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, 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.