THE FOURTH DIMENSION 



387 



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 tbat 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 



M 



N 



"a." 



\ 



N 

 Fig. 9. 



"b" 



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 

 Tig. 9. Here "a" and "b" are symmetrical tetrahedrons, 3 in length 



3 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 ABE, 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. 



