Animal life exhibits the phenomenon of the right-and left-handed symmetry of solids.  This is exemplified in the human body, wherein the parts are symmetrical with relation to the axial plane.  Another more elementary type of symmetry is characteristic of the vegetable kingdom.  A leaf in its general contour is symmetrical:  here the symmetry is about a line—­the midrib.  This type of symmetry is readily comprehensible, for it involves simply a revolution through 180 degrees.  Write a word on a piece of paper and quickly fold it along the line of writing so that the wet ink repeats the pattern, and you have achieved the kind of symmetry represented in a leaf.

With the symmetry of solids, or symmetry with relation to an axial plane, no such simple movement as the foregoing suffices to produce or explain it, because symmetry about a plane implies four-dimensional movement.  It is easy to see why this must be so.  In order to achieve symmetry in any space—­that is, in any given number of dimensions—­there must be revolution in the next higher space:  one more dimension is necessary.  To make the (two-dimensional) ink figure symmetrical, it had to be folded over in the third dimension.  The revolution took place about the figure’s line of symmetry, and in a higher dimension.  In three-dimensional symmetry (the symmetry of solids) revolution must occur about the figure’s plane of symmetry, and in a higher—­i.e., the fourth dimension.  Such a movement we can reason about with mathematical definiteness:  we see the result in the right- and left-handed symmetry of solids, but we cannot picture the movement ourselves because it involves a space of which our senses fail to give any account.

Now could it be shown that the two-dimensional symmetry observed in nature is the result of a three-dimensional movement, the right-and left-handed symmetry of solids would by analogy be the result of a four-dimensional movement.  Such revolution (about a plane) would be easily achieved, natural and characteristic, in four space, just as the analogous movement (about a line) is easy, natural, and characteristic, in our space of three dimensions.

OTHER ALLIED PHENOMENA

In the mirror image of a solid we have a representation of what would result from a four-dimensional revolution, the surface of the mirror being the plane about which the movement takes place.  If such a change of position were effected in the constituent parts of a body as a mirror image of it represents, the body would have undergone a revolution in the fourth dimension.  Now two varieties of tartaric acid crystallize in forms bearing the relation to one another of object to mirror image.  It would seem more reasonable to explain the existence of these two identical, but reversed, varieties of crystal, by assuming the revolution of a single variety in the fourth dimension, than by any other method.