A New Era of Thought

by Charles Howard Hinton

Fig. 39.	Fig. 41.
Fig. 40.

(vi)

The octa-tessaract is got by cutting off every corner of the tessaract. If every corner of a cube is cut off, the figure left is an octa-hedron, whose angles are at the middle points of the sides. The angles of the octa-tessaract are at the middle points of its plane sides. A careful study of a tetra-hedron and an octa-hedron as they are cut out of a cube will be the best preparation for the study of these four-dimensional figures. It will be seen that there is much to learn of them, as—How many planes and lines there are in each, How many solid sides there are round a point in each.

A Description of Figures 26 to 41.

													Z		X		Y		W
Z I	.	X I	.	Y I	.	W I	-		26	is a	section	taken	1	.	1	.	1	.	1
									27	„	„	„	11⁄2	.	11⁄2	.	11⁄2	.	11⁄2
									28	„	„	„	2	.	2	.	2	.	2

													Z		X		Y		W
Z II	.	X II	.	Y II	.	W I	-		29	is a	section	taken	1	.	1	.	1	.	1⁄2
									30	„	„	„	11⁄2	.	11⁄2	.	11⁄2	.	3⁄4
									31	„	„	„	2	.	2	.	2	.	1
									32	„	„	„	21⁄2	.	21⁄2	.	21⁄2	.	11⁄4

The above are sections of a tessaract. [Figures 33] to [35] are of a tetra-tessaract. The tetra-tessaract is supposed to be imbedded in a tessaract, and the sections are taken through it, cutting the Z, X and Y lines equally, and corresponding to the figures given of the sections of the tessaract.

[Figures 36], [37], and [38] are similar sections of an octa-tessaract.