THE PYTHAGOREAN SOLIDS: by F. J. Dick, M. Inst. C. E., M. Inst. C. E. I.

Students of The Secret Doctrine and of ancient teachings such as those of Pythagoras, the Kabala, and the sacred books of different races and epochs, are often puzzled by the frequent references to Number, and to elementary plane forms like the circle, triangle, and square. It may be surmised that these symbols refer to meta-physical forces of various orders concealed within the "atom" and within nature generally. For nature is built, obviously enough, upon some internal principles of structural harmony. Without discussing the many avenues of thought suggested by a study of the five regular solids, the main features of these forms may be briefly summarized.

In the first place, they may be all considered as generated by Twelve Points on the surface of the Sphere, at equal adjacent distances, or by six diameters of the sphere mutually inclined at angles whose tangent is 2, the number of the octave in music. Joining each of the twelve with every other point, we have 66 lines, of which 36 are internal. Six of the latter being diameters, there remain 30, intersecting at 20 points, which give the 30 edges of the internal dodecahedron. The 30 outer, or external lines of the 66, form the edges of the icosahedron.

Joining one set of alternate corners of the Dodecahedron by 12 lines, a cube appears. So far, there are 33 points defined, including the center of the sphere. Joining opposite corners on each Cube-face by 12 lines, two interlaced tetrahedrons appear. These define, by their intersection, 6 new points and 12 new lines forming the Octahedron, beautifully poised in the heart of the Sphere.

Thus only 39 points, including the central point, are needed to define the Pythagorean solids, only one solid form being repeated, the Tetrahedron, which in fact is seen to repeat itself ten times. For between the interlaced Tetrahedron corners and the eight faces of the included Octahedron, eight smaller Tetrahedrons are seen.

The interlaced Tetrahedrons suggest the origin of the plane symbol—the interlaced triangles; but the full beauty of the symbol does not appear until we notice that the axis of symmetry of the Tetrahedrons coincides with the diagonal of the Cube, and that the orthographic projection of all these on a plane perpendicular to the diagonal gives a perfect hexagon with the interlaced triangles in the center. The interlaced Tetrahedrons—one a reflection of the other—in fact define the eight corners of the Cube. The Tetrahedron is "3," and the Cube is "4" (or 6). So we see one way in which the "three fall into the four," and why it is a septenary, and a decad, as well as a three, or a four, according to the various aspects and interrelations considered of the electric, rotary, magnetic, or vibratory forces symbolized by the various lines.

Science has already reached the speculation that the hypothetical carbon "atom" has a tetrahedronal form. Let us look at this Tetrahedron with the eye opposite the middle of an edge and in line with the center. The two opposite edges now form the Cross, composed of two equal lines, but separated by a space. One is reminded of an electric wire, and a magnetized needle placing itself at right angles to, although at some distance from, the current in the wire. Thus the opposite edges, whether as rotational vectors or in some other way, indicate a connexion with the dual forces of attraction and repulsion. The Tetrahedron, a triangular pyramid, may be a Fire-symbol. In any case the following passage is suggestive:

When the molecules of salt, clustering together, begin to deposit themselves as a solid, the first shape they assume is that of triangles, of small pyramids and cones. It is the figure of fire, whence the word "pyramids"; while the second geometrical figure in manifested nature is a square or a cube, 4 and 6; for, "the particles of earth being cubical, those of fire are pyramidal" truly—(Enfield). The pyramidal shape is that assumed by the pines—the most primitive tree after the fern period. Thus the two opposites in cosmic nature—fire and water, heat and cold—begin their metrographical manifestations, one by a trimetric, the other by a hexagonal system. For the stellate crystals of snow, viewed under a microscope, are all and each of them a double or treble six-pointed star, with a central nucleus, like a miniature star within the larger one. (The Secret Doctrine, II, 594.)

The number Five penetrates the whole system of the Five solids in a remarkable way. Thus there are 24 pentagons visible, and by joining other corners of the Dodecahedron, Five Cubes are seen, which of course produce Five Octahedrons, and twice that number of principal interlaced Tetrahedrons. Five has been said to be the Number of Life.

Confining ourselves to one rectangular system, we find Four axes of symmetry for the Tetrahedrons and Three for Cube and Octahedron. Thus there are really 73 principal lines in the complete system defined by the 39 points. A study of the three principal orthographic projections shows that the circle should be divided into 3, 4, 5, 6, parts, and the products of these, or 360 degrees. Certain angles are found in abundance, such as 36, 60, 72, 90, 108, 144; and their combinations and products by 10 and 12, and their multiples, give figures bearing a strong resemblance to the various cyclic periods of eastern chronology. Periodic orbits are vibrations on a large scale.

Twice the perimeter of an Icosahedron-face divided by the perimeter of a Dodecahedron-face is 3.1416, the value of π used in all ordinary scientific and constructional work.

The actual error is so small that if both were accurately made of copper at the same temperature, the Icosahedron-face would only have to be brought rather more than one degree Fahrenheit below the temperature of the other for the π value to be absolutely correct. Accuracy of this sort is unattainable outside of specially equipped laboratories. So the Pythagorean solids may be said to "square the circle."