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It is created but not creative; it is imagined but not imaginative. Organic architecture is both creative and imaginative. It is non-Euclidian in the sense that it is higher-dimensional that is, it suggests extension in directions and into regions where the spirit finds itself at home, but of which the senses give no report to the brain.
Commenting upon the significance of Lobatchewsky's and Bolyai's work along the lines of non-Euclidian geometry, Hinton says, "By immersing the conception of distance in matter, to which it properly belongs, it promises to be of the greatest aid in analysis, for the effective distance of any two particles is the result of complex material conditions, and cannot be measured by hard and fast rules."
One form of the modern non-Euclidian geometry assumes that a similar theorem is true for the space in which our universe is contained.
Geometricians have always sought to found their science on the most logical basis possible, and thus have carefully and critically inquired into its foundations. The new geometry which has thus arisen is of two closely related yet distinct forms. One of these is called NON-EUCLIDIAN, because Euclid's axiom of parallels, which we shall presently explain, is ignored.
He knew no more than a firefly about rays or about race or sex or ennui or a bar of music or a pang of love or a grain of musk or of phosphorus or conscience or duty or the force of Euclidian geometry or non-Euclidian or heat or light or osmosis or electrolysis or the magnet or ether or vis inertiae or gravitation or cohesion or elasticity or surface tension or capillary attraction or Brownian motion or of some scores, or thousands, or millions of chemical attractions, repulsions or indifferences which were busy within and without him; or, in brief, of Force itself, which, he was credibly informed, bore some dozen definitions in the textbooks, mostly contradictory, and all, as he was assured, beyond his intelligence; but summed up in the dictum of the last and highest science, that Motion seems to be Matter and Matter seems to be Motion, yet "we are probably incapable of discovering" what either is.
The space intervening would grow smaller, at last being filled up. The earth would then be so expanded as to fill all existing space. This, although to us the most interesting form of the non-Euclidian geometry, is not the only one.
In the other form space is assumed to have one or more dimensions in addition to the three to which the space we actually inhabit is confined. As we go beyond the limits set by Euclid in adding a fourth dimension to space, this last branch as well as the other is often designated non-Euclidian.
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