The shape of the world - Enrique Macías Virgós

14.2

The shape of the world.

(Appendix 2) The five Pythagorean solids.

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A “platonic solid” is a convex regular polyhedron (Figure 1). This means that their faces are equal regular polygons and that in all the vertices the number of incident faces is the same. In addition, the well-known Euler's formula is fulfilled:

V – A + C = 2,                      (Equation 1)

where V is the number of vertices, A the number of edges and C the number of faces.

Fig. 1. The Platonic solids

(Credit: Максим Пе [CC BY-SA 4.0] https://commons.wikimedia.org/wiki/File:Platonic_solids.jpg)

In three of the solids the faces are equilateral triangles:

- the tetrahedron: V = 4, A = 6, C = 4,

- the octahedron: V = 6, A = 12, C = 8,

- the icosahedron: V = 12, A = 30, C = 20;

in the cube or hexahedron, the faces are squares, V = 8, A = 12, C = 6; and in the dodecahedron the faces are regular pentagons, V = 20, A = 30, C = 12

These solids, which Sagan calls "Pythagoreans", have been known for thousands of years. For Plato (427 BC-347 BC) they represented the elemental components of the world, respectively: fire, air, water, earth and the shape of the universe.

The amazing beauty of these bodies lies in their mirror and rotational symmetry, which has attracted the interest of many artists throughout history. Their forms also appear in Nature: there are minerals that crystallize in the form of a polyhedron, such as the pyrite in Figure 2, and it is possible that the observation of these crystals is at the origin of Plato's ideas. They also appear in the arrangement of the atoms of certain molecules [1], in the skeletons of radiolaria [2] and in the capsid of some viruses [3].

Fig. 2. Pyrite crystal

(Credit: Rob Lavinsky, iRocks.com [CC-BY-SA-3.0]

https://commons.wikimedia.org/wiki/File:Pyrite-135018.jpg)

There are only five platonic solids. A proof already appears in the Elements of Euclid (325 BC-265 BC). Here, instead of the algebraic proof that was given in Sagan's original book, we will do another one, based on the fact that, at each vertex, the sum of the interior angles of the incident faces must be less than 360º.

On a regular n-sided polygon, each interior angle measures

a = 180 º - 360º/n,

in a polyhedron, if r is the number of edges (and also faces) that affect a vertex, we have

r • a < 360º,

as we can see if we imagine that we squash the vertex and open the faces until we put them on the same plane. Since n cannot be less than 3, we see that

- for n = 3 we have r • 60 < 360, so r = 3,4,5;

- for n = 4 we have r • 90 < 360, so r = 3;

- for n = 5 we have r • 108 < 360, hence r = 3,

- for a higher n, no r is valid, since r cannot be less than 3.

On the other hand, since each edge is bordered by two faces, when counting the number of edges we will have n • C = 2A; and since each edge has two vertices, we shall have r • V = 2A. If we use these formulas, Equation 1 becomes

1/r + 1/n = 1/A + 1/2,

which gives us, in each case, the correct value of A, and from there those of V and C. With this we have proven that there are no more regular polyhedra.

In mathematics, these polyhedra are related to group theory, due to their symmetries; with graph theory, since a graph is obtained by projecting its edges onto the plane of one of its faces; and, above all, with Topology, since the deep understanding of Equation 1 led Poincaré (1854-1912) to boost this new specialty, when he introduced the notions of homotopy and homology. Equation 1 is also true for semi-regular polyhedra, for example platonic solids with truncated corners (Figure 4), already studied by Archimedes (287 BC-212 BC); and for prisms and antiprisms, where faces are regular polygons, but of two different types.

It is also true for some regular non-convex polyhedra, such as the starry polyhedron, in Figure 3, which is an illustration by Leonardo da Vinci (1452-1519) for the book Divina proportione, by Luca Pacioli (1445–1517). This polyhedron is a "stellation" of the icosidodecahedron and has V = 62, A = 180, and C = 120 triangles.

Fig. 3. Starry polyhedron

(Credit: Leonardo da Vinci [Public domain]

https://commons.wikimedia.org/wiki/File:De_divina_proportione_-_Dodecaedron_Abscisum_Elevatum_Solidum.jpg)

From the topological point of view, the essential thing of a polyhedron is that its faces form a closed, that is, limited and borderless, surface. The edges are thus lines drawn on the surface, which intersect at the vertices. On the right side of Figure 4 it is clear that a polyhedron is essentially a sphere, with some decomposition of its surface into regions. As we will see below, this idea is crucial for the proof of Equation 1.

Fig. 4. Truncated icosahedron

(Credit: Aaron Rotenberg [CC BY-SA  4.0]

https://commons.wikimedia.org/wiki/File:Comparison_of_truncated_icosahedron_and_soccer_ball.png)

This equation, stated by Euler (1707-1783) [4], has many different proofs. Coxeter (1907-2003) gave in [5] the following one: we go over the edges of the polyhedron so that we pass only once through each of the V vertices. In this way we will have a “tree” that has V - 1 branches, which are the traversed edges (in Figure 5 some edges are repeated). Each missing edge in the tree limits two faces and therefore we can draw a path that connects the C centers of the faces. This path is connected, that is, it can be reached from one face to any other, since otherwise there would be two faces separated by a closed circuit of the tree, but by definition a tree has no circuits.

On the other hand, our path cannot have circuits either, since it would separate the surface into two parts, with some vertices of the first tree in each one, which can not be true. Thus, the second path is also a tree, and has C - 1 branches. Now, each edge of the polyhedron is a branch of one of the two trees, so

(V - 1) + (C - 1) = A,

which is Euler's formula.

Fig. 5. Proof of Euler's formula

(Credit: Pixelmaniac pictures (Leave a reply) [Public domain] and own elaboration.

https://upload.wikimedia.org/wikipedia/commons/c/ce/Foldable_tetrahedron_%28blank%29.svg)

The essential property used in the previous proof, highlighted by Poincaré [6], is that in a sphere and therefore in all the polyhedra we are considering, any simple closed curve drawn in it will disconnect it. The sphere is said to be "simply connected," or, what is the same, to have "gender zero." Said more loosely, the sphere has no "handles."

In contrast, a torus (Figure 6) has a handle, that is, its gender is g = 1. For a polyhedron drawn on a surface of gender g, the "Euler-Poincaré formula" will give us

V – A + C = 2 - 2g.

Fig. 6. Toroidal polyhedron.

(Credit: Dr. Ozan Yarman [CC BY-NC-SA 3.0] https://www.nodebox.net/code/index.php/Mark_Meyer_%7c_Parametric_surfaces_%7c_torus)

These ideas led, at the beginning of the 20th century, to the classification of all surfaces, and more recently to the generalization, by Perelman (1966-), of this classification to other dimensions [7].

In Carl Sagan's original Cosmos, Platonic solids appear twice: when he recounts the ancient Greeks' ideas about the elements; and when he comments on Kepler's theory (1571-1630) on the orbits of the planets. The merit of these speculations was to establish the principle that it is possible to explain the universe through mathematical models. Centuries later, the objective of Mathematics continues to be to imagine, from geometric and topological methods such as those we have seen, all possible theoretical worlds and then discard those that do not conform to the laws of astrophysics.

References:

[1] S. Álvarez , Polyhedra in (inorganic) chemistry, Dalton Trans. 2209-2233 (2005), https://www.doi.org/10.1039/b503582c

[2] M. Mallo Zurdo, Sistemas radiolarios. Geometrías y arquitecturas derivadas. Tesis doctoral. Universidad Politécnica de Madrid, 2015.

[3] Viral Zone https://viralzone.expasy.org/8577

[4] L. Euler, Elementa doctrinae solidorum, Euler Archive – Obras completas. E230 (1758). https://scholarlycommons.pacific.edu/euler-works/230

[5] H.S.M. Coxeter, Regular polytopes. Methuen & Co. Ltd. London, 1948.

[6] H. Poincaré, Sur la généralisation d’un théorème d’Euler relatif aux polyèdres.  Comptes Rendus Acad. Sci., t. 117, p. 144-145 (17 juillet 1893). http://analysis-situs.math.cnrs.fr/

[7] Virtual and manipulative geometrical and topological games, Juegos Topológicos del Mago Moebius.  https://topologia.wordpress.com/2009/03/04/la-conjetura-de-poincare/

Bibliography:

(1) Henar Lanza González. Matemática y física en el Timeo de Platón. Poliedros regulares y elementos naturales. Praxis Filosófica Nueva serie, No. 40, enero-junio 2015: 85 – 112.

(2) Marjorie Senechal Editor. Shaping Space: Exploring Polyhedra in Nature, Art, and the Geometrical Imagination. Springer, 2013.

(3) J. R. Weeks.  The shape of space, Second Edition, Marcel Dekker, 2002.

(4) E. Cabezas Rivas y V. Miquel Molina. Demostración de Hamilton-Perelman de las conjeturas de Poincaré y Thurston, La Gaceta de la RSME, Vol. 9.1 (2006), 15–42.

Enrique Macías Virgós.

Ph.D. Mathematics. 

Professor of Geometry and Topology.

Departament of Mathematics,University of Santiago de Compostela.