1913-14.] 
Projection-Model of the 600-Cell. 
253 
XIX. — Description of a Projection-Model of the 600-Cell in Space 
of Four Dimensions. By D. M. Y. Sommerville, M.A., D.Sc., 
Lecturer in Mathematics, University of St Andrews. (With a Plate.) 
(Read May 4, 1914. MS. received June 1, 1914.) 
§ 1. In 1880 Stringham 1 proved that in space of four dimensions there 
exist six and no more regular rectilinear figures, whose boundaries are 
regular polyhedra. The same result was arrived at independently by 
Hoppe 2 in the following year. In 1883 Schlegel 3 gave an extensive 
investigation of the same problem, and constructed projection-models of 
the six regular figures, which were exhibited at the Magdeburg meeting 
of the Society of German Naturalists in 1884. This series of models was 
published by the firm L. Brill of Darmstadt and is obtainable from their 
successors, Martin Schilling in Leipzig. 
The models are constructed of brass wire and silk threads, and represent 
projections of the figures in ordinary space in such a way that there is no 
overlapping of boundaries. In each case the external boundary of the 
projection represents one of the solid boundaries of the figure. Thus the 
600-cell, which is the figure bounded by 600 congruent regular tetrahedra, 
is represented by a tetrahedron divided into 599 other tetrahedra ; 20 
tetrahedra meet at every vertex and 5 at every edge; 12 edges meet at 
each point; the total number of vertices is 120. At the centre of the 
model there is a tetrahedron, and surrounding this are successive zones of 
tetrahedra. The boundaries of these zones are more or less complicated 
polyhedral forms, cardboard models of which, constructed after Schlegel’s 
drawings, are also to be obtained from the same firm. 
§ 2. The model which was constructed and exhibited by the present 
writer represents an exact stereographic projection of the 600-cell, i.e. the 
centre of projection is taken on the circumscribed hypersphere, and in fact 
is one of the vertices of the figure. The projection of this vertex would 
therefore be at infinity, and the 12 edges which meet there would be 
represented by lines proceeding to infinity from the vertices of the regular 
icosahedron, which is the outermost accessible boundary of the projection, 
1 “ Regular Figures in ^-dimensional Space,” Amer. J. Math., 3, 1-14. 
2 “ Regelmassige linear begrenzte Figui en von vier Dimensionen,” Arch. Math., Leipzig, 
67, 29-44. 
3 “ Theorie der homogenen zusammengesetzten Raumgebilde,” Halle, Nova Acta Acad. 
Leo'p., 44, 343-459. 
