# Page:Popular Science Monthly Volume 83.djvu/391

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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

Fig. 8.

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

Fig. 9.

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

1. 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.