Profiting from uncertainties about whether Plato already knew the dodecahedron, Austrian philosopher and artist Renate Quehenberger, proposes that the “fifth configuration” that Plato is vaguely alluding to in his Timaeus, is not the dodecahedron, but the epitahedron: a form rediscovered by herself and described in a paper in 2014 (*).
In his book Timeaus Plato describes how the Demiurge created the world out of simple forms. For the ancient Greeks, the world was composed of the 4 elements: fire, air, water and earth. According to Plato, each of these elements were composed out of elemental geometrical forms. The most elemental of elemental forms is “the triangle which has its hypotenuse twice the lesser side” (= a triangle with angles of 90°, 60° and 30°). Using such triangles one can compose regular triangles (= triangles with equal sides and equal angles of 60 °). Using these regular triangles one can subsequently create three regular volumes: the tetrahedron, the octahedron and the icosahedron, which, according to Plato make up fire, air and water respectively. Plato introduces the isosceles triangle with a right angle ( = hence angles of 90°, 45° and 45°) to compose squares. With six squares he composes the cubes which make up the element earth. The tetrahedron, octahedron, icosahedron and cube are regular solids and are now called Platonic solids (Cat. Nr. e18). After a lengthy description of how the elements are composed of the regular solids, Plato finishes with the enigmatic sentence: “There is yet another fifth configuration which the Demiurge uses for the delineation of the universe”. With hindsight, it is tempting for us to conclude that this fifth configuration must be the dodecahedron made of twelve regular pentagons; it is in fact the remaining of the five and only five Platonic solids. It is however very strange that Plato does not describe the dodecahedron. It might be that he was not familiar with the dodecahedron, that he had only heard of it. There is in fact a discussion among historians whether the dodecahedron was discovered before Plato by the Pythagoreans or only later, when it was described by Euclide.
A pentagon can be composed of 5 isosceles triangles with angles of 54°, 54° and 72°. So the the dodecahedron could be composed of 60 such triangles. This construct based on triangles is in line with the construction of the other platonic solids. However it does not reflect anything of the most peculiar characteristic of the pentagon: namely that it is based on the golden ratio: phi = 1,618 ...
By drawing two diagonals in a pentagon, one can see that it can be composed by two golden isosceles triangles (1,1,phi) plus one golden isosceles triangle (1,phi,phi), (the values in brackets refer to the normalized lengths of the three sides). The dodecahedron can thus be formed with 24 (1,1,phi)’s and 12 (1,phi,phi)’s. This is much less that 60 and still pretty ideal, one could say. However the newly or re-discovered epitahedron would be composed of only 4 (1,1,phi)’s and 4 (1,phi,phi)’s, which is simpler, and hence more ideal. Furthermore, crossing two such epitahedra does create the dodecahedron (as shown in Cat. Nr. 108).
Quehenberger furthermore shows that the epitahedron appears frequently in projections of 4 and 5 dimensional hyperobjects (Cat. Nr. 7) into 3-dimesional space.
All this makes Quehenberger conclude that her epitahadron, rather than the dodecahedron, is in fact the “fifth configuration” that Plato is alluding to, and that with this she has solved a 2500 yrs old riddle.
More technically speaking, the epitahedron is a golden truncated pentagonal pyramid. It has 7 sides ( 3 (1,1,phi)’s, 3 (1,phi,phi)’s plus one side which is composed of one (1,1,phi) and one (1,phi,phi)). Renate’s 11 yrs old son, who already showed an interest in his mother’s adventures in geometry, suggested that "the shape must have a proper Greek name" and mispronounced the word "heptahedron" for the 7 -sided polyhedron his mother had found as "epitahedron" . That was very OK for Renate because epi in Greek means also “beyond” and it is a word that appears in the enigmatic sentence in Plato’s Timaeus: “eti de ousês sustaseôs mias pemptês, epi to pan ho theos autêi katechrêsato ekeino diazôgraphôn” or “There is yet another fifth configuration the demiurge uses it for the delineation of the universe) (Plato, Timaeus, 55c4-6).
(*) Quehenberger, R. (2014) A newly found golden heptahedron named epitahedron. Symmetry: Culture and Science, 25, 177-192.
for Queheberger's computer animations look here
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