255 
1913-14.] Projection-Model of the 600-Cell. 
any zone are numbered ±n. Thus, e.g., B 3 and — B — 3 denote opposite 
vertices of the 600-cell. For the mesial zone — E is the same as E, and 
— E 3 is the same as E 3. 
In zones B and D, the icosahedra, the vertices are numbered 0 ; 1, 2, 3, 
4, 5; -1, -2, -3, -4, -5; 0. 
In zone C, the dodecahedron, they are numbered 1, 2, 3, 4, 5 ; 1 , 2 , 3 , 
4 , 5 ; - 1 , - 2 , - 3 , - 4 , - 5 ; -1, -2, -3, -4, -5. 
In zone E, the icosidodecahedron, they are numbered 1, 2, 3, 4, 5; 
1 , 2 , 3 , 4 , 5 ; 13, 24, 35, 41, 52, -13, -24, -35, -41, -52 (or 31, 42, 53, 
14, 25); etc.; i.e. the 10 vertices of the equator of this zone are represented 
by the pairs of numbers which represent the vertices of the adjacent rings 
to which they are connected, and 31 is the same as —13. 
§ 4. In order that reference may be made to Schoute’s tables, Table I. gives 
the numbers in the present system, which correspond to Schoute’s numbers. 
No. 4 of his system, which is on the axis of w, is taken as centre of projec- 
tion. Then — 4 is — A, the centre of the model. The numbers corresponding 
to the negative numbers of Schoute’s system are obtained by changing B, 
C, D into — B, — C, — D, and changing the sign of the number. 
Table II. gives the edges of the 600-cell. 
Table III. gives the co-ordinates of the vertices, according to Schoute, 
but the plane of x, y, z has been moved so as to pass through A, i.e . w has 
been changed into 2 (e + lj-m The symbol e = j5. 
Table IV. gives the lengths of the edges of the projection. 
§ 5. In the projection a number of groups of points become coplanar. 
These are the projections of points which lie in the same hyperplane 
passing through the centre of projection. The groups of points in the 
original figure which are so projected form zones of the same form as the 
zones B, C, D, E, and their centres lie in the zones B, G, D, E respectively. 
Thus, taking the point BO as centre, we have the icosahedron — 
A; B 1, 2, 3, 4, 5; C 1, 2, 3, 4, 5; BO, j 
which is projected into a plane figure. 
With centre Cl we have first the icosahedron — 
B 0, 3, 4 ; C 1, 2, 5 ; B 0, 3, 4 ; E 1, 3, 4, 
and next the dodecahedron — 
A; B - 1, 2, 5; C 3, 4, 2, - 3, - 4, 5 ; 
-Cl; - B 0, 3, 4 ; E, 2, 5, 2, 5, 13, 14, 
and this last is projected into a plane figure. 
