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Thus in the four-dimensional geometry of all event-particles there is a two-dimensional locus which is the locus of all event-particles on points lying on the straight line r. I will call this locus of event-particles the matrix of the straight line r. A matrix intersects any moment in a rect. Thus the matrix of r intersects the moment M in a rect ρ.
The helix of many helices would become four-dimensional, and superficial space would change to solid space: each tiny circle of intersection would become a sphere of the same diameter, describing, instead of loops, helices.
"I adopt the hypothesis that that which thinks in us has an ample experience, of which the intuitions we use in dealing with the world of real objects are a part; of which experience, the intuition of four-dimensional forms and motions is also a part.
The phrase is used here in a different sense from that in which the mathematician uses it, but oddly enough four-dimensional geometry provides the symbols by which some of these occult and mystical ideas may be realized by the rational mind.
Yet they would all co-exist for the three-dimensional man of the present day. The suggestion appeared to be that, in exactly a similar way, events which to the three-dimensional man can only be perceived normally in sequence, would co-exist for a four-dimensional being. This would mean practically the annihilation of Time, as giving sequence.
We have got to determine the meaning of a space-point in terms of the event-particles of the four-dimensional manifold. There is only one way to do this. It is rather disconcerting to find that a point in space is not a simple entity; but it is a conclusion which follows immediately from the relative theory of space.
When at the end of a day he lays aside his clothes that two-dimensional sheath of the three-dimensional body it is in full assurance that his body in turn will be abandoned by the inwardly retreating consciousness, and that he will range wherever he wills during the hours of sleep, clothed in his subtle four-dimensional body, related to the physical body as that is related to the clothes it wears.
This regular solid of four-dimensional space consists of sixteen cells, each a regular tetrahedron, thirty-two triangular faces, twenty-four edges and eight vertices. It is the correlative of the octahedron of three-dimensional space. First it is necessary to establish our four axes, all mutually at right angles.
Thus, on the hypothesis that our space is traversing four-dimensional space, and that the forms of our space are cross-sections of four-dimensional forms, the unity and harmony of nature would be accounted for in a remarkably simple manner. The above exercise of the imagination is a good preparation for the next demand upon it.
This amounts to nothing less than reducing a thing from three dimensions to two, in order to know it thoroughly. Now the flux of things corresponds to the four-dimensional aspect of the world, and with this the reason finds it impossible to deal. As Bergson has so well shown, the reason cuts life into countless cross-sections: a thing must be dead before it can be dissected.
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