Notes in progress © 1998-2002 Alan McAllister

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This is supposed to be the basic geometry of the universe, the underlying patterns of all existence. The following derivation is said to parallel the creation of the world given in Genesis.

First there is the void. Then spirit reaches out in six directions, up, down, back, forth, right and left, creating three axis' which allows measurement and movement. Filling out the axis's results first in a square and then in an octagon, two base to base pyramids. (In planar figure this also gives the hexagon, and by reprojecting into 3 dimensions the cube.)

Straight lines are male, curves female, so spinning the octagon to create a sphere generates female from male. Spirit moved to the surface of the sphere and then began to make more spheres. In planar form six more can fit around the circle (Spirit rested on the seventh day). In space perhaps there are also six, one centered out each axis? These are all interlocking, i.e. the center of each is on the circumference of the others.

By rotation this generates a tube torus? which appears to be a torus with a zero inner radius, e.g. a donut without a hole. This is used to generate a figure inside a tetrahedron which in turn generates the various Hebrew letters, by projection back to a plane. [Stan Tenen]. The Greek and Arabic letters can also be formed by a similar process.

Adding two more layers of circles gives two more figures. The first is the `egg of life' which can be seen as a set of eight non-overlapping spheres in star tetrahedral form. The second is the `flower of life' which in the plane is seven non-overlapping circles bounded by a large circle. This can be used to generate the `fruit of life' by inscribing half radius circles about the centers of the larger circles, then adding a ring of six to fill in between the seven. The figure comprised of the smaller circles is the `fruit of life'.

Adding male lines back to the `fruit of life' by connecting all the centers gives Metatron's Cube. This contains 4 of the 5 Platonic solids. These are the only figures inscribable inside a sphere and made up of similar equilateral surfaces, i.e. all angles and edges are equal.

The platonic solids are the Terahedron, the Cube, the Octahedron, the Dodecahedron, and the Icosahedron. All except the Dodecahedron are found by erasing some lines from Metatron's Cube. The Platonic solids, plus the sphere, were used to symbolize the five elements and the void. Cube = Earth, Icosahedron = Water, Terahedron = Fire, Octahedron = Air, Dodecahedron = Ether, Sphere = Void.

Note that these figures are the only ways to pack spherical atoms into crystals.

In life, the egg and sperm form a `vesica pisices' (two circles overlapping, the first day of Genesis), then a zygote, then a tetrahedron with four cells, and a star tetrahedron with eight (the egg of life). The to sixteen, and a cube inside a cube, the last symmetrical configuration. Becomes a hollow sphere, then the poles connect to form a tube.

Discussion of Fibonacci numbers, the Golden Mean, and Leonardo's Human Proportion drawing. Plants grow according to the Golden Mean and create spirals using the Fibonacci numbers.

A `tree of life' figure, including some of the centers in the `seed of life', is the kabala diagram.

The Fibonacci series can be used to define both the Golden Mean and Pi, the two main irrational numbers. This is related the way in which the Fibonacci rectangle relates to the Golden rectangle, and the approximation of the Logarithmic (Fibonacci) Spiral to that based on the golden mean. One of the prime significances of all this is that these series and ratios show up so often in nature.

The basic theme is that in the ancient world musical scales were strongly correlated to mathematical knowledge, and also to astronomical and cosmological systems. The theory of integer ratios, was music. The relationship between arithmetic (ideal) and musical (real) systems, served as a basis for the philosophical relationship between the absolute and the relative. The arithmetic systems included both linear (number theory), and circular (degrees of arc). [MI]

The limitations of ancient mathematics to define the ratios of exact musical tones (equal-temperament), in particular the limitation to the use of integer ratios produced a variety of musical tunings, which were different attempts to approximate the theoretical tunings. Mathematically this involved the approximation of the square root of two, by various integer ratios. The finer the tuning, the larger the ratios. A whole series of musical mandalas are formed by arranging the powers of the prime numbers 2, 3 and 5 in combinations up to a certain maximum number. [MI]

A focal point of the discussion is the mismatch of g# and ab, opposite to the reference D. In the various tunings this gap, or overlap, has different sizes. It is equivalent to the errors in the various approximations to the square root of two. There is a correspondence between this gap and that between the actual and various calendar years, as based on the sun and the moon. [MI]

A great deal of the ancient writings that seem quite obscure may be read as musical allegory. It is shown that the generation of the notes of the musical scale maps allegorically onto the cosmological generation of the manifest universe in the Vedas, and at a later time in Plato's writings. This methodology can also be applied to Sumerian, Babylonian, Egyptian and Hebrew writings. [MI]

Start with 1, the Father, the undifferentiated, etc. Using 2, the Mother/daughter, define the octaves, powers of two, which are an identity transformation, and define the matrix within which the scales come into being. Then the male (divine and human) 3 and 5, are used to define notes, children, within the octave. The divine 3 gives fixed notes that mark the tetracords (fifths within the octave), while the human 5 generates the movable notes interior to the teracords (major and minor thirds), via pentatonic scales. [MI]

A related text.

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Last updated April 17, 1998.

© Alan McAllister