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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.
Around it, at the three points of a triangle, were three similar, but smaller stones. Engraved lines ran from each of the stones to the center, and other lines connected the outer three in a triangle. The effect was as though one were looking down at the apex of a regular tetrahedron.
His theory was this: "Around the orbit of the earth describe a dodecahedron the circle comprising it will be that of Mars; around Mars describe a tetrahedron the circle comprising it will be that of Jupiter; around Jupiter describe a cube the circle comprising it will be that of Saturn; now within the earth's orbit inscribe an icosahedron the inscribed circle will be that of Venus; in the orbit of Venus inscribe an octahedron the circle inscribed will be that of Mercury."
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.
From the triangle of which the hypotenuse is twice the lesser side the three first regular solids are formed first, the equilateral pyramid or tetrahedron; secondly, the octahedron; thirdly, the icosahedron; and from the isosceles triangle is formed the cube. Let us now assign the geometrical forms to their respective elements.
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.
It is a well known fact that in the microscopically minute of nature, units everywhere tend to arrange themselves with relation to certain simple geometrical solids, among which are the tetrahedron, the cube, and the sphere.
Each of the four depressed areas forms a side of our tetrahedron and is occupied by an ocean. The ridges and the areas immediately flanking the oceans form the continents. The side which we may think of as the base contains the Arctic Ocean. The ridges surrounding it are broad and flat. Large parts of them stand above sea-level and form the northern portions of North America, Europe, and Asia.
On the strength of this conceit, he supposed that the distances of the planets were regulated by the sizes of the five regular solids described within one another. "The Earth is the circle, the measurer of all. Round it describe a dodecahedron; the circle including this will be Mars. Round Mars describe a tetrahedron; the circle including this will be Jupiter.
"Therefore, I will make my first move in office now, and suggest that this plaque signify, not the Coordinator, and first power of your country, but Counselor and first friend in all things in which I can serve you. "The tetrahedron you have chosen; so let it be. The apex is out of the plane of the other points, and I am out of this galaxy.
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