI-5 to 7 and plane PLN4 defined by vertices V~-5-7 match each other.

The two structures STRlb and STR2b are then joined together at the two interfaces il and i2 by connecting the vertices Vl-2 and V2-1 together, as well as the vertices Vl-3 and V2-4, all the whale maintaining the planes PLNl and PLN2 equal, as well as the planes PLN3 and PLN4 equal, which results in the new single synthesized partially complete structure STR3 in Fig. 97-99.
In Fig. 100 struts Sl to 4 are created by creating a} strut S1 between the two top vertices Vl-9 and V2-9 at the top of STR3 creating the new panels ~l and J2, b) strut S3 between the bottom two vertices Vl-8 and V2-$ creating the new panels Ll and L2, c) strut S2 between the two front vertices Vl-5 and V2-5 creating the new panel Kl, and dj strut S4 between the two back vertices Vl-6 and V2-6 creating the new panel K2. The panel families J, L, K are shown at the bottom of the Figure.
The connected vertices result in the completed invention, an lc:osahexahedron, in Fig. 101-103, showing the regular panels in transarent (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. l I6.
In Fig. 117-119 the non-regular panels are showed in various two-dimensional special relationships which can be summarized as following.
~) ~ l~ig. lal., sty si=sa, a~~:~ty, b> sa~.t sa=s~, ana C) sz=si ~- a In Fig. 120 forward the variable R is assigned to the length of Sl 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 extxeme 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 atso 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 STR3 is oriented, the vertical oriented struts all are size T, whereas the horizontal oriented struts S 1 to 4 all follow the above relationships, f} 3 Panel J's in Fig. 123 fit perfectly dimensionally inside 1 Panel L as in Fig.
124.
i0 All internal angles are listed in Fig. 12b and can be calculated using standard mathematical geometry since all triangle side lengths are known.
(3f structural significance is the orientation of the plane PLNS 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 fii.rther in Fig. 127-134.
In an analysis of Plane Top PT first, in Fig. 127 & 131 looking dome from above are two identical 5 sided polygons known as pentagons, PNl and PN2, which are the same as the two top and bottom planes of a regular lcosahedron, as can be seen in Fig. 85 & 87.
Plane Top PT is derived as in Fig. 94 by connecting vertices VI-2 to V2-1 and vertices Vl-3 to V2-4. This results in a geometric shape of Plane Tvp as in Fig. I29. The location of Vl-9 (and V1-8) in STRI is the same as vertex VTXl in Fig. 128 which is also the intersection oflines F1L1 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 VTXl and 2 in Fig. I29 & 134 (& 133).
Plane $ottom PB is made up of the same two Pentagon components in an inverse way, as in F'ig.
133 where the vertex in Fig. 94 Vl-5 is connected to V2-5 by the distance S, similarly for the vertex Vl-6 to V2-b, resulting in the pattern for Plane Bottom in Fig. 133 Looking down on the structure STR3 in Fig. 101-103 results in the same pattern as in Fig. 134, showing the overlapping views of Plane Top, and Pian Bottom, with the interaction of R in Fig.
I29, and S in. Fig. 134, which is extremely novel and intriguing.
A ftirther 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 tl~e new values P in Fig. 136 which is the perpendicular line from T looking in the downward plane across to the vertex VTx:2 in Fig. 132. The value Z in Fig. 13'7 is the line segment from VI-2/V2-1 in Fig. 94 to the extension of the line segment T which also defines 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 anal~.~zed 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 I. 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.
I42 & 143 is made up of exactly 3 Panel rs in Fig. 140, d) the Angle Q in Fig, 140 & 147, e) the Angle W in Fig. 14I & 148, ~ the bottom corner angles of Panel L in Fig. 142 is equal to (7 -I- V
in Fig. 149.
In Fig.150 is a representation looking down on STR3 with P'T 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 T op PT (or similarly PB) with a Large edge of both sides of the long side of the rectangle adjacent m 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 symtnet~rically on the right side.and is calculated in Pig. 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 tThe 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 verif.cation of the value, using ZZ and T.
Pig. 155 is a review of the relationship between S and R.
'The length FP in Fig. 150 and 156 is not relevant to the native lcosahedron 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 Pl'. 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 VTXl or VTX2, the distance to the nearest wall comer at the connection between 2Z 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 snnuning with the line segment S.
Pinally, a major discovery in the invernion is that the relationship he~cve~
the enclosing s~ctuire width, SW in Pig.154, and T, the unit ~ of the base sides of the subcornponent triangles malaing up 20 panels of the structure, i~ a p ~ ~ ,~..in~ea:~, otherwise known as tf~e "Fibona~ Number", or "Golden. Ratio". i~Vhieh means that viewed in wartous other ways is the:
same relation as PHI. Le., an ate equivalent relation is that ~~ ~ .per.
In Fig. 159-173 axe analyses of horizontal front views of STR3 summarized in Fig. 160 and developed as follows. Fig. 159 shows how intriguingly the intersection of Strut S3 in Fig. 101 with Plane $ottom P$ as in Fig. 274, creates a perfect square as also shown in Fig. 172 elemea_zt WW with side dimensions of R. A similar square is created in perfect symmetry in the front view in the vertical plane from this bott<>m square where the Strut Sl in Fig. 101 intersects with Plane Top PT in Fig. 274. In aesthetic terms this is useful in using STR3 in applications as a macro space structure for habitation or warehousing in portraying balance ergonomically.
This is proven by the derivation of the value I-3FJE in Fig. 159 .& 166 as being equal to R
Tf the relation R to T, or other dimensions native to the inven tion 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 STR3 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. 16~;
(hence the Not Front indicator since the plane is not perpendicular to front).
The practical dimension TT 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. Mere it is also used to support the development of the various angles in Fig. 163 of BB, angle from ceiling to roof i~ZClination, angle MM, angle fronn wall to vertical, angle QQ which is tile 90 degree angle onset of the wall to the ground, and angle B which is the angle of prime importance in the entire structure of which derivation and primary understanding is required in the pr<xess 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 MM in Fig. 163 & 167, proceeding to Angle BB iii Fig. 163 8r:164, finally to arrive at Angle 3~ in Fig.163 and Fig.169.
Angle B can be rounded to 138.2 degrees as in Pig. 52-60.
Dimension C G in Fig. 163 is slightly different to TT 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 TT. So T'7C is the panel.
height, GG is the vertical height, which are slightly different as shown in Fig. 171.
The total vertical height of the stnzcture TH shown in Fig. 1 ~ 2 as the addition of the vertical dimensions of WW ~ 2 + AA is derived in Fig. I68 but suinrning previously known dimensions as related to T.
A further intriguing discovery in the invention ias t~ m.latxonship in Fig.170 showing xhat t~
ratio between. the vertical distance between the trap (PT) and bottom. plane (PB) as iu Fig. 2y4, CC~ (or the height of AA), to T, is PF~I<
Further, it is fnuad in Fig.1~0 and suar~nnacized in Fig. I73 that the ratio of the height of tire previously mentioned pt square WVfT, to ttze height of the rye AA~, is PHI.
The synthesized 2b sided polyhedron structure thus derived, is known as an x~osAHEx~3E~aRaN
from Ieosa - 20 + Hexa 6 = 26. ~~xi alternate nomenclature for 6 is Hexa, resulting i~ci IC08.A~F.XAFiEDRC3N
These are acceptable references in general terms, but to be more precise, the structure is symmetrical in 20 panels, then another two more Jl and J2, r:.qual to themselves but not to flue other 20 panels, as in Fig. 100, then two more are different again I~.1 and Ii,2 in Fig. 100, and finally another two more which are the Quadrilaterals Ll and :L~ in Fig. 100 as well.
To complete the definition may also be included the 3 internal planes PT, EQ, and PB as in Fig.
27~ as they define the orientation of how to connect the two fused Icosahedrons in Fig. 94-9~.
So a more precise reference for the structure is:
2~+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"
ICOSADODODOHLDRON
or two being "Di"
ICOSADff3IDTHEDRON
Finally, there is a whimsical identification which refers to the significance of a ~6 sided structure identified in literature as being defined as: "A fictitious structure". Also which of note, having ~C
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 it's final resting place after proving too much a burden for th~~
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:
sRa..
~s Of lesser note there is an alternate 2C sided polyhedron structure which will be briefly described, which is related to the invention in that it also has 2b sides, is simiD.arly made from the fusion of two Icasahedrons, 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.
Tt 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 aligmnent 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 iCOSA~37EX~~HEDR02~T (andjdr ICOSA~SSF.~~AHF~JROhi}
is completely acceuate since the 6 extra panels are all identical.
To construct this structure is similar to the ICOSADUODUODUOHEDRON in that twa base I~OSAHEDRONS 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 tlZe internal planes intact, but rather depends completely on the joining at 3 co-planar vertices instead.
Meaning also, that after joining the twa together, no extra connection struts are required.
In Fig. 1'74 is a slight off angle front view of an Icosahedron. In Fig. 175 is a Front view.
Note that an alternate method of n raking both this structure and the Icosaduoduoduohedron are presented in Fig. 17b, in that rather than follow the process outline previously shown in Fig. 91-..._ _.._____._..,.~ ~~"..~~,~~".F~~...~ _', 93, & I04-IIS, an alternate method is to simply take E7NE Icosahedron, arid split it along a natural equatorial that follaws it's strut connections.
This results in the two half Icosahedrons in Fig. i76 and also creates the same resultant structurres 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 &
I78 to arrive at the orientation in Fig. I79, which is identical to the result of the process in Fig.
94.
From this point it would be possible to align the internal planes P'T, EQ, and PB as described and arrive at the synthesis of the Icosaduoduoduohedron.
But to create this alternate structure the two structures STRl and STR2 are joined differently.
First are some views of the nature of each half Icosahedron 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 Imsahedrons to make this alternate structure requires aligning them so that the Axis AX4 for both halfs, is equal, as in Fig. I86-189, next: one half must be rotated about .Axis A~4 I80 degrees as in Fig. I90 & I91, whereupon the two halves can be joined at 3 native vertices as in Fig. I92. To finish, the Struts SUI, SUZ and SU3 in Fig. I93 &
I94 must be connected, resulting in the complete structure in Fig. I94, a pure Icosahexadron.
Further wireframe views are in Fig. 195-I97. 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. 2I0-221.
It is a novel structure that has geodesic strength, is symmetrical m 2 planes (facing front, left to right, and top to bottom, as in Fig. 2i3, 220).
The structure also exhibits the similar trait in the Icosaduodu~duohedron of "hybrid-symrr~etry", as in Fig. 2I4, and semi-asymmetry in Fig. 210, 211, 212, 215, 216, 2I7, 218, 219, and 221.

Continuing on with the primary invention, the Icosaduoduoduohedron, which from this point forth will be referred m 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 b substructures shown in Fig. 249-273 farther 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 SI of Fig. IOi of Fig. 237 STR-L.
Thus results in the substructure iXLJL in Fig. 237, a novel structure which defines physically the nature of the interface between two Icosahexahedrons in this plane further shown as T'lane MPT in Fig. 247 and Fig. 246 with 3 units stacked vertically, and also in Fig. 238, 239, 24(3, Lit 24I.
Next, there is a natural vertical surface which is Panel KI 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 3I) space, the common vertices of joined Icosahedrons 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 & 243.
A variation on interfaces that flips variaus units around in various planes is explored in Fig. 24.4, 245, and 248.
Such orientations and interfaces will have use in micro structures allowing a large base unit, the Icosahexahedron, but with many planar connection points, which will tend to create a novel material with properties of strength.
as At a molecular level the synthesis of two Icosahedrons into one fused one will create a nevv 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 Icosahe~ahedron are listed in Fig..
249-273.
Fig. 249-252 are views of the HEXAL~L7C~CAP (6 + 2}, named for being a 6 common plus 2 extxa sided pyramid that is defined as the slice created by Plane Top PT in Fig. 274 resulting in the tvp component in Fig. 275..
Fig. 253-256 are views of the I~ECADUOSECF (I0 + 2), the middle section created by slicing the Icosahexahedron at both PT and PB in Fig. 274, which has IO identical sides, with two K
panels as in Fig. I00, which appears as an elliptical (or oval) shaped, i2 sided drum-kit also shown as the middle section in Fig. 275.
Fig. 257-260 are views of the TETR.ADUOCAP {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 pyramidl which is native to the Tcosahedi:on and represents the unchanged parts of the Fused Icosahedxons making up the invention.
Fig. 265-268 are views of the TETRAUNIL7NICAP (4 -~- 1 -t- 1} sided pyramid which are the substructures resulting from the interface of 4 Equilateral triangles, a J
Panel, and a K Panel from Fig. I00.
Pig. 269-273 are views of the INTEHE~.AI'E:NTACAP (integrated 6 and 5 sided) pyramid, which is the result of part of a Pentacap with one L Panel in lEig. 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 Heatacap (6 sided pyramid) Note that the 6 identified substruc.-tures 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 Icasahedrons together along IrT 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 Icosahexadron has a natural equatorial, E~, as in Fag. 27b, 277 and defined in Fig. 274. ECM
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 ~ 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 AI and A2 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 fiu~ther below.
The Plane Bottom PB in Fig. 280 t~ 28I 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 Std axe definedl in more detail previously in Fig. I52 r~ I54. The Angles BI and B2 are further described in Fig.
282.
Again in a practical application, F'B serves as a very convenient and useful ground floor-line as described flu then below.
The Equatorial EQ in Fig. 283 & 284 is a hybrid of Plane Top I'T and Plane Bottom T'B, since struts interfacing the upper and lower plane pass though the e~quatarial definition points. This results in a plane that is I2 sided, is elliptical (or oval) shaped, and has dimensions that at half ~caf T
or half of S { R ). The length of ECM is SL - YY as in Fig. I50 and has the same width as both 7L'T
and PB.

Again EC~ serves as a very convenient and usefixl floor or ceiling line as described further below.
Note that intriguingly, bPT, FCC arid PB afil have the sine width, this means that dae paaels K in Fig.14Q, are pezfecdy completely vertical planes.
This is directly useful in building constnzction 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 rnore 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 nay be too overwhelming in it's von-orthodoxy.
Various combinations of slicing and removing substructures Firom the Icosahexahedron result .in various navel configurations, some of which are well suited to building construction applications.
The first variation, considered the least desirable, is in by turning the Icosahexahedron upside-down (bottom-up), slicing it at IyT and removing the HEXA7~C70CAP 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 C~uadrilateral 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 F3:EXA~UOCAP in the roof pasit:ion, 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 TETRA.DUOCAP 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 ECM 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 Et;~ 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 altow a standard doorway.
For comparison purposes the upside-dawn 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 vera~cal support blocking making a vertical doorway,, c) a novel aesthetic e~ffecr 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 fB, the window configuration, floor and ceiling configuraitions in Fig. 314-319.
Note that inherent in using the lcosahexahedron in such applications lend well to Post-and-Beam consnzction 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 strucctuures 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 E(~ 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. 32fi 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 rebating diamond effect, as in Fig. 330 on the front and back side views of the strezcture, 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 contraction 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-paints and have support struts eonnecte 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 TP 1, 2, & 3 are known as °'Tri-panels", being triangular building panels. The elements QPl, 2 &3 are known as "Quadpanels" since they are made up of 4 joined Tri-pane.
Hence a method of building up Quad-panels from Tri-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 QPlb, 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 mi:~ture of both panel types is beneificial since the roof and wall panels do not require openings, hence could make use of the advanges of the QP1, 2, &3 configuration, and there the lower floor window and doorway panels could make use of the QPIb, 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 ira 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.
I00.
Fig. 3?5-379 is the substructure resulting from a QPb employment between a wall and roof interface at the front K~J panel Interface in Fig. 100.
Fig. 380-384 is the substructure resulting in the corner QPb employment between a wall and rod interace at the Pentacap interface in Fig.l 264.
In Fig. 385-392 are various dimensional values for the different panel configurations. Fig. 384 shows the dimensions for a QPla (same as QPl).
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.

This means a way of fitting standard I6" 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 j together into Tri-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 applyang 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 aUT 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 thl~ickness. 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.400-4fi4 an application of a two storey building constn~etion 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 2U
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 C~ Panel and a J Panel, although at the very tap 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 "Vixtuai-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 ThIE 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 larl?e ftzlcruxri 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. 405-4ltl. Fig 4a7 shows two possible modes of filiiing 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 screwy 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 fram 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 litre 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 lcosahexahedron sti~ucaure.
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 Pig. 411-41f~ are shown examples of a two-story free-standing structure with a door entry at front and window ways at the 4 comer locations that allow outward slopitng 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-spaurs since the equivalent to an eaves-trough can be nan 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.
Wind-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.
a~

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 destnzctive vortices underneath eaves-troughs, roof overhangs, and flat surfaces with square flaring back effects which itself causes vor~t~ces. 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 ea~trem~e:
integrity, it becomes a perfect wing, with the expected resultant outcome of flying away sudde~~ly.
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 as actually a very week frame structure.
In any frame structure that does nat 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 far 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 aff 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 cantributing to lift in the structure.
None of this is at issue in the icosahexadron design, in that it is Post-and-Beam: It has great strength of integrity with NO sheathing in place; t$ere is no roof overhang, the connxection between the roof and the wall is of high qty and is the sane unique as every where else in the structure; the lack of eaves-troughing eliminates destructive vor~
forn~ation, and ~e over"all shape is very aerodynamic from a~ angle of oncoming wind In employing the structure as a macro structure, eliminating the eaves-trough/roof overhang h.~,s certain advantages and disadvantages. The advantages are: a) no eaves-traughs 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 s:oof is ventilated through air openings in the downward angle struts of the walls as in Fig. X00 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-hinge 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 View;, and ViewZ to View3.
Hence airflow is up through the bottom triangular panels edges through air openings, through the wall up into the ref and out traditional roof vents at the top of the roof.
The advarnages of all this are a) cathedral ceiling inherent in the ~ allowing use of attic space, b) rai~a.-troughs run at an angle dov~nard and meet at V'.tew3 where a simple dxaun. removes rain, resulting in self-cleaning rain-troughs, c) ~e rain-troughs double as down-spouts. d) since there are two troughs per wall., they can be smaller and. less visible, e~ the overall snvcture becomes i~ery aerodynamic and hence wind-resilient, fj since the intxrEac~ between roof and wall as continuous, last, and not least, the problem office-damming is conaplete~_y_..~~at~ci. in this design, a major achievement in macro building stru~nre design.
Moving on to one final extraordinary feature of the extremely versatile Icosahexahedron is evident in viewing it from the bottom, i.e. upside-down, i.e. in viewing the TETR_AI7L70CAl' in Pig.
25~, where evident is the hour-glass shape made up from the ia~terface ofthe L
Panels in Fig. ~~00.

In looking down upon the TETRADUOCAP hourglass, the angle between the native Equilateral triangles is 72 degrees, i.e. 3bi~/5, as derived in Fig. I35. 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 iXI-iX2, and iX2-iX3, is exactly I44 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 Constellatian 4rion, 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 fixrther in the 3 interface points iXl, iX2, and iX3.
Further, in the novel views of points iX4 to iX.B the following is observed:
a) the quadrilateral defined by the points iX4, iXS, iXb, and i.X7, as viewed off angle, under transfarrnation is the same as the L Panel of Fig. I00 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. I98-221, in transforming the structure in Fig. 102 into the structure of Fig. 198 whereby the ~I panel is mapped to the jX panel, the Kl panel to the KX., and the Ll to the LX
respectively. .As in the pracess of rotating the two structures STRl and STR2 in Fig. 95 about the axis defined by V:l-2 and Vl-3 in Fig. 94 such that in Fig. 102 the short edge of Jl increases, the short edge of KI
decreases, and the short edge of Ll. decreases to zero as the points Vl-8 and V2-8 of Fig. 94 are transformed together as in the described rotation inFig. 98, resulting in the pure Icosahexadran.
,A further study of the interface at iXl as in Fig. 420, shows a depiction of the ancient Universal Symbol of Peace.
... ~.., , ~~~ ro,~-~~.M.~ ... __~-__.__ ___ ..... . ~,~.,~,,4.~,,~~~~~~.,~
w~.m,~,...._.~.___ _ ____. -~._.-.._

Claims (31)

1. A 26 sided,16 vertex, semi-regular polyhedron structure known as an Icosahexahedron.

2. As in Claim 1 where the Icosahexahedron is composed of two Icosahedron structures fused or merged together aligned about 3 internal planes, which are parallel and equidistant to each other.

3. As in Claim 2 where 20 sides are identical equilateral triangles, 2 sides are isosceles triangles, 2 sides are isosceles triangles where the third side is equal to twice the size of the third side of the previous 2 triangles, and 2 sides are quadrilaterals of the dimension that would allow perfect inclusion of the former 2 triangles as a placement of 3 in alternating fashion.

4. The interface elements as in Claim 2 where they are non-native to the original Icosahedrons.

5. As in Claim 4 where the interface elements are either dimensions of connecting straight line segments, or angles between planes or panels connected along a straight line segment.

6. As in Claim 5 where the interface elements deviate from the pure definition of Claim 2 but retain 26 sides and 16 vertices in the overall structure.

7. As in Claim 5 or 6 including all the resultant vertices of they structure and their locations in physical relative or absolute space relative to each other as a set.

8. As in Claim 5 or 6 where the structure is composed of a solid or semi-solid body.

9. As in Claim 5 or 6 where the structure is composed of a wireframe body.

10. As in Claim 5 or 6 where a representation of the wireframe body is extruded outward into a thickness resulting in a hollow representation that is a shell defined by the inner vertices.

11. As in Claim 5 or 6 where a representation of the wireframe body is extruded inward into a thickness resulting in a hollow representation that is a shell defined by the outer vertices.

12. Any shape or structure that is a substructure of the structure of Claim 5 or 6 made up of two or more connected or not connected panels but that are not in the shape of any configuration of connected or not connected panels of a Pentagonal Pyramid.

13. A transformation of Claim 3 where the first 2 differently sized isosceles triangles are increased in size of the third side and the second 2 differently sized isosceles triangles are decreased in size of the third side, so that they equal, making the 2 quadrilaterals transform into equal triangles as they, as well such that the 20 identical equilateral triangles retain configuration as two joined half Icosahedrons joined at interface points by the said 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 vertice, thereby reducing the vertice count of this version of the Icosahexahedron which in still retaining 26 sides, now has one less vertex to total 15.

14. As in Claim 5 or 6 where the structure is used as a macro building structure.

15. As in Claim 5 or 6 where the structure is used as a micro structure.

16. As in Claim 5 or 6 where the structure is used as a molecular structure.

17. As in Claim 5 or 6 where the structure is used in an artistic representation.

18. As in Claim 5 or 6 where the dimension of the interface elements relate to each other in ratios of different spectral vibration frequencies such as sub-sonic, music, broadband, microwave, light, or higher

19. As in Claim 2, 3 uniquely shaped internal planes made up of: a) an equatorial plane perfectly dissecting the structure about natural vertex connections resulting in a 12 sided, elliptical polygon shape, and b) two different planes defined by natural vertex connections of the structure which are equidistant to the equatorial plane in opposite directions, both of which are different 6 sided elliptical polygons, and transform mathematically into each other through relationship with the equatorial.

20. As in Claim 5 the three different shapes resulting from interfacing two Icosahedrons into the invention, where a) one shape is an isosceles triangle with two equal sides T
and a third side equal to 1.0512T, b) a second isosceles triangle with two equal sides T and a third side 1.0512T/2, and a quadrilateral made from 3 of the latter triangles arranged alternatingly point to point.

21. As in Claim 14 - 17 where the invention is dissected about one of the 3 internal planes where the adjacent wall of the structure makes an angle of 79.1876 degrees to any flat surface extending outward beyond the plane.

22. As in Claim 21 where the invention is dissected in the same plane as the equatorial but in any direction still remaining within the boundaries of any vertex of the structure, in either direction perpendicular to the plane of the equatorial.

23. As in Claim 14-17 where the panels of the invention are divided at the midpoints of their sides resulting in internal panels.

24. As in Claim 23 where the panels are divided at any point along their sides resulting in internal panels of various shapes.

25. As in Claim 23 in a macro structure where the internal panel struts are joined at the vertices with a simple bolt fastener pattern.

26. As in Claim 25 where the internal panels are joined together at their vertices with a simple bolt fastener pattern resulting in any of the panels of Claim 5 or 6.

27. As in Claim 10 or 11 where the gap resulting from extrusion of panels into a shell thickness is converted into a hollow triangular tube by the placement of a longitudinal plane of some thickness, or of a thickness matching the thickness of the extrusion resulting in a solid triangular beam.

28. As in Claim 27 where the gap is filled with spaced triangular supports which fit the insertion angle of the triangular gap such that one side of the triangular gap is open in the direction of the extrusion but supported at the distance of each placement of the supports.

29. As in Claim 14 where the structure is used in wind-resilient applications due to its aerodynamic shape.

30. As in Claim 3 a transformation of the invention into the shape of the Star Constellation Orion as viewed from Earth.

31. As in Claim 30 the shapes, angles, and dimensions resulting.