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Sacred Geometry

21 20 haLFLingS and thirdLingS defined by triangles and squares An equilateral triangle opposite top left, or two nested squares opposite top right both do the same thing the circle inside each of these figures is exactly half the size of the surrounding circle. This is a geometrical image of the musical octave, where a string length or frequency is halved or doubled. Appropriately, the threedimensional equivalent of the triangle, the tetrahedron, defines the next fractional proportion, one third, as the ratio of the radius of the innermost sphere to that of the containing sphere opposite bottom left. Two nested cubes, or two nested octahedra, or an octahedron nested in a cube opposite bottom right all produce one third too. The geometric third is musically equal to an octave plus a fifth in harmonic notation. Thus two dimensions quickly define a half, and three dimensions a third. Another fascinating third is shown below. ad triangulum an equilateral triangle defines two circles, one half the size of the other ad quadratum two nested squares define two circles, again one half the size of the other ad cuboctum a cube and octahedron again define an insphere one third of the outsphere ad tetratum the sphere inside a tetrahedron is one third of the size of the sphere outside
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