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OLD AND NEW PICTURES OF THE 600-CELL

The 600-cell is one of the six regular polytopes which exist in four-dimensional (4D) Euclidean space. It is enclosed by 600 regular tetrahedrons (*), which meet in twos at 1200 faces, in fives at 720 edges and in 20s at 120 vertices. The vertex figure is a regular icosahedron. The 720 edges form together 72 flat regular decagons.

Like all points in 4D Euclidean space, the 120 vertices can be represented by quaternions, even by unit quaterions if one takes the radius of the circumscribed hypersphere as the length unit and its centre at quaternion zero  (0 + 0i + 0j + 0k). When one furthermore identifies an arbitrary vertex with quaternion unity (1 + 0i + 0j + 0k) the 120 vertices turn out to form a finite subgroup of the group of unit quaternions under quaternion multiplication, the so-called binary icosahedral group. This group contains elements of orders 1, 2, 3, 4, 5, 6 and 10. As a consequence, the left- and right-isoclinic rotational symmetry groups of the 600-cell contain rotations of the same orders.
A further consequence of this is that the full rotational symmetry group of the 600-cell also contains simple rotations of orders 12, 20 and 30 (**).

Reference: Oss, Salomon Levi van: Das regelmässige 600-Zell und seine selbstdeckenden Bewegungen. Verhandelingen der Koninklijke [Nederlandse] Akademie van Wetenschappen, Sectie 1 Deel 7 Nummer 1 (Afdeeling Natuurkunde). Amsterdam: 1899.

This gallery concentrates on the dodecagonal aspect of the 600-cell. The original drawing (AD 1899) to Van Oss's paper is compared and contrasted with modern computer-generated pictures (AD 2011).
The dodecagonal aspect corresponds with simple rotations through 30° and 150°, which appear when isoclinic rotations of opposite senses through 60° and 90° are composed (**).


FOOTNOTES

(*)
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A PROOF that the 600-cell is indeed enclosed by 600 regular tetrahedrons:
From a list of vertex coordinates we infer that we have a regular polytope at hand in which 20 regular tetrahedrons meet at each of the 120 vertices. This makes 120x20 = 2400 vertices of tetrahedrons. Therefore there are 2400 / 4 = 600 tetrahedal cells.

(**)
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There is the general theorem on 4D rotations about a fixed point which states that any 4D rotation can be decomposed...

(1) in two ways into a pair of isoclinic rotations of opposite senses: through angles here denoted by α and β, and through angles α+180° and β+180°, and also...

(2) into a pair of simple rotations in completely orthogonal planes through angles α + β and α − β, respectively, where α and β are the isoclinic rotation angles defined in (1) above.
For non-isoclinic rotations both angles α and β are nonzero, the angles α + β and α − β are different, and decomposition (2) is unique; for isoclinic rotations one of the angles α and β is zero, and decomposition (2) is not unique; there exist infinitely many pairs of completely orthogonal invariant planes. More information in the Wikipedia lemma on 4D rotations.





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