Catalan Solid and Its Generalization

Jen-chung Chuan

Introduction

A Catalan solid is a dual polyhedron to an Archimedean solid. A list of 13 Catalan solids together with their Archimedean duals can be found in Wikipedia: https://en.wikipedia.org/wiki/Catalan_solid In this talk, we are to show the most efficient ways to construct all 13 dual pairs of polyhedrons, under the environment provided by the dynamic solid geometry software Cabri 3D. We then proceed to show with animations how each of such dual pairs can be distorted to form other generalized dual-pair polyhedrons families. Here we adopt the notion that two polyhedrons A and B are said to form a (generalized) dual pair if they have the same number of edges and each edge of A intersects a unique edge of B orthogonally in space.

Results

Archimedean Solid	Castalan Solid	Duality	Orthogonality Preserving Distortion
truncated tetrahedron	triakis tetrahedron
truncated cube	triakis octahedron
truncated cuboctahedron	disdyakis dodecahedron
truncated octahedron	tetrakis hexahedron
truncated dodecahedron	triakis icosahedron
truncated icosidodecahedron	disdyakis triacontahedron
truncated icosahedron	pentakis dodecahedron
cuboctahedron	rhombic dodecahedron
icosidodecahedron	rhombic triacontahedron
rhombicuboctahedron	deltoidal icositetrahedron
rhombicosidodecahedron	deltoidal hexecontahedron
snub tetrahedron	pentagonal icositetrahedron
snub dodechedron	pentagonal hexecontahedron

Orthogonal Preserving Distortion of
Regular Poyhedron and Its Dual

Orthogonal Preserving Distortion of Tetrahedron and Its Dual

Orthogonal Distortion of Cube-Octahedron

Orthogonality Preserving Distortion of Dodecahedron-Icosahedron

Orthogonal Preserving Distortion of Other Pairs

Distortion

Triangular Cupola

Some Distorted Triangular Cupola Has Dual

Square Cupola

Some Distorted Square Cupola Has Dual

Pentagonal Cupola

Some Distorted Pentagonal Cupola Has Dual

Triangular Orthobicupola

Triangular Orthobicupola-Trapezo-Rhombic Dodecahedron Distorted Symmetrically

Process of Distortion

Step 1. Construct the most symmetric pair of dual polyhedrons.

Step 2. Since all edges are tangent to a same sphere S, their faces meet S in two orthogonal families of circles.

Step 3. By taking inversion of circles with respect to S, two orthogonal families of circles are transformed to two new orthogonal families of circles.

Step 4. Construct the two polyhedrons corresponding two the circles.