Leonhard Euler discovered in the 18th century that for polyhedra in which any face is a hexagon or a pentagon and every vertex is of degree three there must be exactly twelve pentagonal faces. This is referred to as Euler's Twelve Pentagon Theorem. For the proof of this theorem see Euler. There is a dual to this theorem to the effect that for a polyhedron with only triangular faces and all vertices are of degree five or six there must be exactly twelve vertices of degree five. For the proof of this theorem see Euler2.

There are other theorems of these sorts. First let us consider a few specific cases. Consider polyhedra with only triangular and square faces. Let the number of these faces be denoted as f₃ and f₄, respectively. Let the number edges be e. All vertices are assumed to be of degree 4 and this number is denoted as v₄.

Euler's formula for polyhedra requires

f₃ + f₄ − e + v₄ = 2

If the edges are counted face-by-face the total will be 3f₃+4f₄. But each edge is counted exactly twice so the total is equal to 2e. Therefore

3f₃ + 4f₄ = 2e

If the edges are counted vertex-by-vertex the total will be 4v₄. But that also is equal to 2e. Thus

4v₄ = 2e
or, equivalently
2v₄ = e
and hence
v₄ - e = −v₄

The first equation involves the term v₄-e. When −v₄ substituted for (v₄-e) in the first equation the result is

f₃ + f₄ − v₄ = 2

The second and third equation can be combined to give

3f₃ + 4f₄ = 4v₄

If the preceding equation is multiplied by 4 the two equations to be satisfied are

4f₃ + 4f₄ − 4v₄ = 8
and
3f₃ + 4f₄ − 4v₄ = 0

This is a set of two equations in three unknows. Typically for such a situation no definite result can be derived. However, subtracting the second equation above from the first gives

f₃ = 8

In words, any polyhedron whose faces are either triangles and squares and whose vertices are all of degree 4 must have exactly 8 triangles. Two of the Archimedean polyhedra satisfy the conditions of this proposition. One is the Cuboctahedron, having 8 triangular faces, 6 square faces, 24 edges and 12 vertices. Another is the small rhombicuboctahedron having 8 triangular faces, 18 squares, 48 edges and 24 vertices. One of the Platonic polyhedra also sastisfies the conditions although the number of its square faces is zero. That is the octahedron with 8 triangles, 0 squares, 12 edges and 6 vertices.

There is another Archimedean polyhedron made up entirely of triangles and squares but with degree five vertices. It is called the snub cube. The analysis of its case does not produce a definite result. The equations for its case are:

f₃ + f₄ − e + v₅ = 2
3f₃ + 4f₄ = 2e
5v₅ = 2e

These can be converted into

2(v₅-e) = −3v₅
2f₃ + 2f₄ − 3v₅ = 4
3f₃ + 4f₄ − 5v₅ = 0

And finally to:

4f₃ + 4f₄ − 6v₅ = 8
3f₃ + 4f₄ − 5v₅ = 0

Subtraction of the second equation above from the first does not lead to a definite value for one of the variables. Instead

f₃ − v₅ = 8

So, only for certain polyhedra can a conclusion analogous to Euler's Twelve Pentagon Theorem be drawn.

A Generalization of Euler's Twelve Pentagon Theorem

Consider a polyhedron made up of n-gons and m-gons with all vertices of degree k. The equations to be satisfied are then

f_n + f_m − e + v_k = 2
nf_n + mf_m = 2e
kv_k = 2e

Thus

2(v_k-e) = −(k-2)v_k
2f_n + 2f_m −(k-2)v_k = 4
nf_n + mf_m − kv_k = 0

To eliminate f_m the last equation can be multiplied by 2 and the preceding equation by m to get

2mf_n + 2mf_m −m(k-2)v_k = 4m
2nf_n + 2mf_m − 2kv_k = 0

The number of vertices and the number of one type of face will be simultaneously eliminated only if

m(k-2) = 2k
or, equivalently
k = 2m/(m-2)

For Euler's Twelve Pentagon Theorem m=6 and k=3. As can be seen these values satisfy the equation. For the case of the triangles and squares considered previously m=4 and k=4. If m=3 then k has to be 6, but the interior angle of any polygons is too large to bring six polygons together at one vertex. Therefore m cannot be 3.

The value of m cannot be 5 because 10/3 is not an integer. Likewise m cannot be 8 because 16/6 is not an integer and the same holds true for any value of m greater than 8. Thus m has to be 4 or 6.

When m is 4 then the subtraction of the equations yields

2(4-n)f_n = 16
or, equivalently
f_n = 8/(4-n)

There is an integral value of 8 for n=3 but for no other value of n. This is the case which was considered first above; i.e., polyhedra of triangles and squares with vertices of degree four.

When m is 6 then k must be 3 and the subtraction of the equations yields

2(6-n)f_n = 24
or, equivalently
f_n = 12/(6-n)

There is an integral solution for n=5; i.e, f_n=12. This is the Euler twelve pentagon case. Examples of this case are the dodecahedron with 12 pentagons and 0 hexagons and the truncated icosahedron with 12 pentagons and 20 hexagons.

dodecahedron

truncated icosahedron

The cases of a polyhedra made up of triangles and pentagons with vertices of degree 4 or 5, the icosideodecahedron and snub icosidodecahedron, fit in with the Euler case. In both cases these polyhedra have 12 pentagonal faces.

icosidodecahedron

snub dodecahedron

There are also solutions to the equation f_n = 12/(6-n) for n=4 and n=3. (There are no polygons for n<3 so the solutions for n=2 and n=1 are not relevant.)

The case for n=4, m=6 and k=3 is that of the truncated octahedron with 8 hexagons and 6 squares. It is also the case of the cube with 0 hexagons and 6 squares. The value of f₄=12/(6-4) is 6.

cube

truncated octahedron

The case of n=3, m=6 and k=3 is that of the truncated tetrahedron with 4 hexagons and 4 triangles. It is also the case of the tetrahedron with 0 hexagons and 4 triangles. The value of 12/(6-n) for n=3 is 4.

tetrahedron

truncated tetrahedron

For the polyhedra involving triangles and squares (the case of n=3, m=4 and k=4) there are the examples of the cuboctahedron with 8 triangles and 6 squares, the small rhombicuboctahedron with 8 triangles and 18 squares. It is also the case of the octahedron with 8 triangles and 0 squares.

octahedron

cuboctahedron

small rhombicuboctahedron

Thus polyhedra of triangles and squares with vertices of degree 4 must have exactly 8 triangles. There is another polyhedron made up of triangles and squares but with the degree of the vertices being 5 instead of 4. This polyhedron, the snub cube has 32 triangles and 6 squares.

There is the case polyhedra made up of triangles and hexagons, the case of n=3, m=6 and k=4. This is the the case of the of the truncated tetrahedron with 4 hexagons and 4 triangles. It is also the case of the tetrahedron with 0 hexagons and 4 triangles. The value of 12/(6-n) for n=3 is 4.

tetrahedron

truncated tetrahedron

So polyhedra made up of triangles and hexagons and vertices of degree 4 must have exactly 4 triangles.

There are Archimedean polyhedra involving three types of faces. These are not considered in this analysis. There are also cases of the truncated dodecahedron which has triangular and decagonal faces and the snub dodecahedron with triangular and pentagonal faces. These do not have analogs of Euler's Twelve Pentagon Theorem.

The dual theorems concerning polyhedra with one type of face but vertices of possibly two different degrees is dealt with elsewhere.

Conclusions

The theorems are of a conditional sort. If there exists polyhedra made up of only triangular and square faces and having vertices of only degree four then there must be exactly eight triangular faces. There are at least three polyhedra satisfying these conditions.

Any polyhedron whose faces are either squares or hexagons and whose vertices are all of degree three must have exactly six square faces. The cube and the truncated octahedron are examples of this case.

Any polyhedron whose faces are either triangles or hexagons and whose vertices are all of degree three must have exactly four triangular faces. The tetrahedron and the truncated tetrahedron are examples of this case.