92 



KNOWLEDGE & SCIENTIFIC NEWS. 



[May, 1904. 



The Do\ible 



Stereoscopic Projection 



of the Eight-Cell. 



By G. H. Bryan. Prof. Sc. D., F.R.S. 



In connection witli Mr. Benham's paper on " The Super- 

 Solid," it will be noticed that the diagrams of pairs of 

 connected cubes, even when seen through a stereoscope, 

 fail to convey the impression of being the projections of 

 a regular figure. 



A much better idea of the regular character of the 

 " super-cube " or "eight-cell," as it is called by most 

 writers, and of its connection with four-dimensional 

 space can be acquired by choosing the jilane of projection 

 in such a way as to give the dia^jram a more symmet- 

 rical form, and by using two different stereoscopic pro- 

 jections instead of one. 



space containing the first, second and third dimension, 

 the other view^ represents the aspect of the same " eight- 

 cell " projected in a space containing the first, second 

 and fourth dimension. 



Either of the two aspects shows a solid figure, which 

 is symmetrical but not perfectly regular. It is not difficult, 

 however, to convince oneself that the four-dimensional 

 figure of which the two aspects are simultaneous projec- 

 tions is regular. 



In regard to the fact that in either view two of the 

 vertices (not the same two) appear inside the solid pro- 

 jection, a comparison of the two aspects will show that 

 they are not really inside, but only look so owing to the 

 direction of projection. This property is exactly analo- 

 gous to the fact that if we draw the trace of a cube by 

 projection on a plane, the projections of two of the ver- 

 tices will be inside the polygon formed by the projec- 

 tions of the remaining vertices. It is only when the 

 cube is viewed as a solid, or studied by means of its pro- 

 jections on different planes, that we become aware that all 

 the vertices he on the boundary of the cube. 



-A. complete account of the regular figures possible in 



In the anne.xed series of diagrams the central figure 

 represents a symmetrical plane projection of the " eight- 

 cell." It is not the only projection which is symmetrical, 

 but it is a convenient one in which the edges and sides 

 are well separated, and are nowhere near overlapping in- 

 conveniently. 



When this figure and the figure to :he left of it are 

 viewed together through a stereoscope, the fines will 

 stand out in relief, giving the impression of forming a 

 solid figure in w'hich the point H is nearest the observer, 

 and K is furthest always. The points C, P appear to h& inside 

 the solid, and to be in the straight line joining E and N. 



Now let the central and the right hand figure be 

 brought into view in the stereoscope, and it will be ob- 

 served that the whole aspect of the figure has altered. 

 This time P is at the front of the figure and C is at the 

 back, while the points H and K which were previously 

 the nearest and furthest points appear to be inside the 

 figure in the straight line joining Q and B. 



As the same central figure is used in both cases, the traces 

 of the two stereoscopic solids on the plane of the paper 

 are, to all intents and purposes, the same. If, as assumed 

 they both represent ditt'erent aspects of the same figure 

 the distances of the different points from the plane of the 

 paper in the first place must be entirely independent of 

 the distances from the plane of the paper in the second 

 case. These distances therefore correspond to different dimen- 

 sions of space. 



In fact, if the first stereoscopic view represents the 

 projection of a four-dimensional " eight-cell " in a solid 



four-dimensional space, corresponding to the five regular 

 solids enumerated in our text books of elementary solid 

 geometry, is given by Mr. S. L. Van Oss in the Trans- 

 actions of the Amsterdam Academy for 1899. The 

 largest number of faces a regular solid can have is 20, 

 the figure being known as an icosahedron, but in four- 

 dimension space, the maximum number of boundaries is 

 600, and the projections of the "600 cell" shown in Mr. 

 Van Oss's diagrams are very beautiful and symmetrical. 

 An interesting variation of the experiments described 

 in this paper may be made by cutting out the two 

 extreme figures and placing them simultaneously in the 

 stereoscope, then inverting one of them and again placing 

 in the stereoscope. In this manner two other aspects of 

 the eight-cell w^ill be seen. The scale of stereoscopic 

 relief will, howe\er, be different to what it was in the 

 previous observations, but this will not much matter. 



N-rays and Smell. 



Thh controversv concerning the objective reality of the 

 N-rays suggests "that to the proverb concerning the difficulties 

 of accounting for taste, we shall have to add other maxims 

 about the difficulties of accounting for sight and smell. On 

 the oue hand. M. Blondlot, Professor Charpentier, and M. 

 Edouard Meyer continue in their respective spheres of investi- 

 gation to add new facts each week— by means of papers read 

 before the Academie des Sciences — to the common knowledge 

 of the N-rays. On the other hand, Professor J. G. McKendrick 

 and Walter Colquhoun, as well as other observers in Great 



