Page:Popular Science Monthly Volume 83.djvu/391

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.