Intertransformability

Year: 2010 Authors: John A. Hiigli

Core claim

A subdivision of tetrahedra and octahedra into smaller blocks enables freely interconvertible geometric states and further related polyhedral transformations.

Topics

polyhedral transformation, geometric block system, intertransformability, structural forms

Domains

solid geometry, polyhedra, tessellation, geometric design, construction systems, visual form transformation

Methods

block subdivision, state transformation, rotational reassembly, polyhedral decomposition

Media

tetrahedral blocks, octahedral blocks, patent drawings

Paper text

The text below is the locally extracted OCR/Markdown version of the paper. Raw PDF files remain local and are not published here.

Bridges 2010: Mathematics, Music, Art, Architecture, Culture

Intertransformability

John A. Hiigli

164 West 83rd Street,

Apt 1R New York, NY, 10024, USA

E-mail: john@jardingalerie.org

Abstract

Following is a brief description of a single transformation of structural forms into other forms that are possible with a building block system described in a United States Patent granted October 5, 1993:

GEOMETRIC BUILDING BLOCK SYSTEM EMPLOYING SIXTEEN BLOCKS, EIGHT EACH OF ONLY TWO TETRAHEDRAL SHAPES, FOR CONSTRUCTING A REGULAR RHOMBIC DODECAHEDRON.

A geometric block system for constructing a regular rhombic dodecahedron having twelve identical rhombic faces, the system consisting of eight identical first blocks each being one-quarter of a regular tetrahedron and eight identical second blocks each being one-eighth of a regular octahedron, as illustrated below in the first drawing for USPTO Patent # 5, 249,966.

Hiigli

The Transformations

Figure I: Duo-tetrahedron

Figure II: Cuboctahedron

Figure III: Cuboctahedron

Figure: Duo-tetrahedron

Two blocks in four sets of sixteen blocks in each set, thus consisting of thirty-two ¼ T blocks and thirty-two 1/8 O blocks may be used to create two basic forms: in the first state (State “A”), the result is a “duotetrahedron” formed from an octahedron surrounded by eight tetrahedra (as shown in FIG. I). In the second state (State “B”), the result is a cube octahedron (please see Fig. II). Note that these two States A and B are freely interconvertible. In a variant on the transformation process involving the “duotetrahedron” shown in FIG. I, the complex cube shown in FIG. IV can be broken down into the “duotetrahedron” shown in FIG. I plus three octahedra. In a parallel transformation, the complex cube of FIG. III breaks down into the cuboctahedron of FIG. II, with a remaining octahedron. We can in fact divide the cube in either phase into two halves by separating along any one of three orthogonal planes. After the two halves have been separated, each half is rotated 180 degrees around a vertical axis, and the two halves are brought back together to reveal the other phase. That is, the duo-tetrahedron of “State A” freely may be converted into the cube octahedron of “State B” and vice versa. This phenomenon of “intertransformability” is the direct result of the subdivision of the tetrahedron into fourths and the octahedron into eights. Further subdivision and additional transformations are possible. In addition to the transformations described above these respective geometric State A and State B configurations can be converted into:

(a) eight tetrahedra plus four octahedra; (b) a cuboctahedron plus one octahedron; (c) a double tetrahedron plus three octahedra; (d) four isosceles dodecahedra and four octahedra; (e) eight cubes each formed from a tetrahedron consisting of four ¼ T blocks surrounded by four 1/8 O blocks; (f) four rhombic dodecahedra; (g) two complex tetrahedra, each made of four tetrahedra, each of which is formed from four ¼ T blocks, surrounding an octahedron formed with eight 1/8 O blocks, with two remaining octahedra.

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