8/7/2019 MathPoster

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C 2005 Gregg F

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The RHOMBICUBERhombic Dodecahedron

(A)Imagine a checkerboard where the blacksquares are solid cubes and the white squaresare empty cubes. Then place another layer ontop with the solid cubes on top of the emptycubes, etc. This could be called a 3Dcheckerboard. Nowdraw fourlinesdiagonallyin the empty cubes. These all intersect at thecenters and form pyramidal shapes off eachface. If you expand the solid cubes into thepyramidal shapes adjacent to each face, youform modules with twelve diamond shapedfaces we call the RHOMBICUBE. Originallycalleda RhombicDodecahedron,it isclassedas a Catalan solid first described by an 18thcentury Flemish mathematician of that name.

(C)When the points are cut off of the diamondfaces uniformly, we have the TRUNCATEDRHOMBICUBE (smal l rhombicubocta-hedron). Inadditionto thetwelve squarefacesin place of the diamond faces we have sixsquare faces that relate to the original cubefacesandeighttrianglesthatoccur inwardfromthe original cube corners making a total of 26faces. This form is classed as an ArchimedeanSolid, in which the faces are different thoughregular while the vertices are identical. It wasrediscovered in the 15th century by Pacioli andeven drawn by Leonardo. In the 20th century,postBuckminsterFuller,itwasshownbyseveralpeople that the TRUNCATEDRHOMBICUBE,when placed in the 3D array, nests with otherslike itself plus a variety of similarly configuredforms. In the basic checkerboard array it nestswith cubes (C) and tetrahedrons (T).

This family of shapes, including the other

different combinations following in Sections(D)thru(G) arecalled SpaceFillersandwerefirst comprehensively compiled by Robert E.Wilson in his Handbook of Structure Part I:Polyhedra and Spheres, 1968, from DouglasAdvanced Research Laboratories. The firstTRUNCATED RHOMBICUBE by Fleishmandates to 1972. It was transformed to a rhombicdodecahedron and subsequently used inspace filling arrays in 1995.

The RHOMBICUBE

(E)If we turn the six cubic faces of theTRUNCATED RHOMBICUBE into octagonsand the eight triangle faces into hexagons wearrive at the GT (GREAT TRUNCATED)RHOMBICUBE. Noticehowthe twelveangledsquaresremain,butarespacedapartasifcubeshad been inserted. This form also has 26 faces,the same as the TRUNCAED RHOMBICUBE.

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The Truncated Rhombicube

The GT Rhombicube

Forms of the RHOMBICUBE based cluster-structures

(B)Because the faces of these RHOMBICUBESare aligned at 45 degrees to the original cubefaces, the exact dimensions of the faces can bed e t e r m i n e d e a s i ly b y u t i l i z i n g t h ePYTHAGOREAN THEOREM. Note that if wehave constructed the diamonds using a two(2) unit cube, the short axis of each diamondis 2, the long axis is and the edge lengthis . Another interesting property of themoduleisthattherearetwoorientationswherecross sections result in regular square orhexagonal grids.

(D)When the TRUNCATED RHOMBICUBES areplaced as if the original sol id cubes are moreadjacent to one another, the smaller cubesoccur off the angled faces and different largerfourteenfacedformswithsix squaresandeighttriangles, cuboctahedrons, now occur off thecorners in between. The drawing indicates aframelikesuperstructureofthese smallerformswhen the larger T RHOMBICUBES are left asvoided areas in between.

(F)The GT RHOMBICUBE can also be placed ineither the tightly adjacent array or thecheckerboard array as before. In the adjacentarray (F) the small cubes again fit in between atthe edges but a second fourteen faced form, atruncated octahedron (TO), fits at the corners.Again note that the drawing indicates aframelike superstructure with the larger GTRHOMBICUBES left as voided areas inbetween.

(G)In the checkerboard array the spaces acorners are filled with an eight faced forma truncated tetrahedron, and the origempty cube spaces become a third foufaced form, a truncated cube (TC), wioctagons and eight triangles. If the hexapanel of the TT becomes a triangle, thewill meet edge to edge and octagons wilbetween. Including prisms, Wilson lisdifferent space filling arrays.

(T) (CO) (TO) (TT) (TC)

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think big(A) (B) (C) (D)

(E) (F) (G)

(O) (C)

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