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Proceedings of the Royal Society of Edinburgh. [Sess. 
In the model these infinite edges have been omitted, so that the model is 
to that extent incomplete. The projection of the vertex which is opposite 
the centre of projection forms the centre of the model, and the successive 
zones of vertices are very simple and regular. Starting from the outside — 
Zone A is the vertex at infinity (1 vertex). 
Zone B is a regular icosahedron (12 vertices). 
Zone C is a regular dodecahedron (20 vertices). 
Zone D is a regular icosahedron, whose vertices are not joined to one 
another. In the model these vertices are joined to the vertices of zone C 
by wires painted black, forming pyramids on the faces of the dodecahedron 
(12 vertices). 
Zone E, the mesial zone, is the semi-regular polyhedron called the 
icosidodecahedron, which is bounded by 20 triangles and 12 pentagons 
(30 vertices). 
Zone — D is similar to zone D, and its vertices are joined by black wires 
to the vertices of zone — C, forming pyramids on the faces of a dodecahedron 
(12 vertices). 
Zone — C is similar to C, i.e. a regular dodecahedron (20 vertices). 
Zone — B is similar to B, i.e. a regular icosahedron (12 vertices). 
Zone — A is the centre (1 vertex). (Total number of vertices 120.) 
The edges which join up the vertices of each zone are of brass wire, 
and, with the exception of the edges joining zones C, D, and — C, — D, the 
edges joining different zones are of differently coloured silk threads. All the 
threads which join the vertices of the same two zones, and those of the cor- 
responding zones on the other side of the mesial zone, are of one colour. 
§ 3. The model has been constructed so that the radius of the circum- 
scribed hypersphere is 8 cm. 
In making the calculations for the lengths of the edges great use has 
been made of Schoute’s valuable paper, 1 which gives the co-ordinates of the 
120 vertices in the most symmetrical form, and tabulates the connecting 
edges. In Schoute’s system of numbering, the 120 vertices are numbered 
from 1 up to 60, and from — 1 to — 60 for the opposite vertices. With 
reference to the special arrangement of the vertices in the stereographic 
projection, whereby one vertex is singled out as centre of projection, 
another system of numbering allows of a more compact table of connecting 
edges. The vertices of each zone are numbered separately, and so also are 
the rings or zones of vertices of each zone. A pair of opposite vertices of 
1 “ Regelmassige Schnitte und Projectionen des Hundertzwanzigzelles und Sechs- 
liundertzelles im vierdimensionalen Raume,” Amsterdam, Verh. K. Akad. Wet. (le. Sect.), 
II. No. 7 (1894). 
