Catalan Solid and Its Generalization

Jen-chung Chuan




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: 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.



Archimedean Solid

Castalan Solid



Orthogonality Preserving Distortion­

truncated tetrahedron

triakis tetrahedron


描述 : Server HD:Users:jcchuan:Desktop:ATCM 2015:truncated tetrahedron and its dual  triakis tetrahedron_html.png

Orthogonality Preserving Distorsion of Truncated Tetrahedron-Triakis Tetrahedron_html

truncated cube

triakis octahedron


描述 : Server HD:Users:jcchuan:Desktop:ATCM 2015:truncated cube and its dual triakis octahedron_html.png

Orthogonality Preserving Distorsion of Truncated Cube-Triakis Octahedron_html

truncated cuboctahedron

disdyakis dodecahedron


truncated cuboctahedron and its dual disdyakis dodecahedron_html

Orthogonality Preserving Distorsion of Truncated Cuboctahedron-Disdyakis Dodecahedron_html

truncated octahedron

tetrakis hexahedron


truncated cuboctahedron and its dual disdyakis dodecahedron_html

Orthogonality Preserving Distorsion of Truncated Octahedron-Tetrakis Hexahedron_html

truncated dodecahedron

triakis icosahedron


truncated dodecahedron and its dual triakis icosahedron_html


truncated icosidodecahedron

disdyakis triacontahedron


 truncated icosidodecahedron and Its dual disdyakis triacontahedron_html


truncated icosahedron

pentakis dodecahedron


truncated icosahedron and its dual pentakis dodecahedron_html



rhombic dodecahedron


cuboctahedron and its dual rhombic dodecahedron_html

Orthogonal Preserving Distorsion of Cuboctahedron-Rhombic dodecahedron_html


rhombic triacontahedron


icosidodecahedron and its dual rhombic triacontahedron_html

Orthogonality Preserving Distorsion of Icosahedron-Rhombic Triacontahedron_html


deltoidal icositetrahedron



deltoidal hexecontahedron


rhombicosidodecahedron and its dual deltoidal hexecontahedron_html

Orthogonality Preserving Distorsion of Rhombicosidodecahedron-Deltoidal Hexecontahedron_html

snub tetrahedron

pentagonal icositetrahedron


snub dodechedron

pentagonal hexecontahedron




Orthogonal Preserving Distortion of
Regular Poyhedron and Its Dual

Tetrahedron and Its Dual_html

Orthogonal Preserving Distortion of Tetrahedron and Its Dual

Orthogonal Distorsion of Cube-Octahedron_html

Orthogonal Distortion of Cube-Octahedron

Orthogonal Distorsion of Dodecahedron-Icosahedron_html

Orthogonality Preserving Distortion of Dodecahedron-Icosahedron




Orthogonal Preserving Distortion of Other Pairs



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.