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QA: What is Gnomonic Expansion-Contraction-Spiral

“There are certain things,” said Aristotle, “which suffer no alteration save in magnitude when they grow…” He was referring here to the phenomenon the Greek mathematicians called the gnomon and the type of growth based upon it, known as gnomonic expansion. Hero of Alexandria defined it as follows: “A gnomon is any figure which, when added to an original figure, leaves the resultant figure similar to the original.” … growth by accretion or accumulative increase, in which the old form is contained within the new. (Taken from Robert Lawlor’s Sacred Geometry).

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Gnomon <—-> brain, as it evolved through evolution


The Spirit of Motion

Here, we shall consider motion inside the Logic Vacuum, namely the Logic-Information of Motion. The Logic of Motion has to do with the potential of movement, namely the stage before motion, as expressed in a Gnomon or a Form, e.g., triangles, rectangles, pentagrams, pentagram star, and so on.


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The above gnomon illustrates the relationship of the Pythagorean formula a 2 + b 2 = c 2. Shown above is the gnomonic increase from the square surface area of 4 to the square of 5 where the gnomon of the larger square 5 is equal to ¼ of the initial square of 4.

Next, the gnomons of a pentagram and pentagram star.

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The Logic of motion in the above diagram has to do with the golden mean and the phi (φ) proportions. See Gyorgy Doczi, The Power of Limits and Robert Lawlor’s Sacred Geometry for detailed discussions on golden mean and the phi proportions.

The Soul of Motion

With the pentagrams, the gnomonic spiral can be generated, which is shown below.


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This spiral represents the Soul of Motion, i.e., motion in the information-energy stage. In addition, the above diagram expresses the information crossover from the gnomonic pentagram (Logic-Information) onto a gnomonic spiral (Information-Energy).


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Great Pyramind of Cheops at Gizeh. Cross section shows that aporthem and half the base are in a golden section relationship

The Particle of Motion

This is the collapse of Information-Energy onto Energy-mass stage. This is the manifestation stage, manifested into a space physical form as shown below.


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This slice of a celery bunch shows a spiral arrangement of stalks accumulating as a pentagram star.

In the words of Lawlor:

One mathematical characteristic is that all figures which grow by gnomonic expansion create intersections upon which spirals can be drawn. These forms, as Jill Purce has so beautifully shown in “The Mystic Spiral,” are everywhere in nature: spiraling trunks of huge eucalyptus trees, the horns of rams and reindeer, our skeletal bones, mollusc shells, particularly the Nautilus pompilius which follows the spiral derived from the Golden Proportion. Spirals can be found in the successive florets of the sunflower, in the outline of a cordiform leaf in a lock of hair or a snake coil or an elephant trunk, an umbilical cord or in the cochlea of the inner ear.

The Holon of Motion

The following diagram shows all three stages coming together.


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The distribution of leaves around a central stem is governed by the Fibonacci Serie: 3 leaves in 5 turns, 5 leaves in 8 turns

This conclude the formulation for the Holon of Motion, Spirit–Soul–Physical motion.


Again, as Lawlor puts it:

The Golden Mean spiral, in which the geometric increase of the radial arms is equal to φ… To Pythagorean eyes, however, this form embodies the dynamics of the rhythmic generation of the cosmos, and through its harmonic principle, represents universal love.

The series that generate φ are referred to as the Fibonacci Series. Three of these series are illustrated below.

  • 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610 …
  • 1, 3, 4, 7, 11, 18, 29,47,  76, 123, 199, 322, 521, 843, 1364 …
  • 1, 5, 6.11, 28, 45, 73, 118, 191, 309, 500, 809, 1309, 2118, …

The Fibonacci Series is such that any two successive terms tend to be approximately in relation to one another as 1:φ, and any three successive terms are as 1 : φ : φ x φ … etc. (Lawlor). Via these and other φ series that the Inflationary Universe is explained within the Matter-Being Paradigm. Obviously, gnomonic contraction is the reverse process of gnomonic expansion.


Some physicists have explored the notion that if the (dimensionless) fundamental physical constants had sufficiently different values, our universe would be so radically different that intelligent life would probably not have emerged, and that our universe therefore seems to be fine-tuned for intelligent life.

One such constant binding proton with electron to form atoms is 1836 (=proton mass/electron mass), which also produced the proton-electron dipole. But 1836 reduces to the number 9 in Base-9, which binds the pre- with post-heaven octagons. Thus, 1836 like the fine-structure constant (137), binds Matter with Being just as the light constant binds the physical constants to free will. And φ binds the Logic Vacuum with the Information Vacuum to form Physical Constants. One thought is that these bindings are pre-determined by the Thought Octagon at symmetry breaks between pre- and post-octagons as the Universe’s Involution Logic, which is  pre-coded in God’s Vacuum Computer.

[Continued in postings, Physical Constant Generated by PHI (φ), and How Light Constant Derives other Physical Constants]