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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.
The same would hold true in cases of possession and obsession; for if the bastion of the hand can thus be captured, so also may the citadel of the brain. Certain familiar forms of hypnotism are not different from obsession, the hypnotizer using the brain and body of his subject as though they were his own. All unconsciously to himself, he has called into play four-dimensional mechanics.
The two boundaries must have a common portion which is in fact a continuous three-dimensional locus of event-particles in the four-dimensional manifold. A three-dimensional locus of event-particles which is the common portion of the boundary of two adjoined events will be called a 'solid. A solid may or may not lie completely in one moment.
Psychic phenomena indicate that occasionally, in some individuals, the will is capable of producing physical movements for whose geometrico-mathematical definition a four-dimensional system of co-ordinates is necessary.
Now this coincides remarkably with the idea implicit in all higher-space speculation, that the figures of solid geometry are projections on a space of three dimensions, of corresponding four-dimensional forms. All ornament is in its last analysis geometrical sometimes directly so, as in the system developed by the Moors.
The reader is referred to Hinton's book, The Fourth Dimension, for an extended development of this idea. What follows is a brief summary of his argument. First, he examines the characteristics of a vortex in a three-dimensional fluid. Then he conceives of what such a vortex would be in a four-dimensional medium of analogous properties.
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.
The totality of event-particles will form a four-dimensional manifold, the extra dimension arising from time in other words arising from the points of a timeless space being each a class of event-particles.
We shall find that this discovery of definite unique properties defining perpendicularity is of critical importance in the theory of congruence which is the topic for the next lecture. I regret that it has been necessary for me in this lecture to administer such a large dose of four-dimensional geometry.
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.
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