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The plane figures principally employed in determining architectural proportion are the circle, the equilateral triangle, and the square which also yields the right angled isosceles triangle. It will be noted that these are the two dimensional correlatives of the sphere, the tetrahedron and the cube, mentioned as being among the determining forms in molecular structure.
They are universal: for when I say it is impossible to be and not to be; the whole is bigger than a part of it; a line perfectly circular has no straight parts; between two points given the straight line is the shortest; the centre of a perfect circle is equally distant from all the points of the circumference; an equilateral triangle has no obtuse or right angle: all these truths admit of no exception.
Our operation with Jupiter Equilateral is no different from an independent miner's operation. We aren't different kinds of people." Tawney smiled. "When you get right down to it, we're both exactly the same thing ... scavengers in space, vultures picking over the dead remains to see what we can find. We come out to the asteroids, and we bring back what we want and leave the rest behind.
These took up positions at equidistant points, lines connecting which would have formed an equilateral triangle, the little cluster of huts surrounded by the sleeping Makalakas being in the centre. The dogs, tired of barking at the different parties of Makalakas which had arrived during the night, did not make so much of a disturbance as might have been expected under the circumstances.
Also one sees by experiment that when grinding down the crystal on a rather rough stone, directly on the equilateral solid angle, one verily finds much facility in reducing it in this direction, but much difficulty afterwards in polishing the surface which has been flattened in this manner.
To make, out of cardboard, a tetrahedron like one given to him, is a problem which will interest the pupil and serve as a convenient starting-point. In attempting this, he finds it needful to draw four equilateral triangles arranged in special positions.
However often the operation is performed, the parts remain identical with the original figure, having all its extraordinary properties, the Equilateral Triangle appearing everywhere, whereas no other rectangle can have this curious property.
At the second blow he put out its right eye, and at the third its left; and we had all the pleasure to see the physeter bearing those three horns in its forehead, somewhat leaning forwards in an equilateral triangle. Meanwhile it turned about to and fro, staggering and straying like one stunned, blinded, and taking his leave of the world.
But if things were distinguished into species, according to their real essences, it would be as impossible to find different properties in any two individual substances of the same species, as it is to find different properties in two circles, or two equilateral triangles.
The shock of the arrest, the realization that the Captain had been serious when he reeled off the charges lodged against them ... they had been certain it was some kind of ill-planned joke until they saw the delegation of Jupiter Equilateral officials waiting at the port to greet Merrill Tawney like a man returned from the dead.
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