On the Last theorem of Euclid, the Platonic solids, and the uniqueness of the Tetrahedron

Symmetry is when, despite some (methodical and orderly) changes, some things remain the same {constant}.

According to W. Shakespeare, G. Julius Cæsar [Gaius] said:
`I could be well moved, if I were as you:
If I could pray to move, prayers would move me:
But I am constant as the northern star,
Of whose true-fix'd and resting quality
There is no fellow in the firmament.
The skies are painted with unnumber'd sparks,
They are all fire and every one doth shine,
But there's but one in all doth hold his place:
So in the world; 'tis furnish'd well with men,
And men are flesh and blood, and apprehensive;
Yet in the number I do know but one
That unassailable holds on his rank,
Unshaked of motion: and that I am he,
Let me a little show it, even in this;
That I was constant Cimber should be banish'd,
And constant do remain to keep him so.'

The last proposition of Euclid's XIII books of `the Elements' states that there are exactly five Platonic solids (cf. art. 1 below for the meaning); in many, or even, all ways these are the most symmetrical solids in the universe; Plato in his `Timæus' had earlier considered them the atoms of the five elements. A demonstration of the Last theorem follows.

1. Definition: A solid which has for all its faces, identical {congruent} regular polygons; and for all its vertices, identical characteristics, is called a Platonic solid or regular polyhedron. (A polygon which has identical {equal} sides {edges} and angles is called a regular polygon; thus, an equilateral and equiangular polygon.)

Nota bene: Since the faces are all identical, and the sides of each face likewise, the sides of a Platonic solid are all identical too, and this truth about the sides need not be stated particularly. Thus, a Platonic solid is symmetrical in its vertices, sides, and faces.

2. A regular polygon has all angles equal; thus all vertices have identical characteristics, and all of them must be convex (thrusting out), since all cannot be concave (caving in) or all of them flat.

Consider a convex polygon. Looking at each vertex, it is clear that there is an internal angle (through the interior of the polygon) and an external angle, the supplement to the internal angle, formed at the vertex by a side and an adjacent side produced (this is through the exterior of the polygon since the vertex is convex). e. g.: In an equilateral triangle, i. e., a 3-sided regular polygon, the internal angles are each 60; and the external angles, each 120.

The sum of all the external angles is determined as follows: imagine a thread wound over the perimeter of the polygon; now, unwind the thread while keeping it taut by detaching it from successive vertices; at each vertex, the thread revolves through an angle pivoted at the vertex, the external angle; when the unwinding is complete, the thread would have turned through a full circle; the unwinding from vertex to vertex is a partition of the revolution of the thread in a full circle among the vertices of the polygon, with each vertex being apportioned as its share its external angle; thus, the sum of all the external angles is 360. Another way to show this is to roll a straight line fully through the perimeter of the polygon without slipping and again observe the revolution of the line.

Let n be the number of sides (or vertices) of the convex polygon. The sum of all the internal and external angles (which are supplementary, i. e., form linear pairs) is 180 per vertex and n*180 for all vertices. Thus the sum of all internal angles is n*180-360 = (n-2)*180; and for a regular polygon, each internal angle is (1-2/n)*180 (which can also be shown by observing that each external angle is 360/n, i. e., an equipartition of the revolution of the thread (or rolling of the line) in a full circle among the vertices of the polygon, and that the internal angle is its supplement).

The sum of all the internal angles can also be shown by dissecting the convex polygon into n-2 triangles, each of which has internal angles adding up to 180; simply draw n-3 diagonals from any vertex.

In this second demonstration, to show why any triangle has internal angles adding up to 180, consider one of its vertices, and draw a line through it parallel to the opposite side; the angles at the other vertices are equal to the alternate interior angles on either side of the angle at the vertex under consideration, which all add up to a straight angle (the line drawn), 180; another way to show this is to apply the thread unwinding (or line rolling) to the triangle, as described in the first demonstration.

Nota bene: I do not know if the funicular (i. e., thread unwinding) or line rolling demonstration for the sum of the external angles of a convex polygon, and particularly for the triangle, as needed in the second demonstration of the same, is at all known and noted; I did discover it myself.

3. Next, consider a regular polyhedron. Looking at each vertex, it is clear that the angles meeting there must add up to less than 360 if either the vertex thrusts out fully, i. e., is convex, or caves in fully, i. e., is concave (such a vertex, either convex or concave, has planes passing through it with all the incident edges lying on one side); a sum of 360 would be given by a flat vertex; and a greater sum would be given by a saddle vertex, i. e., one where it is possible to draw angles through the vertex along the faces meeting there which would respectively thrust out and cave in at the vertex, with the geometry at the vertex resembling a mountain pass or an equestrian saddle on which a rider sits upon a horse (such a vertex has no plane passing through it with all the incident edges lying on one side); there is also the heterogeneous vertex, which is not convex, concave, or saddle, and yet has planes passing through it with all incident edges lying on one side; since all vertices have identical characteristics, all of them must be convex, since all cannot be concave, or all of them flat, or all saddles, or all heterogeneous.

3. Next, consider a regular polyhedron. Looking at each vertex, it is clear that the angles meeting there must add up to less than 360 if either the vertex thrusts out, i. e., is convex, or caves in, i. e., is concave; a sum of 360 would give a flat vertex, and a greater sum would give a saddle vertex, i. e., one where it is possible to draw angles through the vertex along the faces meeting there which would respectively thrust out and cave in at the vertex, with the geometry at the vertex resembling an equestrian saddle on which a rider sits upon a horse; since all vertices have identical characteristics, all of them must be convex, since all cannot be concave, or all of them flat, or all saddles.

4. At least 3 polygons meet at the vertex, and thus (from art. 3), the internal angles of the polygons are each less than 360/3 = 120.

5. By art. 2, the possible internal angles are:

n                     -> 3    4    5     6
180-360/n     -> 60  90  108  120

where the last instance (hexagon) barely fails to meet the condition in art. 4.

Thus the possible regular polygons constituting the faces of a regular polyhedron are triangles, squares, and pentagons.

6. Finally, consider the possible ways in which regular polygons can meet at a vertex of a regular polyhedron, so that the angles add up to less than 360 (art. 3):

3 * 60    = 180 -> three triangles meet at a vertex    -> tetrahedron (4 faces)
4 * 60    = 240 -> four triangles meet at a vertex     -> octahedron (8 faces)
3 * 90    = 270 -> three squares meet at a vertex     -> cube {hexahedron} (6 faces)
5 * 60    = 300 -> five triangles meet at a vertex      -> icosahedron (20 faces)
3 * 108  = 324 -> three pentagons meet at a vertex -> dodecahedron (12 faces)

In contrast:
6 * 60   = 360
4 * 90   = 360
4 * 108 = 432
all fail (the first two, barely) to meet the condition in art. 3.

Thus there are at most five Platonic solids.

Nota bene: The instances of regular polygons meeting at a vertex which fail barely (yield flat vertices), 6 triangles, 4 squares, and 3 hexagons (the last one from art. 5), lead to planar tessellations by regular polygons, the only ones possible, as can be seen from the deductions just described. The hexagonal tessellation is used by bees for the honeycomb. Regular tessellations are similar to regular polyhedra, except that the vertices are flat, planar, and a planar pattern rather than a solid is formed.

7. These regular polyhedra can all be constructed and do exist, and thus there are at least (and thus, by art. 6, exactly) five Platonic solids. This is Euclid's Last theorem.

From a stereometric study of the Platonic solids, the dihedral angles, i. e., the angles between faces at each side, are:

number of faces -> 4                   6             8                          12                    20
dihedral angle     -> arccos(1/3)  arcsin(1)  180-arccos(1/3)  180-arctan(2)  180-arcsin(2/3)
                                 = 70.529      = 90       = 109.471            = 116.565       = 138.190

The angles are all inverse trigonometric functions of simple (harmonic) ratios of the first three numbers: 1, 2, 3.

Nota bene: Considering polyhedra which can tessellate space, it is clear that, at a side, the dihedral angles must add up to 360. Among the Platonic solids, this is only possible with 4 cubes, and thus: from the Platonic solids, only the cube can tessellate space. Aristotle, a student of Plato and also a teacher of the Hegemon Alexander III of Macedon, asserted in `de Cælo' that regular tetrahedra can tessellate space; this was refuted only centuries later, by Bishop Paul von Middelburg of the Diocese of Fossombrone, Professor at the University of Patavium {Padua}: in 1481, he observed that, were such a tessellation possible, the packing (around a vertex or a side) would yield a larger regular polyhedron constituted of triangular faces, which can only be an octahedron or an icosahedron by the Last theorem; such a construction is impossible, as can be verified by elementary calculations.

Affixing the corresponding pyramids to the polygonal faces gives rise to stellated polyhedra; there are only five such regular stellated polyhedra (as a rider).

8. The regular tetrehedron is special and particular even among the five Platonic solids: It is the only solid in the universe for which each vertex is equally distant from and symmetric with respect to all other vertices (and not merely the adjacent ones); also, each face is placed symmetric and identically with respect to all other faces (and not merely the adjacent ones); (note though in the case of sides that, for each side, not all the other sides are placed symmetric and identically with respect to it; there is one side which is peculiar, `opposite' to it, while also being `perpendicular', though they do not intersect); and thus, the most and uniquely symmetrical, exhibiting harmony, equilibrium, and balance unlike any other in the cosmos; as Cæsar said: with `no fellow in the firmament'.