Archimedean Solid

Exploration 1: Transforming Platonic Solids

Material: Play dough, plastic knife, digital Platonic solid figures (ref. below). Instructions:

Form a tetrahedron, cube, and an octahedron using play dough.
Cut off every vertex on each solid formed.
Observe the changes that happened in each solid in terms of vertices, edges, and faces.

Concept focus: The process of cutting off the vertices is called truncation. When Platonic solids are truncated, they become truncated Platonic solids.

Truncated Tetrahedron

What happens to a truncated tetrahedron?

The original tetrahedron has 4 vertices, 6 edges, and 4 triangular faces. When every vertex is truncated... Faces: The 4 original triangular faces become 4 hexagons, while the truncation process creates 4 new triangles at the corners, totaling 8 faces. Vertices: Each of the 4 original vertices is replaced by 3 new ones, resulting in 12 vertices. Edges: The 6 original edges are shortened, and 3 new edges are created at each of the 4 truncated corners, totaling 18 edges.

Truncated Cube

What happens to a truncated cube?

The original cube has 8 vertices, 12 edges, and 6 square faces. When every vertex is truncated... Faces: The 6 original squares become 6 octagons, while the truncation process creates 8 new triangles at the corners, totaling 14 faces. Vertices: Each of the 8 original vertices is replaced by 3 new ones, resulting in 24 vertices. Edges: The 12 original edges are shortened, and 3 new edges are created at each of the 8 truncated corners, totaling 36 edges.

Truncated Octahedron

What happens to a truncated octahedron?

The original octahedron has 6 vertices, 12 edges, and 8 triangular faces. When every vertex is truncated... Faces: The 8 original triangles become 8 hexagons, while the truncation process creates 6 new squares at the corners, totaling 14 faces. Vertices: Each of the 6 original vertices is replaced by 4 new ones, resulting in 24 vertices. Edges: The 12 original edges are shortened, and 4 new edges are created at each of the 6 truncated corners, totaling 36 edges.

Truncated Dodecahedron

What happens to a truncated dodecahedron?

The original dodecahedron has... When every vertex is truncated...

What happens to a truncated icosahedron?

The original icosahedron has... When every vertex is truncated...

The 4 Cubocta Family

Besides truncating a cube, the parent cube can be transformed through four different ways. *The transformation can be observed using the applet below (credits to Roman Chijner). A. Midpoint truncation (rectification) of the cube - Cuboctahedron Cut off the corner exactly to the midpoint of each edge. The cube will now look like...

12 vertices
24 edges
14 faces (8 triangles, 6 squares)

B. Truncation of a cuboctahedron - Truncated Cuboctahedron / Great Rhombicuboctahedron Slightly truncate the vertices, edges, and faces simultaneously. The solid will now look like...

48 vertices
72 edges.
26 faces (6 octagons, 8 hexagons, and 12 squares)

C. Expansion of the cube - Rhombicuboctahedron / Small Rhombicuboctahedron Imagine pulling the 6 square faces of the cube outward from the center and filling the gaps (cantellate the cube). Or truncate the truncated cuboctahedron exactly to the midpoint of each edge. The solid will now look like...

24 vertices
48 edges.
26 faces (8 triangles and 18 squares)

D. Rotation & Snubbing of the cube - Snub Cube Pull and twist the faces of the cube. The gaps are then filled with equilateral triangles. Or, twist the faces of the rhombicuboctahedron and fill the gaps with triangles. The solid will now look like...

24 vertices
60 edges.
38 faces (32 triangles and 6 squares)

Note (ref. applet below): Move the slider 't' to observe the transformation from a cube into a cuboctahedron. To form a snub cube, move the slider 'a' until the faces are twisted and gaps are filled.

The 4 Icosidodeca Family

Start with a Dodecahedron (12 pentagons) or an Icosahedron (20 triangles) as the parent. Since the Dodecahedron and Icosahedron are duals, performing the following transformations to either one results in the exact same result. *The transformation can be observed using the applet below (credits to Roman Chijner). A. Midpoint truncation (rectification) of the dodecahedron- Icosidodecahedron Cut off the corner exactly to the midpoint of each edge. The cube will now look like...

30 vertices
60 edges
32 faces (20 triangles, 12 pentagons)

B. Truncation of a icosidodecahedron - Truncated Icosidodecahedron / Great Rhombicosidodecahedron Slightly truncate the vertices, edges, and faces simultaneously. The solid will now look like...

120 vertices
180 edges
62 faces (30 squares, 20 hexagons, 12 decagons)

C. Expansion of the cube - Rhombicosidodecahedron / Small Rhombicosidodecahedron Imagine pulling the faces of the dodecahedron outward from the center and filling the gaps (cantellate the parent). Or truncate the truncated Icosidodecahedron exactly to the midpoint of each edge. The solid will now look like...

60 vertices
120 edges
62 faces (20 triangles, 30 squares, 12 pentagons)

D. Rotation & Snubbing of the dodecahedron - Snub Dodecahedron Pull and twist the faces of the dodecahedron. The gaps are then filled with equilateral triangles. Or, twist the faces of the rhombicuboctahedron and fill the gaps with triangles. The solid will now look like...

60 vertices
150 edges
92 faces (80 triangles, 12 pentagons)

Note (ref. applets below): Move the slider 't' to observe the transformation from a dodecahedron into a truncated icosidodecahedron, and finally into a rhombicosidodecahedron . Then, move the slider 'a' until the faces are twisted and gaps are filled to see the snub dodecahedron.

Summary of The 13 Archimedean Solids

The 5 Truncated Platonic Solids

Truncated Tetrahedron: 4 triangles, 4 hexagons.
Truncated Cube: 8 triangles, 6 octagons.
Truncated Octahedron: 6 squares, 8 hexagons.
Truncated Dodecahedron: 20 triangles, 12 decagons.
Truncated Icosahedron: 12 pentagons, 20 hexagons (e.g., football).

The 4 Cubocta Family

Cuboctahedron: 8 triangles, 6 squares.
Truncated Cuboctahedron / Great Rhombicuboctahedron: 12 squares, 8 hexagons, 6 octagons.
Rhombicuboctahedron / Small Rhombicuboctahedron: 8 triangles, 18 squares.
Snub Cube: 32 triangles, 6 squares.

The 4 Icosidodeca Family

Icosidodecahedron: 20 triangles, 12 pentagons.
Truncated Icosidodecahedron / Great Rhombicosidodecahedron: 30 squares, 20 hexagons, 12 decagons.
Rhombicosidodecahedron / Small Rhombicosidodecahedron: 20 triangles, 30 squares, 12 pentagons.
Snub Dodecahedron: 80 triangles, 12 pentagons