*THE FOURTH DIMENSION*

distances perpendicular to the *xy* 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 2-space only as a section of 3-space, and a *duodim* is purely an imaginary being to us; and we know of 1-space only as a section of 2-space (and therefore of 3-space), and the *unodim* is imaginary. We have seen that a duodim might interfere with life in 1-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 2-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 "*a*" or "*b,*" safe from 2-space intruders, but a tridim could help himself to anything he pleased without breaking the sides of the vault. By analogy, a 4-space being could do many things in 3-space impossible to man and entirely inexplicable to him. No 3-space safe or vault would be secure from a 4-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 "*a*" and "*b*" are symmetrical tetrahedrons,^{[1]} in length

- ↑ A model of "
*a*" and "*b*" can be readily constructed as follows:Cut out the figure (Fig. 10) from a piece of cardboard, perforated along the lines

*AB, BC, CA,*and having*AF = AE, CE = CD and BD = BF.*Fold over the triangle*ABF, ACE, CBD*till the points*F, E*and*D*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.