Catalan Solid and Its Generalization
Jenchung
Chuan
http://140.114.32.246/d2/ATCM%202015/
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 dualpair 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 








































snub tetrahedron 
pentagonal icositetrahedron 




Orthogonal Preserving Distortion of 
Orthogonal Preserving Distortion of Tetrahedron and Its Dual 
Orthogonal
Distortion of CubeOctahedron 
Orthogonality
Preserving Distortion of DodecahedronIcosahedron 
Orthogonal Preserving Distortion of Other Pairs 


Distortion 
Triangular
Cupola 

Triangular
Orthobicupola 
Triangular
OrthobicupolaTrapezoRhombic 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.
1.
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.