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As it is well known, every platonic solid has a dual (obtained by interchanging vertices and faces), which also happens to be a platonic solid. I would like to know who was the first person to discover this, or in which period it probably happened.

As we can possibly argue from this carved stone figures, platonic solids were already known long before ancient Greece (~600BC). So it's plausible the discovery wasn't recorded. On the other hand, in the Wikipedia article we read:

The ancient Greeks studied the Platonic solids extensively. Some sources (such as Proclus) credit Pythagoras with their discovery. Other evidence suggests that he may have only been familiar with the tetrahedron, cube, and dodecahedron and that the discovery of the octahedron and icosahedron belong to Theaetetus, a contemporary of Plato. In any case, Theaetetus gave a mathematical description of all five and may have been responsible for the first known proof that no other convex regular polyhedra exist.

And proceeds to note that

Euclid completely mathematically described the Platonic solids in the Elements, the last book (Book XIII) of which is devoted to their properties.

Which makes me believe that the discovery was either made by Euclid or before him.

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According to New World Encyclopedia people may have showed some awareness of the duality already in prehistoric times: "Stones carved in shapes showing the symmetries of various polyhedra have been found in Scotland and may be as much a 4,000 years old. These stones show not only the form of various symmetrical polyhedra, but also the relations of duality amongst some of them (that is, that the centers of the faces of the cube gives the vertices of an octahedron, and so on). Examples of these stones are on display in the John Evans room of the Ashmolean Museum at Oxford University." However, as Marius Kempe pointed out, the stones in question were altered to look more like Platonic solids. And even interpreting "polyhedron" loosely icosahedron is still missing among them:"while the symmetry of some of the Platonic solids is present, these are not polyhedra... no balls with twenty knobs have icosahedral symmetry. Many of the carved stone balls have six knobs, and thus have approximate octahedral symmetry in their knob placement. Some have four knobs and tetrahedral symmetry." While duality is not restricted to Platonic solids, and some idea of it is possible without icosahedron, it is unclear what exactly "relations of duality amongst some of them" refers to.

There are differing accounts on whether Greeks knew of all five regular polyhedra before Theaetetus. But even for Theaetetus the duality, if he noticed it, was likely little more than an afterthought or a curious fact rather than a construction/principle. When Archimedes later discovered semi-regular polyhedra, a.k.a. Archimedean solids, as far as we know neither he, nor anybody else for over two thousand years thought to swap their faces and vertices and see what happens. There was so little interest in polyhedra that the work of Archimedes did not survive the middle ages, and we only know of it from a mention in Pappus's Collection. Only in 1430 Paolo Uccello depicted two new (non-convex) regular polyhedra in a mosaic on the floor of San Marco cathedral in Venice, and Kepler described them mathematically only in 1619 Harmonices Mundi, two more were found by Poinsot in 1809. But again neither Kepler nor Poinsot made any use of duality.

The first use of duality as a principle appears to be in Catalan's Memoire sur la Theorie des Polyedres of 1865, where he constructed the duals to Archimedean solids, and noted that they are not semi-regular (faces are not regular polygons). These are now called Catalan solids. And only Brückner in his classic 1900 textbook Vielecke und Vielflache: Theorie und Geschichte, that summarized everything known about polyhedra at the time, defines what we refer to as "duality" explicitly. George Hart assembled annotated bibliography on polyhedra that may be useful for further digging.

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  • $\begingroup$ As you will see from my comments on the other answer, the Scottish stones are not (all) regular polyhedra, nor do they exemplify all regular polyhedra. As for what Theaetetus or Archimedes thought, so little survives that I think it is unwise to impute what they did not know or consider. As Netz has argued in his books, it is so difficult to write down mathematics in the Greek style that a great deal of what they knew was probably not even written down, nevermind transmitted through the millenia. $\endgroup$ – Marius Kempe Apr 12 '15 at 20:53
  • $\begingroup$ @Marius Kempe Duality applies to non-regular polyhedra just as well as to regular ones. Pappus is usually very comprehensive in what he reports, if Archimedean duals were considered he would have mentioned it. And we can be fairly sure that nobody between Kepler and Catalan considered them. $\endgroup$ – Conifold Apr 12 '15 at 21:15
  • $\begingroup$ I really like this answer, but I think the controversy around the Scottish stones, as well as @MariusKempe link (math.ucr.edu/home/baez/icosahedron), or the paper cited in it (which I didn't read) should be mentioned. My conclusion from all this is that it's highly probable Pythagoreans didn't know about duality, since they didn't know all 5 solids, but most likely Theaetetus did, since he knew all of them, and was probably the discoverer. I'd also like to add that the 4 elements are dual to each other in certain ways, so this geometrical duality may have inspired Plato's view. $\endgroup$ – hjhjhj57 Apr 14 '15 at 1:23
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    $\begingroup$ @Javier Done. I am not sure about Theaetetus. Did he notice that if one connects face centers of one one gets another? Maybe, but it sounds like something Euclid might put into book XIII, or Pappus might mention. Kepler doesn't seem to note it either despite his interest in "mystical connections". Sometimes it seems to us "they couldn't have missed it", but maybe they could. $\endgroup$ – Conifold Apr 15 '15 at 3:28
  • $\begingroup$ Definitely. I'll read the Dialogue one of these days to see if I can find something. $\endgroup$ – hjhjhj57 Apr 15 '15 at 3:32
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Kepler knew about the duality: he described the five solids as two pairs male/female and called the self-dual tetrahedron 'androgyne' (See Stephenson's book The Music of the Heavens).

There is a good ground to suspect that indeed Theaetetus already knew about the duality. If, as the story is told, the series tetrahedron, cube, dodecahedron was already known, he might have made his discovery by noting the duals; and the self-duality seems to hint that there are no more to be found - which prompted him to invent the proof. Apparently he told Plato who, as his Timaeus suggests, seems to be aware of both the finite number of solids and their dualities (see e.g. here).

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