328 On the Use of the Globe 



In representing the cube by great circles parallel to its 

 faces, we have divided the hemisphere into four equal and 

 similar triangles. If we regard this diagram as the spherical 

 projection of a polyhedron where all the faces, whether 

 independent or not, are represented where in fact the faces 

 have been delineated by drawing small circles round their 

 poles, and arcs of great circles through the intersections of 

 these circles to give the edges, we find that we have here the 

 radial projection of the regular octahedron. The trans- 

 ference of the faces of the cube parallel to themselves until 

 they coincide with planes of great circles, has the effect of 

 transferring the corners of the cube parallel to themselves to 

 the centre. If the extremities of each set of three edges 

 which go to form a corner are connected by arcs of great 

 circles, the diagram produced on the globe is the one which 

 we have been considering. As was pointed out in the paper, 

 the corners of the cube when radiating from the centre of the 

 sphere delineate the projection of the octahedron on it. This 

 fact can be expressed by saying that the central representation 

 of the cube is identical with the radial projection of the 

 octahedron. 



Models exhibiting the central grouping of corners, and the 

 forms thereby produced, are quite easily made out of card- 

 board or stiff paper, and are very instructive. 



Let us consider the regular tetrahedron. The normals to 

 its faces, four in number, are found to be inclined to each 

 other at an angle of 109^ across an edge. The poles of 

 these faces when placed on the globe form a group of four 

 points symmetrically arranged, each being separated from its 

 neighbour by an arc of 109^. Draw the great circles of 

 which these points are the poles, and call any one of them 

 No. o. Take its intersection with any other and call it 

 node (o), the opposite node is (o, i'), and the great circle 

 which intersects No. o in these nodes is No. i, diameter (o, i) 

 being parallel to and representing the edge made by Nos. o 

 and i. Another pair of nodes are found in azimuths 120 

 and 300. Their diameter represents the edge (o, 2), and the 

 great circle represents face No. 2. Similarly the nodes of 



