A New Era of Thought

by Charles Howard Hinton

He would always have to bear in mind that the ratio of the lengths of the Z, X, and Y lines is the same from Corvus to the sectional plane as from any other point to the sectional plane. Thus, if he were taking a section where the plane cuts Arctos and Cuspis at one unit from Corvus and Dos at one-and-a-half, that is where the ratio of Z and of X to Y is as two to three, he would see that Dos itself is not cut at all; but from Cista to the point on Dos produced is half-a-unit; therefore from Cista, the Z and X lines will be cut at ²⁄₃ of ¹⁄₂ unit from Cista.

It is impossible in writing to show how to make the various sections of a tessaract; and even if it were not so, it would be unadvisable; for the value of doing it is not in seeing the shapes themselves, so much as in the concentration of the mind on the tessaract involved in the process of finding them out.

Any one who wishes to make them should go carefully over the sections of a cube, not looking at them as he himself can see them, or determining them as he, with his three-dimensional conceptions, can; but he must limit his imagination to two dimensions, and work through the problems which a plane-being would have to work through, although to his higher mind they may be self-evident. Thus a three-dimensional being can see at a glance, that if a sectional plane passes through a cube at one unit each way from Corvus, the resulting figure is an equilateral triangle.

If he wished to prove it, he would show that the three bounding lines are the diagonals of equal squares. This is all a two-dimensional being would have to do; but it is not so evident to him that two of the lines are the diagonals of squares.

Moreover, when the figure is drawn, we can look at it from a point outside the plane of the figure, and can thus see it all at once; but he who has to look at it from a point in the plane can only see an edge at a time, or he might see two edges in perspective together.

Then there are certain suppositions he has to make. For instance, he knows that two points determine a line, and he assumes that three points determine a plane, although he cannot conceive any other plane than the one in which he exists. We assume that four points determine a solid space. Or rather, we say that if this supposition, together with certain others of a like nature, are true, we can find all the sections of a tessaract, and of other four-dimensional figures by an infinite solid.

When any difficulty arises in taking the sections of a tessaract, the surest way of overcoming it is to suppose a similar difficulty occurring to a two-dimensional being in taking the sections of a cube, and, step by step, to follow the solution he might obtain, and then to apply the same or similar principles to the case in point.

A few figures are given, which, if cut out and folded along the lines, will show some of the sections of a tessaract. But the reader is earnestly begged not to be content with looking at the shapes only. That will teach him nothing about a tessaract, or four-dimensional space, and will only tend to produce in his mind a feeling that “the fourth dimension” is an unknown and unthinkable region, in which any shapes may be right, as given sections of its figures, and of which any statement may be true. While, in fact, if it is the case that the laws of spaces of two and three dimensions may, with truth, be carried on into space of four dimensions; then the little our solidity (like the flatness of a plane-being) will allow us to learn of these shapes and relations, is no more a matter of doubt to us than what we learn of two- and three-dimensional shapes and relations.

There are given also sections of an octa-tessaract, and of a tetra-tessaract, the equivalents in four-space of an octahedron and tetrahedron.

A tetrahedron may be regarded as a cube with every alternate corner cut off. Thus, if Mala have the corner towards Corvus cut off as far as the points Ilex, Nugæ, Cista, and the corner towards Sors cut off as far as Ilex, Nugæ, Lama, and the corner towards Crus cut off as far as Lama, Nugæ, Cista, and the corner towards Olus cut off as far as Ilex, Lama, Cista, what is left of the cube is a tetrahedron, whose angles are at the points Ilex, Nugæ, Cista, Lama. In a similar manner, if every alternate corner of a tessaract be cut off, the figure that is left is a tetra-tessaract, which is a figure bounded by sixteen regular tetrahedrons.