Vertex Configurations: Platonic Solids, Archimedean Solids, and Johnson Solids

Last week, we explored the world of Euler’s Formula: the relationship between Vertices + Faces — Edges = 2 in any convex polygon. The children were tasked to find polyhedra (solids) in their home and test for vertices, faces and edges. If the solid was concave, Euler’s Formula doesn’t necessarily work.

This week we built on our polyhedra discussion by exploring the famous discovery by Plato around 400 B.C. There are only five convex polyhedra for which each face is the same regular polygon and all vertices are of the same type. These are known as the Platonic polyhedra, or Platonic solids. These were known before Plato (427-348 B.C.) but he and his students are known to have studied them, and details regarding the original discovery of these polyhedra are not known. The five are:

• Tetrahedron - 4 faces, each an equilateral triangle
• Hexahedron (Cube) - 6 faces, each a square
• Octahedron - 8 faces, each an equilateral triangle
• Dodecahedron - 12 faces, each a regular pentagon
• Icosahedron - 20 faces, each an equilateral triangle

The children noticed that the hexahedron and octahedron have the same of number faces as the number of vertices of the other so they can inscribe perfectly in each other. Also, the dodecahedron and icosahedron have the same of number faces as the number of vertices of the other so they can inscribe perfectly in each other. Since the tetrahedron has the same number of faces and vertices, a tetrahedron can inscribe in another tetrahedron.

Then, along comes Archimedes. There are thirteen convex polyhedra for which each face is a regular polygon, with more than one type of regular polygon, and for which all vertices are of the same type. In addition, there are an infinite number of facially regular prisms and antiprisms. These thirteen solids were known to Archimedes (c 290/280 - 212/211 B.C.) but his writings on them were lost, together with the knowledge of their construction. They were gradually rediscovered during the Renaissance. I constructed 11 of the 13 Archimedean Solids and the children noticed that most of them are based on the Platonic Solids. For example, there are truncated tetrahedrons, truncated cubes, truncated icosahedron, truncated octahedron, and truncated dodecahedron, which cut off the vertices of the Platonic solid to create an Archimedean Solid.

Then, I taught the children how to identify polyhedra by their vertex configuration. In geometry, a vertex configuration is a shorthand notation for representing the vertex figure of a polyhedron or tiling as the sequence of faces around a vertex. For uniform polyhedra there is only one vertex type and therefore the vertex configuration fully defines the polyhedron.

A vertex configuration is given as a sequence of numbers representing the number of sides of the faces going around the vertex. The notation "a.b.c" describes a vertex that has 3 faces around it, faces with  a, b, and c  sides.

For example, "3.5.3.5" indicates a vertex belonging to 4 faces, alternating  triangles and pentagons. This vertex configuration defines the  icosidodecahedron. The notation is cyclic and therefore is equivalent with different starting points, so 3.5.3.5 is the same as 5.3.5.3. The order is important, so 3.3.5.5 is different from 3.5.3.5. (The first has two triangles followed by two pentagons; the second is a triangle followed by a pentagon, followed by a triangle, followed by a pentagon.).

It is easier to do the vertex configuration from a picture. So I tasked the children to use the Archimedean Solids I created and find each solid on the worksheet with all of these solids pictured in two dimensions. Then they did the vertex configuration of each. They identified the two solids I could not make do to lack of material. 

Finally, we studied the 92 solids identified by a mathematician named Johnson. In geometry, a Johnson solid is a strictly convex polyhedron, which is not uniform (i.e., not a Platonic solid, Archimedean solid, prism, or antiprism), and each face of which is a regular polygon. There is no requirement that each face must be the same polygon, or that the same polygons join around each vertex. An example of a Johnson solid is the square-based pyramid with equilateral sides (J₁); it has 1 square face and 4 triangular faces.

As in any strictly convex solid, at least three faces meet at every vertex, and the total of their angles is less than 360 degrees. Since a regular polygon has angles at least 60 degrees, it follows that at most five faces meet at any vertex. The pentagonal pyramid (J₂) is an example that actually has a degree-5 vertex.

Although there is no obvious restriction that any given regular polygon cannot be a face of a Johnson solid, it turns out that the faces of Johnson solids always have 3, 4, 5, 6, 8, or 10 sides.

In 1966, Norman Johnson published a list which included all 92 solids, and gave them their names and numbers. He did not prove that there were only 92, but he did conjecture that there were no others. Victor Zalgaller in 1969 proved that Johnson's list was complete.

Of the Johnson solids, the elongated square gyrobicupola (J₃₇), also called the pseudorhombicuboctahedron,^[1] is unique in being locally vertex-uniform: there are 4 faces at each vertex, and their arrangement is always the same: 3 squares and 1 triangle. However, it is not vertex-transitive, as it has different isometry at different vertices, making it a Johnson solid rather than an Archimedean solid.

The children were tasked this week to find the multiple vertex configurations in each Johnson solid.