Shape Computation Lab

Four Eights

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01. Four shapes with equal order of symmetry (a) Shaved octagonal truncated pyramid. (b) Isosceles tetrahedron. (c) Shaved square prism. (d) Rectangular prism.

02. The octagonal prism as the underlying geometric structure of the four shapes of the previous figure. (a) Shaved octagonal truncated pyramid. (b) Isosceles tetrahedron (c) Shaved square prism (d) Rectangular prism.

03. Pictorial representations of the symmetry elements of four shapes. (a) Shaved octagonal truncated pyramid. (b) Isosceles tetrahedron (c) Shaved square prism (d) Rectangular prism.

04. Diagrammatic representations of the four algebraic structures of order 8. (a) Shaved octagonal truncated pyramid. (b) Isosceles tetrahedron (c) Shaved square prism (d) Rectangular prism. The corresponding 7 symmetry classes of order 8 are shown as well.

05. Lattices of subgroups of four shapes with symmetry 8. (a) Shaved octagonal truncated pyramid. (b) Isosceles tetrahedron (c) Shaved square prism (d) Rectangular prism.

06. Some shapes rules instantiating 3d shapes to substitute the labels in the partial order lattices of order 8.

07. Partial order set of parts used for the generation of a design in one of the lattices of order 8.

08. Partial order set of parts used for the generation of a design in one of the lattices of order 8.

09. Axonometric views of a design in the language.

Athanassios Economou

2002

Keywords: Symmetry; Prismatic groups; Configurations; Shape grammars

A constructive program for the generation of three-dimensional languages of designs based on nested group structures is outlined using the four possible algebraic infinite symmetry structures of 3-dimensional space, that is, the cyclic and dihedral groups and their direct product groups, and their corresponding seven prismatic groups in Euclidean space. The pedagogical material and the studies themselves focus on a specific order, here the order 8, to showcase the expressiveness of these structures in design.