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
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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.
