OLD AND NEW PICTURES OF THE 600CELL
The 600cell is one of the six regular polytopes which exist in
fourdimensional (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
socalled binary
icosahedral group. This group contains elements of orders 1,
2,
3, 4, 5, 6 and 10. As a consequence, the left and rightisoclinic
rotational symmetry groups of the 600cell contain rotations of the
same orders.
A
further
consequence of this is that the full rotational symmetry group of the
600cell
also contains simple rotations of orders 12, 20 and 30 (**).
Reference: Oss,
Salomon Levi van: Das regelmässige 600Zell 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 600cell.
The original drawing (AD 1899) to Van Oss's paper is compared and
contrasted with modern computergenerated 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
(*)  A PROOF that the 600cell 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.
(**) 
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
nonisoclinic 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.
