Saturday, September 12, 2009

Great dodecahedron versus Platonic dodecahedron

The dodecahedron is a Platonic solid along long with the tetrahedron, cube, octahedron and icosahedron. The faces of these five highly symmetric, three-dimensional objects are regular polygons (equilateral triangle, square and regular pentagon). A “spherical composition” of a specific number of same-type polygons (12 pentagons in the dodecahedron case) forms a Platonic solid. Each pair of adjacent polygons share an edge. The building polygons are not allowed to intersect, according to a condition imposed by Theaetetus, a colleague of the philosopher Plato, after which the Platonic solids are named. But what if they are allowed to intersect? Marcus du Sautoy “answers” this question with the following historical observation [1]:
To everyone's surprise, in 1809 a new shape had been built out of these 12 pentagons [that were known to build the Platonic dodecahedron]. Theaetetus had insisted that the faces of his shapes should not cut into each other. But what if you relaxed this condition? A mathematics teacher in Paris had found a new way to piece 12 pentagons together to make a new symmetrical shape that was christened the great dodecahedron. Although it looks like a shape built from lots of irregular triangles, it consists of 12 intersecting pentagons. The shape satisfies all the conditions for a Platonic solid except for the fact that the faces cut into each other. How many other strange and beautiful shapes like this might be out there? Three others were soon discovered, and mathematicians began to wonder where the new list might end.
Should these polyhedra with intersecting regular polygons be named anti-Theaetetian solids? Since they look star-shaped, they are otherwise addressed as stellated polyhedra [2].
Whatever they are named, there are only four of them as Augustin-Louis Cauchy (1789-1857) proved by successfully answering the question of the French Academy of Science (founded in 1666), which had dedicated a prize in 1811 for “proving beyond doubt that the five Platonic solids plus the four new solids were all the three-dimensional shapes that you could build from identical regular polygons” [1].

Keywords: solid geometry, nomenclature of polydedra

References and further reading and visualizing
[1] Marcus du Sautoy: Symmetry. A Journey into the Patterns of Nature. First Harper Perennial Edition, Harper Collins Publishers, New York, 2009; pages 164-166.

[2] Chapter VI “Star-Polyhedra” in the book by H. S. M. Coxeter: Regular Polytopes. Dover Publications, New York, 1973.
[3] Image of a great dodecahedron and more at Wolfram

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