Polyhedrality

The study of polyhedra has a long history, one that has intersected with philosophy, mathematics and art - and which, in the modern period, has been associated with physics and chemistry. But the study of these forms has always attracted interest for their own sake, both from an abstract mathematical perspective and for purely aesthetic reasons. The briefest look at the range of Internet links relating to polyhedra show that the enthusiasm for this subject is as strong (and occasionally obsessive) as ever.

Any serious consideration of polyhedra has to begin with the important discoveries of Classical antiquity and the extraordinary achievements of such early mathematician/philosophers, as Pythagorus, Plato, Archimedes and Euclid.

The five regular or ‘Platonic’ polyhedra (the first four of which Plato famously identified with the four elements and two ‘basic’, right-angled triangles. In this scheme the fifth polyhedra, the dodecahedron, was taken to represent the Cosmos itself).

The thirteen semi-regular or ‘Archimedean’ polyhedra.

The mathematical study of polyhedra in the modern period has lead to an extraordinary proliferation of symmetrical forms.

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Stella octangula (two tetrahedra)

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Great Dodecahedron

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Small stellated dodecahedron

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Compound of two dodecahedra

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Great stellated dodecahedron

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Ninth stellation of the icosahedron

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From Max Brückner's 'Polygons and polyhedra: Theory and History, Leipzig: B. G. Teubner, 1900

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From Max Brückner's 'Polygons and polyhedra: Theory and History, Leipzig: B. G. Teubner, 1900

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From Max Brückner's 'Polygons and polyhedra: Theory and History, Leipzig: B. G. Teubner, 1900

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Nested polyhedra 1

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Nested polyhedra 2

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Nested polyhedra 3

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Nested polyhedra 4

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Model of nested Platonic polyhedra 1

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Model of nested Platonic polyhedra 2