*THE FOURTH DIMENSION*

distances perpendicular to the plane are *positive* if measured *above, negative* if measured *below.* This notation enables us to locate any point in our space.

Now we know of -space only as a section of -space, and a *duodim* is purely an imaginary being to us; and we know of -space only as a section of -space (and therefore of -space), and the *unodim* is imaginary. We have seen that a duodim might interfere with life in -space, but the unodim would not know at all what had caused the

interference. We have also seen that a tridim might in a similar way interfere with life in -space. The important point to observe is that in either case the inhabitant of the lower space would not understand what had caused the change.

A duodim could lock up his treasure in circular or polygonal vaults, such as "" or "," safe from -space intruders, but a tridim could help himself to anything he pleased without breaking the sides of the vault. By analogy, a -space being could do many things in -space impossible to man and entirely inexplicable to him. No -space safe or vault would be secure from a -space burglar. He could get a ball out of a hollow shell without breaking the surface, he could get out the

contents of an egg without cracking the shell and enjoy the kernel of a nut without the use of a nut-cracker.

A geometrical illustration similar to those already given is found in Fig. 9. Here "" and "" are symmetrical tetrahedrons,^{[1]} in length

- ↑ A model of "" and "" can be readily constructed as follows:
Cut out the figure (Fig. 10) from a piece of cardboard, perforated along the lines , and having and . Fold over the triangle till the points and meet in a point, thus making one tetrahedron: fold the triangles in the opposite direction and the symmetrical tetrahedron will be formed. The one corresponds to the image of the other in a mirror.