Platonic Solids

A regular polygon is a polygon that has all congruent angles and all congruent sides.  For example, a square is a regular polygon because it has 4 congruent angles and 4 congruent sides, and an equilateral triangle is a regular polygon because it has 3 congruent angles and 3 congruent sides. 

A Platonic solid, or a regular convex polyhedron, is a three-dimensional convex solid that has identical regular polygons for each face.  For example, a cube is a Platonic solid because it has 6 identical square faces.  There are five possible Platonic solids in all: the tetrahedron, the cube, the octahedron, the dodecagon, and the icosahedron.  These five Platonic solids have been known for thousands of years.

  

Platonic Solids (Wikipedia)

Finding all the different possible Platonic solids is a great exercise in logic.  First of all, there must be more than 2 faces around a single vertex, because otherwise a three-dimensional object will not be formed.  Secondly, the angle sum of each interior face angle around a single vertex must be less than 360°.  It cannot be more than 360°, because otherwise not all the polygons can fit, nor can it be equal to 360°, because otherwise a three-dimensional object will not be formed.  For example, since each interior angle of an equilateral triangle is 60°, 6 equilateral triangles around a single vertex would form a two-dimensional tiled surface, because 6 · 60° = 360°, and 7 (or more) equilateral triangles would not fit around a single vertex because 7 · 60° > 360°.  Therefore, only 3, 4, or 5 triangles can share a single vertex and form a polyhedron – the tetrahedron, the octahedron, and the icosahedron. 

Platonic Solid Vertex Defects (Wikipedia)

Similarly, since each interior angle of a square is 90°, 4 squares around a single vertex would form a two-dimensional surface, because  4 · 90° = 360°, and 5 (or more) squares would not fit around a single vertex because 5 · 90° > 360°.  Therefore, only 3 squares can share a single vertex and form a polyhedron – the cube.  Also, since each interior angle of a regular pentagon is 108°, 4 (or more) regular pentagons would not fit around a single vertex because 4 · 108° > 360°.  Therefore, only 3 regular pentagons can share a single vertex and form a polyhedron – the dodecahedron.  Since each interior angle of a regular hexagon is 120°, 3 regular hexagons around a single vertex would form a two-dimensional surface, because  3 · 120° = 360°, and 4 (or more) regular hexagons would not fit around a single vertex because 4 · 120° > 360°.  Therefore, no regular hexagons can be used to form a polyhedron.  Since each interior angle of a regular heptagon, octagon, or larger polygon is greater than 121°, 3 (or more) regular heptagons, octagons, or larger polygons would not fit around a single vertex because 3 · 121° > 360°.  Therefore, no other regular polygon can be used to form a polyhedron, making the tetrahedron, the cube, the octahedron, the dodecagon, and the icosahedron the only possible Platonic solids.

Properties of Platonic Solids

When looking at the different properties of Platonic solids, it is helpful to describe each one with its Schläfli symbol {n, m}, where n is the number of sides of each face, and m is the number of faces around each vertex.  For example, the Schläfli symbol for a cube is {4, 3}, because each face has 4 sides and there are 3 squares around each vertex.  The Schläfli symbol for the tetrahedron is {3, 3}, the octahedron {3, 4}, the dodecagon {5, 3}, and the icosahedron {3, 5}.

Using the above variables n and m, as well as a new variable F representing the total number of faces in the polyhedron, more properties of Platonic solids can be expressed.  Since each face has n number of edges, and each edge is shared by exactly 2 faces, then the total number of edges in the polyhedron is

E = ^n·F/₂

For example, a cube is made up of 6 squares, so F = 6 and n = 4, but each edge is shared by 2 squares, so the total number of edges is E = ^n·F/₂ = ^4·6/₂ = 12.  

Additionally, since each face has n number of vertices, and each vertex is shared by exactly m faces, then the total number of vertices in the polyhedron is

V = ^n·F/_m

For example, a cube is made up of 6 squares, so F = 6 and n = 4, but each vertex is shared by 3 squares, so the total number of vertices is V = ^n·F/_m = ^4·6/₃ = 8.

By substituting E and V into Euler’s polyhedral formula

V + F = E + 2

and simplifying, F can be written as a function of n and m, and after that so can E and V.

01	V + F = E + 2	Euler’s polyhedral formula
02	n·F/m + F = n·F/2 + 2	substitution
03	2nF + 2mF = mnF + 4m	multiply by 2m
04	2nF + 2mF – mnF = 4m	subtract mnF
05	F(2n + 2m – mn) = 4m	factor F
06	F = 4m/(2n + 2m – mn)	divide by 2n + 2m – mn

Therefore,

F = ^4m/_{(2n + 2m – mn)}

For example, a cube has 3 squares around a single vertex, so n = 4 and m = 3, and therefore must have F = ^4m/_{(2n + 2m – mn)} = ^4·3/_{(2·4 + 2·3 – 3·4)} = ¹²/₂ = 6 faces.

Substituting F into E, E = ^n·F/₂ = ^n·4m/_{2(2n + 2m – mn)} = ^2mn/_{(2n + 2m – mn)}, so

E = ^2mn/_{(2n + 2m – mn)}

For example, a cube has 3 squares around a single vertex, so n = 4 and m = 3, and therefore must have E = ^2mn/_{(2n + 2m – mn)} = ^2·4·3/_{(2·4 + 2·3 – 3·4)} = ²⁴/₂ = 12 edges.

Substituting F into V, V = ^n·F/_m = ^n·4m/_{m(2n + 2m – mn)} = ⁴ⁿ/_{(2n + 2m – mn)}, so

V = ⁴ⁿ/_{(2n + 2m – mn)}

For example, a cube has 3 squares around a single vertex, so n = 4 and m = 3, and therefore must have V = ⁴ⁿ/_{(2n + 2m – mn)} = ^4·4/_{(2·4 + 2·3 – 3·4)} = ¹⁶/₂ = 8 vertices.

Using the above formulas:

Solid	Schläfli	Faces	Edges	Vertices
tetrahedron	{3, 3}	4	6	4
cube	{4, 3}	6	12	8
octahedron	{3, 4}	8	12	6
dodecahedron	{5, 3}	12	30	20
icosahedron	{3, 5}	20	30	12

Dual Pairs

From the above formulas and chart, it is apparent that there are some symmetric properties to the five Platonic solids.  The cube and the octahedron are called a dual pair because they have the same number of edges (12), swapped Schläfli symbols ({4, 3} and {3, 4}), and swapped number of faces and vertices (6 and 8).  Similarly, the dodecahedron and the icosahedron are also called a dual pair because they have the same number of edges (30), swapped Schläfli symbols ({5, 3} and {3, 5}), and swapped number of faces and vertices (12 and 20).  The same symmetric properties are true for a tetrahedron and another tetrahedron, so the tetrahedron is called self-dual.

Because the number of faces and vertices are swapped for a dual pair of polyhedra, any polyhedron can be placed exactly inside its dual twin so that each vertex of the inner polyhedron touches the center of a face of the outer polyhedron.

Nested Dual Polyhedra (linkedin)

A dual pair of polyhedra can also be infinitely nested inside each other, alternating every other one.

Nested Cubes and Octahedra (Wikipedia)

The symmetric properties of dual polyhedra can be explained by the formulas for the number of faces, edges, and vertices that were mentioned above.  Dual polyhedra have swapped Schläfli symbols, meaning m (the number of faces around each vertex) and n (the number of sides of each face) are switched.  Switching m and n in the formula for the number of faces F = ^4m/_{(2n + 2m – mn)} results in ⁴ⁿ/_{(2m + 2n – nm)}, which simplifies to the formula for the number of vertices V = ⁴ⁿ/_{(2n + 2m – mn)}, and switching m and n in the formula for the number of vertices V = ⁴ⁿ/_{(2n + 2m – mn)} results in ^4m/_{(2m + 2n – nm)}, which simplifies to the formula for the number of faces F = ^4m/_{(2n + 2m – mn)}. This proves that swapped Schläfli symbols in dual polyhedra necessarily results in swapped number of faces and vertices.  Furthermore, switching m and n in the formula for the number of edges E = ^2mn/_{(2n + 2m – mn)} results in ^2nm/_{(2m + 2n – nm)}, which simplifies to itself, E = ^2mn/_{(2n + 2m – mn)}.  This proves that swapped Schläfli symbols in dual polyhedra necessarily results in the same number of edges. 

The formula for the number of edges in a polyhedra E = ^2mn/_{(2n + 2m – mn)} can also be rearranged algebraically as the sum of reciprocals.  Finding the reciprocal of E yields ¹/_E = ^{(2n + 2m – mn)}/_2mn, and dividing out each term results in ¹/_E = ¹/_m + ¹/_n – ¹/₂, and adding ½ to each side yields the rather elegant equation

¹/_E + ¹/₂ = ¹/_m + ¹/_n

Or, written with negative exponents,

E^-1 + 2^-1 = m^-1 + n^-1

Examples of Platonic Solids

Platonic solids can be found both naturally and artificially.  Some chemical compounds, including crystals, form Platonic solids.  The skeletons of different radiolarian protozoa are shaped as different Platonic solids, including the octahedron, the icosahedron, and the dodecahedron.  In addition, the outer shells of several viruses form Platonic solids, such as the icosahedron-shaped HIV virus.

Circogonia Icosahedra, a Species of Radiolaria (Wikipedia)

There are also plenty of examples of artificially made Platonic solids.  The Rubik’s cube is obviously a cube, but variations of it exist that include the tetrahedron and dodecahedron.  Most dice used in board games are cubes, but there are also dice shaped like tetrahedra, octahedra, dodecahedra, and icosahedra.

Dice Shaped as Platonic Solids

And although not a true Platonic solid because it consists of two different regular polygons, the familiar soccer ball (or football) pattern of hexagons and pentagons is actually a truncated icosahedron.  The 12 pentagons formed were originally the 12 vertices of the icosahedron, and the 20 hexagons formed were originally the 20 triangular faces of the icosahedron.

Truncating an Icosahedron (“How to Make a Soccer Ball”)

It is also worth mentioning that there are only two instances in the Bible where any Platonic solid is described, and both are cubes.  The first cube can be inferred from the description of the Most Holy Place of the temple in 1 Kings 6:20, “The inner sanctuary was twenty cubits long, twenty cubits wide, and twenty cubits high, and [Solomon] overlaid it with pure gold.”  The second cube can be inferred from the description of the New Jerusalem in Revelation 21:16, “The city lies foursquare, its length the same as its width. … Its length and width and height are equal.”  The description of the same shape links the two places; the Most Holy Place in the temple and the New Jerusalem are both the dwelling place of the Most High God.

Conclusion

The simple parameter of the Platonic solid, a convex solid that has identical regular polygons for each face, yields only five different possible polyhedra: the tetrahedron, the cube, the octahedron, the dodecagon, and the icosahedron.  Each Platonic solid follows a formula for the number of faces, edges, and vertices, and have symmetric dual properties.  Lastly, Platonic solids can be found both in nature and in man-made objects, as well as in the Bible.