161 What does the Universe Understand about Building? 
Date: 11/07/04

What Does the Universe Understand about Building?

Can Universe be more than genetic?

Over the years, I found that it doesn't understand 8-bit binary representing our numbers, nor does it understand numbers, "platonic solids", letters, alphabets, mathematics, co-ordinate systems, tori, geometry, primes, physics, time, space, spacetime, higher mathematics, past, future, prayers or any numbering system more than zero and one zero. These abstractions may be in resonance, only, with Universes understanding. In short, Universe is one big, dumb, genetic unconsciousness and the genetic impulse is reproduction. All of Universes higher consciousness, character and culture is, by increasing degrees, from earths minerals, plant, animal and humans. So much for what it doesn't know.

Another author asks the same question and notes what Universe does know:

"The universe does not know numbers but it knows ratios between numbers,
and how to produce results with its own elements."

Several years ago, after considering how the Universe really works, I formed this:

The tetrahedral physics theory will become the tet-double
tet-cubic-octa-dodeca-cuboctahedral-mayan time star theory.

The meaning is? 1 tori-center is the point which is already a 4D object, 2 tori-centers is a line between centers, 3 tori-centers is a triangular surface and 4 tori-centers, a tetrahedron and this is only the beginning of construction of matter. 5 tori in a plane is a pentagram, 6, a hexagon but 5 and 7, 9, 10 and 11 tori grouped as a 3D, close-packed-cluster doesn't have a name. A cluster of 12 tori-centers is a gravity unit called a Vector Equilibrium which can transform between a Dodecahedron and a Icosahedron. And on and on, adding one sphere after another.

Universe only recognizes very simple spiraling of space, sphere-tori-space-paths, ratio including music ratio (Example: 3:4), the place-holder zero (Number: 0), the "measure of number" (Number: 1) and how to produce with its own elements. This, Universe understands.

In the beginning

was the tone and the curve (of space). This curve spiraled around (not circled) until a diameter of Universal Will Power could cross the spiral. In the center appeared the Absolute Self, which instantly transcended its appearance and dis.appeared out of existence, being an abstract IDEA.tion. 2

Stepping over the details of Creation, I want to concentrate on the opening question:

Universe understands ratio

and relationship between a spiral of space and a spiral of space just like it. This is very primitive intelligence. Since I concluded that universe only understands ratio 0 : 0, meaning zero and one zero (the differentiation between "a spiral of space and a spiral of space just like it", it also limits us in understanding this Reality.

So we have to invent a measuring system, just as Science has done, but we will limit it to the digits 1-9, previously noting that our arbitrary numbers other than 0 (zero) and 1 zero, is a total invention and is not recognized by Universe. Let's use them as indicators of a resonant or dissonant condition to 0 and 1. As Universe produced a series of spirals or spherical bubbles by way of the spiraling light, actually spiraling potential of space, at space-velocity not at light-velocity, each bubble having a center, we can now say that a group of 4 connected sphere-centers, if center-connected by line, creates what we call a tetrahedron, the most simple of crystalizations from "empty" space. We are actually seeing spirals of invisible space, spiraling about the invisible surfaces of invisible sphere-tori, which we call "rotation of dimension." With each relative, 90 degree rotation of this invisible, spiraling space/quanta, we say that rotation creates a new dimension. I see no other explanation in existence, that tells me how the beginning rotation, beginning dimension, beginning spheres and Platonic shapes and space are tied to each other. If space does not create form, let me know. If 90 degree rotation applies to something other than space, let me know. If Platonic shapes are not the product of its own connecting centers of groups of close-packed sphere-tori, let me know. If light radiates from anything other than matter, let me know. Does universe understand all this? No, just ratio, 0, 1 and how to produce with its own elements.

Diverging into dimension and quanta

for a moment, recall quotes from three people, that explains some sticky problems.

"Your eye needs at least 11...individual quantum chunks of light coming
from a distant star before your eye can see that star." Fitzpatrick.

"It seems that the entire 10,000 unit torus cycles starting at the 11th dimension is
...total chaos before the 11th dimension...total order and predictability
...above the 11th dimension." J.Iuliano

"If this is 26-dimensional, as required by quantum mechanics for
spinless strings, the 10 closed strings formed by the curling up of the
11-brane proposed by the author in Articles 2 and 5 and Reference
5 around higher, compactified dimensions possess 251 space-time
co-ordinate variables." Article06.pdf, Phillips.

Fitzpatrick notes that 11 quanta is needed before we can see it....Juliano notes total chaos before 11 dimensions.....Phillips notes 10 closed strings are a product of an 11th-brane, of string theory.

Is a quanta and a dimension the same?

Follow simple logic, arrive at amazing conclusions. We've heard that dimension is a 90 degree rotation but what is rotating? It's not light. There is no light in space. Fitzpatrick also guessed it. It's space's path that is spiraling (rotating). Yes, everything Einstein misassigned to light, gravity and mass, belongs to space. Quanta is defined as a discrete packet of energy. Even quanta is misassigned to light. It is space that has the so-called "charge" of energy called + and - and this misassigned charge is actually phase of rotation or precession of the quanta of space 3 4. And so, space is the rotation, space is the "charge", space is the quanta, space is the dimension and dimension applies to every 90 degree spiral of space. Additionally, we can't see what science erroneously calls a "quantum of light" radiating from matter, only, until space rotates 11 dimensions or rotates in a spiral 2 and 3/4 times (90 degrees = 1 dimension x 4 = 360 degrees x 2 = 720 degrees + 90+90+90 = 990 degrees = 11 dimensions (4 dimensions x 2 = 8 dimensions + 3 (90+90+90) = 11 )).

We can conclude that there are unlimited dimensions of space-quanta-rotation. We are only tuned to detect 4 dimensions. Are there 4 dimensions we can see and 7 really small ones (4+7=11) that can fit through the eye of a needle? No. There are 4 visible dimensions we can see and 4 invisible, complimentary partner dimensions we can't see because the other 4 are the invisible half, but still very necessary to the ones we know. Then there are 3 more dimensions beyond those 4, to which is added another 4th dimension...which has, again, 4 invisible, complimentary partner dimensions..... and trillions more beyond that, because dimension is a 90 degree rotation. 11 and 26 dimensions are stop-gap plugs for kids who can't handle unlimited dimension. Universe and humans understand the ratio between 4 dimensional quanta and universe understands the ratio between the quanta of a trillion rotational dimensions. Does universe understand all this detail? No, just ratio, 0, 1 and how to produce with its own elements. Can we use this information for manipulating Ananda's formulae, further on?

"300 times 144,000 minutes of arc per earth grid second
in a vacuum for the speed of light = 43,200,000, this
decimal harmonic of 432 and 24 hours in second 43,200..."

Creation is

space creating close-packed tori spacepaths throughout the non-medium of space. Now we can begin to see what universe sees. It isn't light. There is no light and the close-packed sphere-tori don't move either, but it does precess. 3 Here is a quote about time and ratio:

"Not surprisingly, the same ratios that describe consonant tones in music can be derived from the ratios of angles, faces, edges, vertices and other parameters describing the Platonic solids. For instance, just the ratio of vertices to faces yield the unison (1:1 in the tetrahedron), the fourth (3:4 in the octahedron) and the major sixth (3:5 in the icosahedron). Similarly, the ratio of vertices to edges yield the octave (1:2 in the octahedron) and the fifth (2:3 in the tetrahedron, cube and dodecahedron). Physics has shown us that matter is created, not from fundamental particles, but from waves or energy or vibrations that are brought to focus at a point in spacetime." 2

Ratio is important

because it should tell us what happens when invisible space, not light, rotates past certain points or positions on a sphere-tori. This new information, for example, of "the ratio of vertices to faces yield the unison (1:1 in the tetrahedron)."

So what? What's so special about these ratios?

We are finding out, step by step, how energy is transferred and how unconsciousness becomes conscious. Space/potential, spirals over the surface of the sphere-tori of space. When space/potential reaches these degree-number-points, energy exchanges occur. The potential information of space has a chance to transfer, not move. These transfer points seem special because they are both harmonic, as in musical interval (ratio) and are geometric, as in Platonic "shapes" although these are never visible shapes, in space. Does universe understand all this? No, just ratio, 0, 1 and how to produce with its own elements.

Gematric harmonics of the ATON Institute.

Phi Harmonics in
Faster-than-Light Quantum Tunneling
By Ananda,
ATON Institute, Norway.

Gematric numbers that show up in the text:

"Phi 1.618, 432 or 43,200, 5 cubes at 72 degrees, 5 golden triangles at 72 degrees, 5- 36 degree lines, numbers 36, 72, 108, 144, 180, 216, 252, 288, 324, 360, 396, 432". 1

"43,200, the number of seconds in 12 hours, this is 600 times 72 degree. Hence, make a complex of 20 dodecahedrons". 1

  • Let's try some mathematics to see how it works.
  • 12 hours x 60 min = 720 min x 60 sec = 43,200 sec = 600 x 72 degrees.

..."another 18 times, and one has a stellation PHI complex of 20 dodecahedrons. On each dodecahedron the number sequence continues. i.e. from 2,160 of the first dodeca, we commence to the next stellated dodeca: 2,196; 2,232; 2,268; 2,304; 2,340 etc. With the last pentagon point of dodecahedron 20 giving 43,200 degrees. One could also do this with dodecahedrons of the same scale, making a daisy chain from the north and south face, of 20 dodecahedrons, linked by their faces, and the last 36 degree point of the 20th dodecahedron is 43,200 degrees from the initial first 36 degree base of the first dodecahedron". 1

  • 43,200 s12hrs or degrees or 20 Dodecs x 18 = 777,600?

  • 43200 s12hrs / 2160 = 20 Dodecs
  • 43200 s12hrs / 2196 = 19.672131148
  • 43200 s12hrs / 2232 = 19.35483871
  • 43200 s12hrs / 2268 = 19.047619048
  • 43200 s12hrs / 2304 = 18.75
  • 43200 s12hrs / 2340 = 18.461538462

"300 times 144,000 renders 43,200,000 -- a perfect decimal harmonic of 43,200, which can be comprised of 20,000 stellated dodecahedrons in PHI". 1

"OVERVIEW: 300 TIMES 144,000 MINUTES OF ARC PER EARTH GRID SECOND IN A VACUUM FOR THE SPEED OF LIGHT = 43,200,000, this decimal harmonic of 432 and 24 hours in second 43,200, and the Great Pyramids polar radius ratio to Earth from the multiplication of its original height by 43,200, as well as its base (in multiplication of 43,200) to the circumference of the equator, may be quite interesting. Geometrically these decimal harmonics are extremely interesting". 1

  • 300 x 144,000 moarc/egs = 43,200,000 harmonic of 432.

..." when we multiply the vacuum speed of light of 144,000 nautical miles per grid second by 2.25. The geomatria figure that results works perfectly in the geometrical 3D modeling of the PHI complex. For the resulting 324,000, is 9,000 times that 36 degree for the pentagon lines, and 324,000 in this mapping of degree's fits precisely 150 dodecahedron's, whether in a PHI cascaded stellation relationship with the icosahedron, or by daisy chaining relationships of their pentagon faces". 1

  • 144,000 sol x 2.25 = 324,000 degrees / 36 degrees = 9,000 / 150 Dodecs = 60 degrees each Dodec?

"Indeed, professor Nimtz had succeeded in sending the information that was Mozart's 40th Symphony superluminally, which is not speculation. The 25% mentioned above, if the correct figure, is once again of significance. For 25% of 144,000 nautical miles per grid second, is 36,000 nautical miles per grid second. This is also a 36 degree pentagon decimal". 1

  • 144,000 sol x .25 = 36,000 nmps

"Hence, the 25% tunneling speed exceeding the speed of c, renders 180,000 nautical miles per grid second, in the vacuum. Here we have a decimal harmonic of 180, which is the five 36 degree lines of the pentagon. It is also 2,500 x 72 degree, the golden triangle. Hence, another geometrical PHI complex". 1

  • 5 x 36 degrees of pentagon = 180 degrees = 2,500 x 72 degrees = 180,000 nmps

The ATON Institute has some advanced knowledge about this gematric science even though the authors have not escaped the ruse of light traveling and its existence in space. Since the language plainly describes light and its velocity in a vacuum, demonstrating gematric distances in gematric times, it is clear that ATON is speaking of light. Even this needs correction, for Einstein's velocity of light is actually the velocity of space itself. 1 It was changed to light so it could appear to be measurable. Does universe understand all this? No, just ratio, 0, 1 and how to produce with its own elements.

Does it fit? Platonic numbers and ATON gematria.

From the beginning, the so-called Platonic "solids" are trying to be perfect sphere-tori. Here's some information about the 5 Platonics:

The Tetrahedron. Space assembles 4 spirals near each other:

  • The Tet is its own dual.
  • The ratio of vertices to faces yields the unison and fifth (1:1 and 2:3 in the Tetrahedron). 2

    1. 43,200 x 2/3 = 43,200 x 2 / 3 = 28800 (Double light harmonic).

  • The ratio of vertices to edges yield the fifth (2:3 in the Tetrahedron). 2

    2. 43,200 x 2/3 = 43,200 x 2 / 3 = 28800 (Double light harmonic).

  • # of triangles (: 3) = 4 at 60 degree face angles for corresponding face components. x

    3. 43,200 / 4 = 10800 (Harmonic of gematric 108).

  • Total # of faces = 4 x

    4. 43,200 / 4 = 10800 (Harmonic of gematric 108).

  • Total # of edges = 6 x

    5. 43,200 / 6 = 7200 (Harmonic of gematric 72).

  • Total # vertices (a.k.a. apices) = 4 x

    6. 43,200 / 4 = 10800 (Harmonic of gematric 108).

  • Dihedral angles between polygons with this many sides (3-3) = 70 32' = 70.529 x
  • Central Angle = 109 28' x = 109.471 x

The Cube. Space assembles 6 spirals near each other:

  • The Cube is dual to the Octahedron. Note the commonalities between these two, permitting transformation.
  • The ratio of vertices to faces yields the fifth (2:3 in the Cube). 2

    1. 43,200 x 2/3 = 43,200 x 2 / 3 = 28800 (Double light harmonic).

  • The ratio of vertices to edges yield the fifth (2:3 in the Cube). 2

    2. 43,200 x 2/3 = 43,200 x 2 / 3 = 28800 (Double light harmonic).

  • # of squares (: 4) = 6 at 90 degree face angles for corresponding face components. x

    3. 43,200 / 6 = 7200 (Harmonic of gematric 72).

  • Total # of faces = 6 x

    4. 43,200 / 6 = 7200 (Harmonic of gematric 72).

  • Total # of edges = 12 x

    5. 43,200 / 12 = 3600 (Harmonic of gematric 36).

  • Total # vertices (a.k.a. apices) = 8 x

    6. 43,200 / 8 = 5400 (Harmonic of gematric 54).

  • Dihedral angles between polygons with this many sides (4-4) = 90 x = 90.000 x
  • Central Angle = 70 32' x = 70.529 x

The Octahedron. Space assembles 8 spirals near each other:

  • The Octahedron is dual to the Cube. Note the commonalities between these two, permitting transformation.
  • The ratio of vertices to faces yield the octave (1:2 in the Octahedron). 2

    1. 43,200 x 1/2 = 43,200 x 1 / 2 = 21600 (Harmonic of gematric 2160).

  • The ratio of vertices to edges yield the octave (1:2 in the Octahedron). 2

    2. 43,200 x 1/2 = 43,200 x 1 / 2 = 21600 (Harmonic of gematric 2160).

  • # of triangles (: 3) = 8 at 60 degree face angles for corresponding face components. x

    3. 43,200 / 8 = 5400 (Harmonic of gematric 54).

  • Total # of faces = 8 x

    4. 43,200 / 8 = 5400 (Harmonic of gematric 54).

  • Total # of edges = 12 x

    5. 43,200 / 12 = 3600 (Harmonic of gematric 36).

  • Total # vertices (a.k.a. apices) = 6 x
  • Dihedral angles between polygons with this many sides (3-3) = 109 28' x = 109.471 x
  • Central Angle = 90 x = 90.000 x

The Dodecahedron. Space assembles 12 spirals near each other:

  • The Dodecahedron is dual to the Icosahedron. Note the commonalities between these two, permitting transformation.
  • The ratio of vertices to faces yield the fifth (2:3 in the Dodecahedron). 2

    1. 43,200 x 2/3 = 43,200 x 2 / 3 = 28800 (Double light harmonic).

  • The ratio of vertices to edges yield the fifth (2:3 in the Dodecahedron). 2

    2. 43,200 x 2/3 = 43,200 x 2 / 3 = 28800 (Double light harmonic).

  • # of triangles (: 5) = 12 at 108 degree face angles for corresponding face components. x

    3. 43,200 / 12 = 3600 (Harmonic of gematric 36).

  • Total # of faces = 12 x

    4. 43,200 / 12 = 3600 (Harmonic of gematric 36).

  • Total # of edges = 30 x

    5. 43,200 / 30 = 1440 (Harmonic of gematric light 144).

  • Total # vertices (a.k.a. apices) = 20 x

    6. 43,200 / 20 = 2160 (Unity to gematric 2160).

  • Dihedral angles between polygons with this many sides (5-5) = 116 34' x = 116.565 x
  • Central Angle = 41 49' x = 41.810 x
  • The Dodecahedron is comprised of twelve pentagonal faces. It represents the fifth sacred element. x
  • The angles of the five sides of a Pentagram are at a ratio of exactly 1.618 the Golden Mean ratio, known mathematically as phi. x

The Icosahedron. Space assembles 20 spirals near each other:

  • The Icosahedron is dual to the Dodecahedron. Note the commonalities between these two, permitting transformation.
  • The ratio of vertices to faces yield the major sixth (3:5 in the Icosahedron). 2

    1. 43,200 x 3/5 = 43,200 x 3 / 5 = 25920 (Circumference of earth).

  • # of triangles (: 3) = 20 at 60 degree face angles for corresponding face components. x

    2. 43,200 / 20 = 2160 (Unity to gematric 2160).

  • Total # of faces = 20 x

    3. 43,200 / 20 = 2160 (Unity to gematric 2160).

  • Total # of edges = 30 x

    4. 43,200 / 30 = 1440 (Harmonic of gematric light 144).

  • Total # vertices (a.k.a. apices) = 12 x

    5. 43,200 / 12 = 3600 (Harmonic of gematric 36).

  • Dihedral angles between polygons with this many sides (3-3) = 138 11' x = 138.190 x
  • Central Angle = 63 26' x = 63.435 x

What are the Sphere-Tori doing?

Spiraling space progresses from point, line, surface, solid and beyond to aggregates under harmonics. These sphere-tori rotate, precess and breath "in and out" and the duals have the ability to turn inside-out and outside-in. Breathing is a expansion and contraction of the transverse wavefronts of space. These sphere-tori are the previous wavefronts, layed down from the beginning of the invisible universe. Pure harmonic interaction reigns, in deep space, curved only by the largest Universal tori. The tighter space spirals, the more it becomes like mass. Refer to Ayana 8 for more details on rotation, degree, vector angle and combinatorics. Does universe understand all this? No, just ratio, 0, 1 and how to produce with its own elements.

What Is the most likely topology of Sphere-Tori?

Two popular hypothesis exist, from the Newton/Kepler era.

Nested Sphere-Tori Hypothesis

Hypothesis: Space is completely assembled from the beginning and is sphere-like 4. This seems the most likely hypothesis, from Newton's nested Platonic solids work. As nested assemblages of space expand from the tiniest centers of 4-sphere-tets to its extreme expansion of, for instance, a Snub Dodecahedron with its 80 faces or a crystallized shape of many more faces, space may transmute a trillion times as each resonance instantly searches for and finds a matching resonance. Nested sphere-tori are space impulses expanding over all the vertexes of every more-complicated spatial vortexes called by the name, Platonic solids. The angles of these nested shapes should be the angles at which energy transfers from shape to shape. All these shapes are actually invisible and is the form of vortexing space. The premise is that vortexing space pre-existed these angled, building parts.

Instant Build Sphere-Tori Hypothesis

Hypothesis: Space is continuously being assembled. This hypothesis implies that space is capable of assembling itself from the most basic parts, the tiniest tet triangle groups of 4 space vortexes, adding more and more tet vortexes so as to build or disassemble, immediately, every Platonic shape of space vortexes, as necessary. The premise is that the building parts pre-existed the final vortex of space.

Add the Fibonacci ratios related to music: 5

Note A A D F E C E C# F# C# D F
Interval root octave 4th aug. 5th 5th minor 3rd 5th 3rd 6th 3rd 4th aug. 4th
Fibonacci ratio 1/1 2/1 2/3 2/5 3/2 3/5 3/8 5/2 5/3 5/8 8/3 8/5

. . . Icosa
Major sixth
. .

Can we do anything with this information?


  1. Fibonacci ratios and Music can be used to map one of Fitzpatrick's "A" Laws 6 parameters, which is "frequency".
  2. The information about the 5 Platonics can be used to map Fitzpatrick's parameter, "orientation".
  3. The techniques of the Precession Phase Theory of the Anu Building Block 7 can be used to map the phase relationships of clusters.
  4. "Inertial qualities" or "alignment" applies to how a particular wave can lock onto another wave which represents a wave right next door or represents a wave of a far off source of the local wave. All waves at every level, instantly search for a compatible wave to lock onto. Inertia seems to be how the rest of universe influences a particular wave. What method is best for mapping this lock-on?

Universe recognizes

"Frequency, impedance, phase and alignment".
Put another way:
"Frequency, motion, orientation and inertial qualities". 5


1. Phi and Light
2. The Big Bang
3. Precession Phase Theory of the Anu Building Block
4. The Anu Building Block
5. Binary Mapping of Universal Levels
6. Fitzpatrick's "A" Laws
7. Precession Phase Theory of the Anu Building Block
8. Ayana

Impossible Correspondence Index

Copyright. Robert Grace. 2004


A Model for Describing the Universe


The following text describes a model of the universe.  Fundamental to it is that cosmic phenomena, such as time, electromagnetic waves or particles, gravity and space itself are manifestations of dimensions or geometries. These dimensions are discussed here as the result of being extracted from the traditional set of spatial vectors: length, width, height and the vector usually described as the 4th dimension, time.

Before going any further it is necessary to reassign the traditional numbers so they can be manipulated within this model.  The re-assignment is not arbitrary but key and follows basic mathematical procedure.

Array of Dimensions:


























Mathematical systems are sets of elements that interact according to the rules of an operator.  The operator here is D, a dimensional operator that separates, or divides dimensional combinations by various permutated constituents. D1 is time the first, or primary, or fundamental, or identity dimension, because it acknowledges change without assigning a vector to it.  D2, D3, and D4 are the three familiar spatial dimensions, each with vectors orientated 90 degrees from another.

The values in the table are the complete array of dimensions that are generated in this system.  Any dimension divided by the identity is equal to itself; but the identity divided by any of the spatial dimensions produces different results.   In the case of 2/4, the answer, reduced to lowest terms, becomes 1/2.  This means that the dimensions produced by 2 D 4 and 1 D 2 are the same. There are eleven total distinct dimensions in the system, the same as the number stated by Einstein and others as existing in our universe.  The universe does not know numbers but it knows ratios, and how to produc(e) results with its own elements. This mathematical system may be the most accurate model of how dimensionality is generated in the universe.

Dimension Summary - I

D1 = time - regarding time in the context of this model.
D2, D3>, D4 = length, width, height (spatial dimensions)
D1/4, D1/3, D1/2, D2/3, D3/4, D4/3, D3/2 = fractional, composite dimensions which we still need to describe. Physicists talk about these dimensions being very small, curled up, as it were, inside the others. What does that mean?

Lets start with what we know. D1 is time, or change or motion. Lets agree to use the word motion for this investigation. D2 is a single spatial dimension, a line; D3 is 2 spatial dimensions, a solid object.  Of course, if they exist with D1, as they do in the universe, all these geometries are in motion.  To generate the other 7 dimensions, we will need to divide, and to perform division of dimensions we need to examine the meaning of division as it relates to the physical universe.

It will be useful to think of division as a separation, or extraction. For instance, using numbers to illustrate, if we take the quantity 10 and we would like to extract the fives from this ten, we will end up with two sets of them. Now lets take a simple case using dimensions: 4 D 2 = 2. In words, when we extract 2 dimensions from a solid object (in motion, of course), the result is 2 distinct electromagnetic waves (photons), whose amplitudes are perpendicular to each other (intersecting planes). And how do we induce this dimensional split? By accelerating our solid object (already in motion) to the speed of light (c). This is clearly a catalyst in the process, adding enough instability to the system to cause the bound dimensions to break apart.

I think that each of the eleven dimensions represents one basic aspect, one constituent of the universe.  The attempt by physics to explain phenomena by associating discrete particles with each has gotten out of hand.  It may be more useful and to the point to identify eleven fundamental properties of the universe, each associated with its own dimension. Thus far we have identified three (four if you choose to count the zeroeth dimension, the singularity, which exists out of time, and arguably belongs to the realm of nonexistence, rather than existence).

Please consider this a work in progress. Many ideas still need refinement and clarification. Despite that, the concepts are so exciting that I could not resist posting it.

So, the reader is requested to take the view that this is presented to stimulate thought and provoke discussion rather than as a formal presentation of a finished work.

Dimension Summary II

D1 = the identity dimension, time, or motion
D2 = wave phenomena, the electromagnetic force, or the photon
D4 = ordinary matter, always in motion, or always radiating a temperature.
D3 = before associating phenomena with D3 some preliminary concepts need to be discussed.

D3 Preliminary Discussion

In mathematics, equations are commonly written to describe phenomena in any of the four integral dimensions. The exponent of the variable indicates the degree of dimensionality:  first degree equations have one solution and describe straight lines, second degree equations (with x2) have two solutions and describe planar geometries like circles or parabolas, third degree equations (3 solutions) describe solid objects.  For the fourth dimension, we have a choice:  either write the exponent 2 or write the exponent zero.  Lets do some math and play with a very simple equation.  The equation is x4 = 1.  You say x4 = 1?  That's dumb.  A fourth degree equation has four solutions, but here, all four solutions are the same.  X can only equal 1, right?  Wrong!  You are forgetting how to solve an equation like this. Proceed as follows, treating the equation as if it were an ordinary quadratic:

1. Set the equation equal to zero.
2. Factor the equation into the difference of squares.
3. Let each factor independently equal zero.
4. Both these equations are quadratic. The one on the left is again factorable.
5. Repeat step 3 for this set of equations.
6. The two solutions for these equations are X = 1 (the answer we already know) or X = -1  (the answer we forgot about but which checks since (-1)4 does equal 1.
7. We still must solve the equation on the right side of step 3: X2 + 1 = 0.  It is not factorable, so we will need to use the quadratic formula.

When ax2 + bx + c = 0, then x = 0 +/- Sqrt (b2 - 4ac) / 2a

8. Using the formula, where a = 1, b = 0, and c = 1,

x = 0 +/- Sqrt ((0)2 - 4 (1) (1)) / 2 = +/- Sqrt -4/2

9. Remember that the square roof of -4 is called 2i (right imaginary numbers), so the two solutions for x2 + 1 = 0 are i and -i.
To check: Is (i)4 = 1?  Well, i2 = -1, and (-1)2 = 1 yes.
Is (-i)4 = 1?  (-i)2 = -i2 = -1  also yes.

Therefore, the four solutions for x4 = 1 are +1, -1, i and -i.
Now for the interesting part.  i is the first imaginary number.  Lets list the first five exponents of i:
i0 = 1 (since anything raised to the zero power is 1)
i1 = i (itself)
i2 = -i (by definition)
i3 = i2 x i = -1 x i = -i
4 = 1

Observe that the exponents zero and four give the same result. When we write the imaginary number i, we are really writing i1.  These two symbols each have cosmological significance:

The exponent portion, 1 indicates the first dimension time or motion.  1, the fundamental, imaginary constituent of reality the point. Again, time is going to behave like a spatial dimension, because a point in motion becomes a straight line. Simply by having an imaginary point in motion, we are generating the phenomenon of distance.

When we wanted to break our solid object into simpler dimensions, we had to increase its motion to the speed of light. And yet, nothing can move faster than light. So our D2 entity, the photon (which we can represent mathematically as i2 or 1) is real, but mass-less, and has a velocity of c. When we step down a dimension, a dimension, we reduce exponents by 1, and to reduce the exponent 2 of i2 to 1, to get i1, we can take the square root (in fact, that is the definition of i is that i equals the square root of negative one).  It was the attaining the velocity of c that split D4 into D2 + D2; conversely, reducing motion below c would reunite D2 + D2 into D4.  So if we wish to split D2 into D1 + D1, we certainly cant lower its velocity (this doesn't split, it joins), but we cant increase it either, since c is the maximum velocity in the universe.  Therefore, it stands to reason that the velocity of the point in motion, D1, or i1, is also c.

Ready to begin a discussion of D3?  By using my ideas, we can make some sense of this very abstract notion, and link it to one of the real mysteries of astrophysics. I can think of three ways of getting to D3 from something we know.  We can take D2, the photon and kick it up a notch, and do   D2 + D1 = D3.  In words, this means that we add a new 90 degree vector to our planar electromagnetic wave, producing an abstraction we can refer to as a cubic wave. I don't know about you but I cant visualize that (for a good reason, as you will see). We can also try D4 - D1 = D3.  In words again, we are taking one dimension, say the depth, away from solid matter, to get a plane in motion (once again time as a spatial dimension), but what is a plane in motion? Maybe a tube of some sort. Maybe not. The third option involves negatives. To review: D4, i4 and +1 are all symbols we can use to represent solid matter as we know it.  D2, i2 and -1 are symbols that correlate to electromagnetic waves or photons. The only difference between D4 and D2 is that D2 = D4 x c (remind you of something? E = mc2, perhaps?). Substituting numerals, -1 = (+1) c, and solving c = -1.  In words, c has the effect of negativizing a phenomenon.

We have already referred to cs splitting or destabilizing properties; how it is useful to concentrate on this anti-polarity feature.)  So we can say that the photon is the negative state of matter. A little while ago, we stated that the point in D1or i1, has a velocity of c. What do you suppose would happen if we slowed this point to a sub-light velocity?  If c really does produce an opposite, then i1or i , becomes its negative, i1or -i , which is the value of i3, which is, lo and behold, an alternate symbol for D3.  D3 then (or i,) is the realm of slower-than-light imaginary points.  Since it is sub-light it must have mass, but since it is imaginary, it has no substance.  And this is exactly what our plane in motion is trying to describe, a surface with no thickness and therefore no substance, enclosing on itself trying to form a sphere (or a tube) with no volume. An imaginary figure with mass. Hey, astronomy buffs, what am I describing?  This is dark matter! All that unaccounted-for mass that must exist in the universe but is undetectable.  Its undetectable because its not real.  But because its slowed to sub-light, it has mass. And because these things are negative singularities, similar to the singularity responsible for the big bang and the subsequent universe except hollow inside (like bubbles or macaroni), they are super dense, far more dense even than black holes, so it makes sense, as physicists claim, that this dark matter should account for most of the mass in the universe, much more than visible matter. D3 = dark matter, matter which has been collapsed by one dimension.

Four dimensions, the integral dimensions, have been explained. The seven fractional dimensions remain, and while each is still associated with naturally occurring phenomena in the universe, the job now becomes a lot tougher.

We are going to wade right into the thick of things, but first, a little stage setting. Lets revisit that old trickster dimension, D1, time, which seems so elusive, yet has so many properties, and influences just about everything. To review, time is motion, specifically the motion of an imaginary point at velocity c.  In theory, there was a period of dimensionality (duration is necessarily meaningless here) when a motionless singularity constituted a potential big-bang to begin the universe.  In that instant, when the point acquired motion, the first dimension was born.  But something else was created too, because when the point began to move, it needed a medium to move through, and this did not yet exist. So as consequences of D1, we also got something called space, and something else called distance; things respectively, for the creation to move through, and for it to displace as it went on its journey.  Up to now we have assumed that this distance was a straight line, since this is usually how distance is measured, and since there was nothing (apparently) to deflect a projectile from its course.  Or was there? In that moment in which the singularity burst from non-existence into being, lots of things happened.  One of those things was that the singularitys ambiguous state as a mass-less/infinitely massive thing became unambiguous the instant the new universe existed it was incredibly massive, so that all of a sudden there was this super-massive something sitting the center of a region of space, expanding outward into that region even as it was being created. Imagine that space as a huge net, stretched out evenly, nice and flat. Then into the net we drop a bowling ball.  What happens?  The weight of the ball causes the net to stretch, most noticeably where the ball sits, and less as you get further from it.  But as far as the net extends there is a greater or lesser slope downward toward the center.  If a marble had been sitting on the net before the bowling ball was dropped, it would have been perfectly stationary on the flat net.  But with the bowling ball causing the net to have a different shape, the marble begins to roll towards the center.  The greater the slope the faster the marble rolls. If you cant see the net, it looks like the marble is actually being pulled toward the bowling ball by some mysterious attractive force.  But we know about the net understand that there is no force; the marble is simply rolling down a curved surface caused by the massive object at the center. And the distance the marble rolls is the length of a curved line, not a straight one.

The situation is analogous to the way massive entities shape the space around them, that is, give space a geometry that is not flat, but curved here, stretched there, endlessly twisted and varied.  This geometry of space caused by mass is how Einstein described gravity.  Why at this stage there are physicists still maintaining that gravity is one of the four forces of nature when it is fully explainable as a geometry of space is a bit confusing.  One would think that the failure either to unify it theoretically with the other forces, or to find its associated particle, the graviton, would convince them otherwise, but so far it has not done so. They continue to try to crack the nut of the gravitational force, where no force exists.  It is the geometry of space we are observing, and like other geometries in the universe, it has a dimension associated with it.  But which one?  How doe we decide which of the seven fractional dimensions is responsible for gravity, or rather, for the geometry of space? Lets try an approach that looks for patterns or trends in the qualities we attribute to the various dimensions.

Dimension Summary III

D1 is responsible for time, motion of imaginary points and the distance they travel along gravity influenced lines.
D2 is responsible for wave phenomena, specifically the electromagnetic force.
D3 is responsible for the dark matter which is undetectable but comprises most of the mass in the universe.
D4 is responsible for what we call ordinary matter, both massive and detectable.

What do you notice as you move from Dzero, total nonexistence, up through the dimensions to D4?  I notice that there is a progression from the least substantive, most abstract manifestation; to the opposite of that the manifestation that is wholly concrete and not an abstraction at all.  Maybe by assessing the balance between the abstract and the concrete nature of a phenomenon, it is possible to order it, that is, place it in its proper position in the progression of dimensions, almost as if we were creating a size order.  And if we are lucky, it may additionally be possible to identify elements being utilized by the operator D by deciding which two of the four basic dimensions are interacting to produce the appropriate fraction.  In choosing, we must bear in mind that the seven available fractions all have a value less than 2, and five of them are less than 1.  This means that they all represent significant abstractions.  It is a certainty that none of these dimensions contain massive entities, though it is possible, as with gravity, that their effects are far-reaching enough for us to be aware, or at least suspicious, of their existence.

The geometry of space then.  I dare say this is a very foreign, abstract concept for most of us. The natural tendency is to view space as nothingness.  To learn that it is both a vacuum, and has a definite geometry is a little hard to grasp. Yet most of get a pretty good mental picture of D1, even though the point in motion is completely imaginary.  Is space then a greater abstraction than time?  I imagine that it is, and that were looking for a fraction less than 1. Next, consider gravitys far-reaching effects. Not only matter but also even something mass-less like light, is induced to follow the shape of space. So it would be illogical to choose 2 as our denominator, since that excludes the 4 that contain matter.

There are only two fractions whose denominator is 4, which would account for the matter, and everything contained in all lower dimensions.  What is it then that reacts with either light or matter, to form space?  What thing plays a role in the effect of gravity; appears in all the gravitational principles and equations? Distance!  It is the distance from the distortion causing mass that determines the degree of the curvature of space, or in traditional terms, the strength of the gravitational attraction.  Distance is created by the first dimension, motion.  Therefore the only logical fraction for the dimension of gravity, or the geometry of space in one that points to the interplay of distance (D1) with each of the dimensions effected by that gravity (D4). The conclusion:  gravity = D1/4.

The concern of the next topic will surprise no one, it is that familiar demon come back to haunt us, as it has done time and time again: the speed of light.  My conclusions about how the speed of light relates to dimensionality are nearly ready, but thoughts about how to set down the logical progression and justification of my argument needs more work. So all I can give you at this point is a preview.  In it, the salient highlights are as follows:

1. The speed of light is a misleading term, for while it is the speed of electromagnetic waves; it is also the speed of other cosmic phenomena, including things that have dimensionality less than 2.
2. The speed of light, or c, is indeed constant, but not in the way everyone believes. Constancy in the eyes of humans, and constancy in the eyes of the universe, is not the same thing.  (I have already hinted at this.)
3. The speed of light is a kind of threshold for a change of state of various cosmic manifestations, just as 320 F. is a threshold for the change of state of water.  (Matter can be considered to be frozen energy.) Things at light-speed are mass-less; sub-light things acquire a mass.  Dimensionally speaking, D2 corresponds to this dividing line.
4. The dimension of c is 1/2.  I cannot at present provide the coherent explanation that I would like. Also, a big problem is that a dimension requires a geometry. That is something I have insisted on.  I do not yet have a clear picture of a geometric model for c, nor do I immediately see how to derive one. This, of course, is the challenge. I am working on it.


Mathematics is the language of the Universe.

All things resonate to certain geometric frequencies - sometimes referred to as the Sacred Geometry.

The doctrine of Kabbalah existed at the core of both the Hebrew and the old Arabic religions. It was a traditional (oral) method of handing down moral and occult truths. It was/is a metaphysical system that defines and illustrates the nexus between God and Man by means of symbolism and the alphabetical/numerical system known as Gematria.

Within this ancient and complex system was an understanding of the levels of 'existence' or rather of spiritual forms and forces, of Intelligences that ascended from the microcosmic world of man through a hierarchy of angels and archangels and principalities, of powers, virtues and dominations, of thrones and cherubim and seraphim to God, the omnipotent life force. Within this system was also accounted a Qlipoth of demonic intelligences.

Kabbalism contains an 'experience' of many lifetimes of study and meditation and spiritual development that is still pursued today both within the Jewish faith and in the Western Mystical Tradition of Qabalah. The disciplines of astrology and the Tarot are closely tied to this tradition.

Within this great system of study works the Gematria that links the lineage of number to the letters of the written alphabet(s) and to the language itself. Likewise, each Arabic letter has a numerical value and each number has significance. The names of and words for all things contain numerological meaning , and thence comes a magical linking between all things across time and space.

Groups of letters had significance and were used in amuletic inscriptions. As they also had numerological significance the concept of Magical Squares came about. When and where the signs of the entities or the spirits of the planets and the invention of magical squares came about is not known, but it is probable that they are of Sumerian or Indian origin.

Great significance was attributed to the observed planets moving around us and significant names were given to the planets connecting them with entities and metals and qualities, founding both the science of astronomy and the discipline of astrology. The knowledge of arithmetic required by those who constructed magic squares was considerable and it seems to have formed the foundation of the Arab science of mathematics.

The Kabbalists called the magical squares "Kamea", which translates as "Amulet". They were worn as pendants and the word "Cameo" derives from it.


The role of number in formulating the language of the Universe was perhaps most eloquently described by the Greek mathematician, Pythagoras, and his followers. It was he who tied together the fundamental concepts of ratios, geometries and harmonics. If the concepts of ratios, geometries and harmonics strike one as a little too abstract, they may be substituted with the more familiar terms of number, form and vibration.

Ratios simply describe the relationship between two or more numbers, such as the familiar 2:1 ratio that we associate with the octave in music. In fact, any single number may be represented by a ratio. Even the so-called irrational numbers like p and 2 that are, by definition, infinite may be approximated by a ratio between two whole numbers. Our Universe is described by ratios that are incorporated into the forms (geometries) and vibrations (harmonics), comprising everything that we see, feel and hear. In essence, geometries and harmonics are simply expressions or symbols of number. So monumental was this Pythagorean understanding that it serves as the basis for everything from music theory to subatomic physics.

Whether symbols are spoken in the form of words (vibrations) or written in the form of characters (geometries), they are indeed based on number (ratios). It was the belief of the ancient Greeks, as well as a myriad of other ancient cultures, that number represents the sacred language of God or Spirit or the Creator. Ancient and modern philosophers have repeatedly raised the question of whether "God is number." Ratios between two or more lengths, areas, volumes, angles, spirals or tones were considered to represent a numeric code by which creation is manifested from the infinite unmanifested potential of the Universe that I will refer to as the Absolute.

The well-known Greek concept of logos, which is often used to denote the manifesting potential of the Creator, is actually rooted in the word meaning "proportion." Proportion refers to the relationship among two or more ratios. For instance, 1:2:3:6 or one is to two as three is to six indicates that the same proportion underlies both ratios. One could think of the relationship among numbers, ratios and proportions as being analogous to the relationship among letters, words and language.


In many ancient traditions, numbers were considered divine in their roles as changeless principles underlying a world of matter that was constantly changing. In the Pythagorean tradition, numbers were based on harmonic motion (e.g., swinging of a pendulum or vibration of a string) and an entire cosmology was built on the laws of sound or tonal frequencies inherent in vibrating strings.1 Just in case one were to consider the Pythagorean ideas to be out of touch with modern science, the latest theory that physics has put forth to explain the nature of both matter and spacetime is known as superstring. Superstring theory supposes that particles are represented by vibrating strings that are about 1020 (100 billion billion) times smaller than a proton; such that each subatomic particle is represented by a single string vibrating at a distinct frequency. 2 In this theory, particles of the Universe are not fundamental, but are simply the harmonies created by an almost infinite number of these tiny vibrating strings. Astrophysicists have furthered this notion by postulating the existence of cosmic strings to account for the observed pattern of galaxies. So whether we are talking about quarks or quasars, it all seems to come down to vibration.

Vibration exits as waves that are present in a vacuum (e.g., outer space) as well as in a medium such as air and water. What we refer to as sound or tone is just waves traveling through a medium such as air or water, permitting the vibration to be detected by our senses of hearing and touch. Science tells us that visible light and audible sound are just two examples of a much wider electromagnetic spectrum (including microwaves, X-rays, radio, etc.) that may be distinguished by their unique vibration or wavelength. For the purposes of this discussion, vibration and waves are interchangeable inasmuch as they create and are created by each other. While air and water are ubiquitous media through which waves travel on Earth, this is not the case for most of the Universe. One might inquire as to the medium through which waves travel in the vacuum or void of "outer space." The ancient Greeks would have answered this question by saying that waves travel through the mysterious substance known as aether (i.e., akasha in the Hindu tradition). This aether has been redefined by modern physics as higher dimensions of space that are hidden within the world we observe with our five senses.

Vibration may be defined simply as an oscillation about a reference position (e.g., a pendulum or a pinwheel). Each oscillation is considered a cycle or a change from one state to the opposite state and then back through a common reference point. The number of times that a system moves through this oscillatory cycle per unit time is referred to as its frequency. We commonly use the time unit of one second to quantify frequencies, which are expressed as the familiar "cycles per second" or hertz. In this manner, we have a common reference period for comparing one or more vibrations.

For instance, the tone associated with middle A on the piano has a frequency of about 440 hertz, while red light has a frequency of about 1014 hertz and the rotation of the Milky Way Galaxy has a frequency of about 10-14 hertz. As such, the time required to complete one cycle ranges from less than a trillionth of a second for visible light to millions of years for a galaxy. Understand that there is nothing magical about the units of cycles per second (hertz) and that we could just as easily define our unit of measure as cycles per year or cycles per millennium or cycles per revolution of the planet Jupiter --- the choice is arbitrary.

Any vibration in the Universe may be described by its characteristic frequency or by its wavelength and energy. Therefore, vibration and waves and energy are simply different descriptions of the same thing. That "thing" is what comprises our world and, for the sake of consistency, I will refer to it by vibrational frequency rather than by wavelength or energy. However, keep in mind that vibrations, waves and energy are interchangeable. Every unique expression of that "thing" has a distinctive vibration known as its natural frequency. All natural frequencies include a unique fundamental vibration, as well as a set of harmonics or overtone vibrations. While it is beyond the scope of this article to explain how the harmonics are generated, it is enough to understand that there are no pure tones (i.e., those without harmonics) in nature.

Pythagoras was the first to explain why two or more vibrations or tones, referring to frequencies within the audible range, may result in a sound that is either consonant (i.e., harmonious or pleasing to the ear) or dissonant (i.e., disharmonious or disagreeable to the ear). In essence, the degree of consonance or dissonance is a function of whole number ratios. If whole numbers in the ratio are small, such as 1:1 (unison), 1:2 (octave), 2:3 (fifth) or 3:4 (fourth), then the resulting musical sound is consonant. If the whole numbers in the ratio are relatively large, such as 8:15 (major seventh) or 15:16 (minor second), then the resulting musical sound is dissonant. This rule holds whether we are talking about two different fundamental tones or a fundamental tone and one or more of its overtones (i.e., harmonics). The common thread is number, which underlies all natural frequencies and the countless harmonics that are inherent in them.


Ancient philosophers considered that the orderly principles or Laws of Nature giving rise to the physical world were not only knowable, but describable using numbers. Born from the aether or akasha, the four Elements (i.e., fire, air, water and earth) were considered to be vibratory in nature and represented by number in the form of frequencies or tonal ratios. These same ratios were used to describe the relationship among angles, which give rise to the characteristics (e.g., vertices, faces, edges) of a very special class of 3-dimensional (3-D) geometries known as the Platonic solids. The Platonic solids were believed to make symbolically visible the orderly movement from the Absolute to the endless array of interconnected forms that we know as matter or the manifested world. According to Robert Lawlor, 1 "the Platonic solids serve as essential forms and numbers that interface between the higher and lower realms and that possess, through their analogues with the Elements, the power to shape the material world." In other words, matter is created through the number or ratios contained within these Platonic solids.

The five Platonic solids were considered essential because they are the only volumes which have all interior angles equal and because their rotation or spin generates a sphere. Sacred geometry associates the sphere with the unmanifest and, as such, the Platonic solids are the only angular 3-D geometries that form a perfect interface with the Absolute. All other volumes or 3-D geometries in the Universe are created from these five, which include the tetrahedron (fire), octahedron (air), cube (earth), icosahedron (water) and the dodecahedron (aether). Because they symbolize the creation of matter itself, the Platonic solids are the likely precursors of many familiar written symbols or glyphs. For example, each Platonic solid is composed of faces representing the triangle, square and pentagon, which constitute the template for many glyphs. Similarly, a 2-D representation of the star tetrahedron yields the six-pointed Star of David and symbolizes the yin-yang.

It is through these spinning or rotating Platonic solids that our material world is connected to the Absolute and that time is intrinsically linked with movement in space. Not surprisingly, the same ratios that describe consonant tones in music can be derived from the ratios of angles, faces, edges, vertices and other parameters describing the Platonic solids. For instance, just the ratio of vertices to faces yield the unison (1:1 in the tetrahedron), the fourth (3:4 in the octahedron) and the major sixth (3:5 in the icosahedron). Similarly, the ratio of vertices to edges yield the octave (1:2 in the octahedron) and the fifth (2:3 in the tetrahedron, cube and dodecahedron). Physics has shown us that matter is created, not from fundamental particles, but from waves or energy or vibrations that are brought to focus at a point in spacetime. As such, matter is simply a special manifestation of energy and the two may be converted according to Albert Einstein's famous equation, E=mc2.

Form is no different than vibration, and both are underlain by a sacred code that we refer to as number. The famous British scientist and inventor, Buckminster Fuller, considered there to be only two possible covariables in the design of the Universe.3 The first of these is frequency (for modulating waves or vibrations or energy) and the second is angle (for modulating forms or geometries or matter). But angle and frequency are just different expressions for the same variable --- number. Herein lies the root of the ageless question asking whether God is number. Is God (a.k.a. Creator, Absolute, Spirit, Source) actually number or does number represent a more fundamental expression of nature?


To answer the question of whether number represents something more fundamental, we need to return to our discussion of vibration. Cycles are fundamental to the Universe inasmuch as everything is a vibration. Recall that vibrations are defined as cycles per unit time, but time itself represents just another cycle or portion of a cycle. In essence, the only way in which we are able to describe or quantify a vibration is in terms of a ratio of cycles; otherwise, the concept of a cycle's "duration" is utterly meaningless. Let's examine why this is true. If we use the familiar units of hertz, then all cycles are referenced to the time increment of one second. But what is a second? A second is simply a cycle described by one 86,400th of the Earth's rotation on its axis. Our most accurate measure of a second is the periodic ejection of a particle from the nucleus of a cesium atom. In both cases, a second is nothing more than a cycle or a portion thereof.

Cycles repeat infinitely and are independent of the relative measure we use to quantify them. Whatever we select as our reference cycle will simply change the ratio and thus alter the numeric value of frequency; however, the cycles themselves are obviously unchanged. Hence, ratios are necessary to describe all vibrations because one cycle can only be described in relation to another. Stated most simply, vibrations express the ratio of one cycle to another. While this notion may seem odd to us, many ancient cultures were well aware that all phenomena are cyclic and that cycles are the basis of spirals, vortices and a myriad of other natural forms. If the four Elements, which were believed to be the basis of music, are represented as a repeating cycle of four ascending musical notes, an expanding spiral is created.4 The proportion of each Element used to create matter is described by tonal ratios or musical chords, thus illustrating that it is the ratio of cycles which are coded by the numbers describing vibration and matter.

One of the ancient cultures that left behind the most detailed description of cycles (e.g., seasonal, lunar, solar and galactic) was the ancient Maya of Central America. Their calendrics and understandings of how seemingly unrelated cycles were synchronized to one another is so precise that only now are we able to recognize their significance. The Mayan calendar is derived from a more fundamental 260-unit radial matrix known as the Tzolkin, which codes cycles as number (e.g., ratios, harmonics, geometries) and accurately describes things as diverse as planetary orbits and the sequencing of DNA. These descriptions are facilitated by calculating the relationship among cycles that otherwise appear to be independent.

For example, the Mayan system tracks the waxing and waning of sunspots on the surface of Sun such that shorter cycles (tens of years) are nested within longer cycles (hundreds of years) that are nested within even longer cycles (thousands of years). These shorter cycles accurately predict the solar activity that routinely disrupts Earth's upper atmosphere, affecting telecommunication systems and satellite orbits. The longer periods predict changes in tropical ocean temperatures, global climate trends and human behavior patterns.

The so-called Great Cycle of the Maya is 5,125 years in length. The current Great Cycle is believed to have begun in 3113 bc and will end in 2012 ad, at which time we will enter yet another Great Cycle --- that's the thing about cycles! Interestingly, archeological and geological records dating back to the beginning of this Great Cycle (~3000 bc) suggest a dramatic cultural shift that occurred simultaneously all over the world. 5 Furthermore, it appears that the modern version of the El Nino phenomenon began about the same time, as perhaps did the movement of our Solar System into an interstellar cloud or bubble. It is fascinating to note that science is discovering actual links between these cultural, climatic, solar and galactic events --- all of which are cyclic.

The tabulation of cycles as rhythms gives rise to our perception of time and space. In other words, counting cycles gives rise to 3-D space and linear time and thus relates number (quantity) to vibration and form (qualities). Since both vibration and form are cycles, number is simply a coding of cycles. The idea that our perception of "reality" is based on counting cycles is one that is supported by more than the ancient Greeks and Maya. Recent neurological research has indicated that the common rhythmic pulsing of synchronized nerve cells throughout the body generate the brain wave states (e.g., alpha, theta, delta) that have been recognized and quantified for many years.6 The rhythmic electric output of synchronized neurons throughout the body give rise to our visual perception and to brain functions such as memory and attention.

According to the Mayan understanding of a Universe as cycles within cycles, it is ridiculous to separate time from space when, in fact, both are cyclic. The illusion of our senses tells us that space, as a rather arbitrary 3-D cross section of spacetime, is simply pushed along the axis of linear time. There are those who believe that recent popular interest in the Mayan calendar signals a change in the collective perception of time as cyclic, rather than linear. This may actually permit us to experience a 4-D world rather than a temporally constrained 3-D one.

This is not to say that time is exclusively the fourth dimension because we could have four or more dimensions of space (n) folded upon each other, always leaving time as the n+1 dimension. Astrophysicists postulate that, immediately following the Big Bang, ours was a 10-D Universe (9 space + 1 time). The energy of this new Universe was so intense that this arrangement could not be sustained and, through the breaking of symmetry, quickly split into our familiar 4-D (3 space + 1 time) Universe. The remaining 6 dimensions of space were folded or rolled-up into our 4-D reality, where they may comprise what the mystics referred to as aether. The mathematics of superstrings suggest that unification of physical forces occurs in both 10 and 26 dimensions, the latter of which may characterize other Universes

Perhaps the most poignant of ancient texts regarding the importance of cycles is the so-called Emerald Tablets, which are believed to have been left behind by the great master of ancient Egyptian writing known as Thoth. In his translation of the tenth Emerald Tablet, Doreal writes, "Chanted they the songs of the cycles, the words that open the path to beyond. Aye, I saw the great path opened and looked for an instant into the beyond. Saw I the movements of the cycles, vast as the thought of the Source could convey."7 If the nature of the Absolute is indeed cycles, then number appears to be the sacred code.

- D.L. Marrin

Sacred Geometry and the Structure of Music


Legend recounts how Orpheus was given a lyre by Apollo. By playing his lyre, Orpheus produced harmonies that joined all of Nature together in peace and joy.

Inspired by this Orphic tradition of music and science, Pythagoras of Samos conducted perhaps the world's first physics experiment. By plucking strings of different lengths, Pythagoras discovered that sound vibrations naturally occurred in a sequence of whole tones or notes that repeat in a pattern of seven.

Like the seven naturally occurring colors of the rainbow, the octave of seven tones — indeed, all of Creation — is a singing matrix of frequencies that can be experienced as color, sound, matter, and states of consciousness.

This correlation of sound, matter, and consciousness is important. For as Stanford physicist William Tiller has proved, human consciousness imprints the space and matter of the universe. It is our intent that gives the direction and quality to Creation.

I believe that this matrix of Creation is waiting for us to sound the most harmonious vibrating chord — to sound the universe itself into a perfect, idealized form.

The Music of Atomic Shapes

The Platonic solids, basic shapes of Sacred Geometry, are five three-dimensional geometric forms of which all faces are alike. And each platonic solid represent one of the five elements of creation, as follows:

  1. Tetrahedron — Fire
  2. Cube — Earth
  3. Octahedron — Air
  4. Dodecahedron — Ether
  5. Icosahedron — Water

These five Platonic solids comprise the alchemical dance of the elements and of Creation itself. My introduction to the spiritual power of sound began with an experience of this truth.

This happened many years ago, when I was studying with Michael Helios — who is for me a reincarnated Atlantean wizard. Helios discovered the musical proportions and corresponding tone scales for each of the Platonic shapes. He even tuned his keyboard to specific frequencies in order to achieve exact proportions.

During his presentations, he would play the scales and geometries of each shape, without disclosing to his listeners which geometric shape he was playing. Participants meditated on each piece as he was playing it, and then described which of the shapes they had experienced.

The results were extraordinary. Every Platonic solid was correctly perceived, felt, and "seen" during each of the five musical meditations. As a participant myself, this was my first experience of realizing the power of musical transmission and its potential to specifically re-order creation.

This is exactly what the ancient mystics and scientists had always been telling us!

The Dodecahedron, the Universe, and the Human Form

When Michael Helios played his five compositions, I was most profoundly affected by the Dodecahedron. This shape can be seen to represent the order of the heavens and also the perfect mediation between the infinite and the finite — the sphere and the cube.

So let us look more closely at this as one example of the sacred geometric forms that permeate Creation. Through seeing the simplicity and complexity of the Dodecahedron in its relationship of shape and sound, perhaps we can intuit the rest. And through understanding our relationship to the Dodecahedron, perhaps we can begin to sense our own place within the Divine Song that is Creation.

The Dodecahedron is comprised of twelve pentagonal faces. It represents the fifth sacred element, the divine potentiality known as "ether."

Considering that the Dodecahedron is made up of five-sided faces, it is fascinating that quantum physics researchers in the US and France have recently concluded that, based upon measurements of cosmic waves left over from the so-called "Big Bang," the universe itself is a Dodecahedron!

Besides the fact that there are five platonic solids and five corresponding basic elements of life, it can be shown that the entire human race is joined in these same basic sacred proportions. For the physical body, with the arms and legs spread, is overlaid by a pentagram, with the fifth point being at the top of the head and the reproductive organs at the exact center.

And each of these points also relates to the number five: five fingers at the terminus of each arm, five toes on each leg, and five openings on the face. Additionally, we each possess five senses of physical perception.

So the Golden Mean proportions of the cosmos and our body temples are closely aligned with the harmony of the musical fifth.

If we can imagine the dodecahedronal-pentagonal shape of this One Song that is the universe, together with the pentagram geometry of the human body, we find inherent in both a divine proportion and a potential for harmonic perfection. The universe and humanity ARE singing geometries. And it is we ourselves who embody the geometry of the cosmos!

Phi and the Musical Fifth

To follow this discussion, we first need to know that the Golden Mean and the Pentagram are closely related. For the angles of the five sides of a Pentagram are at a ratio of exactly 1.618 — the Golden Mean ratio, known mathematically as phi.

The fifth is the interval found in most sacred music, and has a powerful harmonizing effect on the human energy system. It is the first harmonic sounded by a plucked string, and is what gives the note its depth and beauty. Its sacred sound is the hallmark of the Gregorian chant. In fact most divinely inspired music, including some New Age music and that of indigenous cultures, is built around the musical interval of the fifth.

This music-geometry connection is well stated by Goethe, who said, "Sacred architecture is frozen music." The same is true of the "architecture" of the human body.

It was Pythagoras who first described the fifth interval that has come to be universally recognized for its beauty. It is "an archetypal expression of harmony that demonstrates the 'fitting together' of microcosm and macrocosm in an inseparable whole. The fifth is a beautiful sound because it demonstrates how the universe works."

And in building the phi proportions, along with those of the other musical intervals, into the designs of cathedrals and temples, the architects also are building in the effects of the musical intervals upon which the sacred proportions are based.

These effects, immediately experienced as harmonious, powerful, and centering, can be experienced first-hand when one enters a Gothic cathedral or an ancient Egyptian temple. Being inside such a space helps us to access other dimensions of consciousness. It is the same experience that is reached through listening to sacred music.

The Circle of Fifths and the Chakras

By applying the principles of progression to the harmonics of the fifth, we come to the Circle of Fifths: a musical sequence that prefigures the harmonic relationships of the human energy system. For the Circle of Fifths delineates the chakra system in the human body.

As we know, each chakra is a spinning wheel. Here, we will also note that each chakra comprises, in both sound and color, a literal mandala of geometries.

The musical tones and colors traditionally associated with the chakras are: C (root) red, G (throat) turquoise, D (belly) orange, A (brow) indigo, E (solar plexus) yellow, B (crown) magenta, and F# (heart) green.

The interweaving of the chakras that we get by applying to them the Circle of Fifths represents a more complex system than the traditional, linear progression. And it is interesting to note that in sound healing, the connections between these "chakra harmonics" reflect a strong correspondence between our issues.

For example, in the Circle of Fifths progression, the root chakra (sexuality, survival, and money) is directly connected to the throat chakra (our expression; speaking our truth). And by working with these two chakras, we find that we can heal survival issues.

The Circle of Fifths and the Fibonacci Sequence

In the Circle of Fifths we see another way in which the musical scale is related to Sacred Geometry, for the musical progression is an exact parallel to the Fibonacci sequence.

As we know, the Fibonacci sequence starts with the number 1, and proceeds by adding the two previous numbers. So the second number in the sequence also is 1, then 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, and so on. And a graph of this sequence almost exactly matches the spiral graph of the Golden Mean sequence. One is finite, the other infinite. "As above, so below."

Fibonacci realized that the natural branching, flowering, and spiraling forms in Nature followed the same uniform laws found in musical scales, for his sequence mathematically predicts all of the intervals that comprise the chords of music.

Note A A D F E C E C# F# C# D F
Interval root octave 4th aug. 5th 5th minor 3rd 5th 3rd 6th 3rd 4th aug. 4th
Fibonacci ratio 1/1 2/1 2/3 2/5 3/2 3/5 3/8 5/2 5/3 5/8 8/3 8/5

Sacred Geometry and the Chanting of Our World

Plato, discoverer of the "Platonic" solids, believed that music was the strongest of all life's influences. In his treatise the Timaeus, he describes the numerical (vibrational-musical) creation of the physical universe and the soul that animates it. He called upon his students to activate the ancient shrines and sacred temples of the earth with sacred song, employing "perpetual choirs" in order to echo the harmonies of the Heavenly Choir.

Plato's Republic describes the cosmos as being held together by eight spinning "whorls," like a giant spinning wheel with eight feminine weavers sounding the fabric of Creation. Each of the whorls contains a planet. And on each planet is a siren who sings her particular note and emits her specific color.

The work of German astronomer and mystic Johannes Kepler (1571-1630) focused on the five Platonic solids, their harmonic ratios, and how these shapes correlated with planetary orbits and sound frequencies. He found the musical tones of individual planets, and the musical scales of planetary movements. As Stephen Hawking reports, Kepler was even able to determine that "four kinds of voice are expressed in the planets: soprano, contralto, tenor, and bass."

In finding the music of the cosmos, Kepler showed that life forms on Earth follow the same harmonic principles as those found in the stars.

Temples of Sound

Similar knowledge has come out of the hermetic tradition, which saw its Western resurgence in the beginning of the second millennium. During this time, hundreds of Gothic Cathedrals were constructed across Europe, all inspired by this Eastern hermetic knowledge that had just been rediscovered by the mystical order known as the Knights Templar.

Excavating Solomon's Temple in Jerusalem, the Knights Templar discovered vaults of hidden artifacts and scrolls that described the alchemical sciences of sacred geometry and architecture and their relationship to sound, astronomy, and genetics. Ancient sacred relics also are said to have been found, including the Ark of the Covenant, the Holy Grail, and secrets pertaining to Mary Magdalene and a Holy bloodline.[3]

Inspired by this material, the great Gothic cathedrals, including Chartres, Notre Dame, Salisbury, St. Denis, and Cluny, were designed and built using the principles of sacred geometry and harmonic acoustics.

Chanting and Millennial Shifts

Sacred music and chant is always with us. But surges in its popularity occur at the crucial millennial turning points. This was so during the beginning of the first millennium, at the inception of Christianity, and during the era of the Grail Romances, which began around 1000 AD. And now today, as we forge a new paradigm and write our "script" for the next one thousand years, chanting has again come into prominence.

The sacred architecture employed in the medieval cathedrals reflected specific acoustical properties that were conducive to the constant rounds of perpetual choirs maintained by the monks.

Author John Michell, who has researched the tradition of "perpetual choirs" in ancient Britain, reports that these choirs were maintained in at least three sites: Glastonbury, Stonehenge, and Llantwit Major in Wales. Together, the sites form the rim of a circle in the landscape, with the center at an old Druid site called Whiteleafed Oak.

Michel found that these sacred sites were equidistant from each other, and that their individual locations corresponded to sunrise points and sacred proportions.[4]

Similarly, sacred sites in other cultures also were laid out in geometric relationship to each other, and were maintained with sacred music and chant, synchronized with the seasons and cosmic cycles.

Emotion, Sound, and Form

Science is only now beginning to discover these interrelationships of sound and matter.

Launching the new science of cymatics, Swiss researcher Hans Jenny (1904-1972) conducted experiments showing that inert powders, pastes, and liquids, when animated by audible, pure tones, would form into flowing patterns that mirrored those found throughout Nature, art, and architecture. He showed that there was a correlation between sound and form — that, in effect, the matter of the universe is a physical manifestation of vibration.

And as several articles in the Spirit of Ma'at have reported, Dr. Masaru Emoto has proven over and over again, through photographing water crystals, that there is a geometrical correspondence between human thoughts and emotions and the very shape of the matter that surrounds us. Water, Emoto has shown, "treated" with love or beautiful music, undergoes molecular change into beautiful, harmonious geometrical forms. And the same happens in reverse: chaotic or "negative" thought and emotion cause the crystals of water to become unformed and unlovely.

These ideas are modern reflections of timeless principles well known to all ancient and indigenous cultures.

As Billy Yellow, a Navajo medicine man, sums it up: "Our task is to chant the world, chant the beauty. The world is a reflection of our chanting."


  1. McIntosh, Stephen Ian, The Harmonic Lyre
  2. F# because each note in the Circle of Fifths creates a fifth interval based upon the preceding note. So each note is four whole tones up from its root. When we get to B, or the crown chakra, since there is no half-tone in the musical scale between E and F, the fifth interval takes us to F#. The half-note progression from B is B#-C, C#-D, D#-E, F-F#.
  3. All of this is reported in great detail in the best-selling book Holy Blood, Holy Grail, by Michael Baigent, Henry Lincoln, and Richard Leigh. See also The Bloodline of the Holy Grail by Laurence Gardner.
  4. Michel, John, Dimensions of Paradise.


The Platonic solids, also called the regular solids or regular polyhedra, are convex polyhedra with equivalent faces composed of congruent convex regular polygons. There are exactly five such solids (Steinhaus 1999, pp. 252-256): the cube, dodecahedron, icosahedron, octahedron, and tetrahedron, as was proved by Euclid Eric Weisstein's World of Biography in the last proposition of the Elements. The Platonic solids are sometimes also called "cosmic figures" (Cromwell 1997), although this term is sometimes used to refer collectively to both the Platonic solids and Kepler-Poinsot solids (Coxeter 1973).

The Platonic solids were known to the ancient Greeks, and were described by Plato Eric Weisstein's World of Biography in his Timaeus ca. 350 BC. Eric Weisstein's World of Astronomy In this work, Plato equated the tetrahedron with the "element" fire, the cube with earth, the icosahedron with water, the octahedron with air, and the dodecahedron with the stuff of which the constellations and heavens were made (Cromwell 1997).

Schläfli (1852) proved that there are exactly six regular bodies with Platonic properties (i.e., regular polytopes) in four dimensions, three in five dimensions, and three in all higher dimensions. However, his work (which contained no illustrations) remained practically unknown until it was partially published in English by Cayley Eric Weisstein's World of Biography (Schläfli 1858, 1860). Other mathematicians such as Stringham subsequently discovered similar results independently in 1880 and Schläfli's work was published posthumously in its entirety in 1901 (Hovinga).

If P is a polyhedron with congruent (convex) regular polygonal faces, then Cromwell (1997, pp. 77-78) shows that the following statements are equivalent.

1. The vertices of P all lie on a sphere.
2. All the dihedral angles are equal.
3. All the vertex figures are regular polygons.
4. All the solid angles are equivalent.
5. All the vertices are surrounded by the same number of faces.

Let v (sometimes denoted ) be the number of polyhedron vertices, e (or ) the number of graph edges, and f (or ) the number of faces. The following table gives the Schläfli symbol, Wythoff symbol, and C&R symbol, the number of vertices v, edges e, and faces f, and the point groups for the Platonic solids (Wenninger 1989). The ordered number of faces for the Platonic solids are 4, 6, 8, 12, 20 (Sloane's A053016; in the order tetrahedron, octahedron, cube, dodecahedron, icosahedron), which is also the ordered number of vertices (in the order tetrahedron, octahedron, cube, icosahedron, dodecahedron). The ordered number of edges are 6, 12, 12, 30, 30 (Sloane's A063722; in the order tetrahedron, octahedron = cube, dodecahedron = icosahedron).

Solid Schläfli symbol Wythoff symbol C&R Symbol v e f Group
cube 3 2 4 43 8 12 6
dodecahedron 3 2 5 53 20 30 12
icosahedron 5 2 3 35 12 30 20
octahedron 4 2 3 34 6 12 8
tetrahedron 3 2 3 33 4 6 4

The duals of Platonic solids are other Platonic solids and, in fact, the dual of the tetrahedron is another tetrahedron. Let be the inradius of the dual polyhedron (corresponding to the insphere, which touches the faces of the dual solid), be the midradius of both the polyhedron and its dual (corresponding to the midsphere, which touches the edges of both the polyhedron and its duals), R the circumradius (corresponding to the circumsphere of the solid which touches the vertices of the solid) of the Platonic solid, and a the edge length of the solid. Since the circumsphere and insphere are dual to each other, they obey the relationship


(Cundy and Rollett 1989, Table II following p. 144). In addition,


The following two tables give the analytic and numerical values of these distances for Platonic solids with unit side length.

solid r R
cube 0.5 0.70711 0.86603
dodecahedron 1.11352 1.30902 1.40126
icosahedron 0.75576 0.80902 0.95106
octahedron 0.40825 0.5 0.70711
tetrahedron 0.20412 0.35355 0.61237

Finally, let A be the area of a single face, V be the volume of the solid, and the polyhedron edges be of unit length on a side. The following table summarizes these quantities for the Platonic solids.

solid A V
cube 1 1

The following table gives the dihedral angles and angles subtended by an edge from the center for the Platonic solids (Cundy and Rollett 1997, Table II following p. 144).

solid (rad) () ()
cube 90.000 70.529
dodecahedron 116.565 41.810
icosahedron 138.190 63.435
octahedron 109.471 90.000
tetrahedron 70.529 109.471

The number of polyhedron edges meeting at a polyhedron vertex is . The Schläfli symbol can be used to specify a Platonic solid. For the solid whose faces are p-gons (denoted ), with q touching at each polyhedron vertex, the symbol is . Given p and q, the number of polyhedron vertices, polyhedron edges, and faces are given by


The plots above show scaled duals of the Platonic solid embedded in a cumulated form of the original solid, where the scaling is chosen so that the dual edges lie at the incenters of the original faces (Wenninger 1983, pp. 8-9).

Since the Platonic solids are convex, the convex hull of each Platonic solid is the solid itself. Minimal surfaces for Platonic solid frames are illustrated in Isenberg (1992, pp. 82-83).

Archimedean Solid, Catalan Solid, Johnson Solid, Kepler-Poinsot Solid, Quasiregular Polyhedron, Uniform Polyhedron

Links search


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Nooshin, H.; Disney, P. L.; and Champion, O. C. "Properties of Platonic and Archimedean Polyhedra." Table 12.1 in "Computer-Aided Processing of Polyhedric Configurations." Ch. 12 in Beyond the Cube: The Architecture of Space Frames and Polyhedra (Ed. J. F. Gabriel). New York: Wiley, pp. 360-361, 1997.

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cite this as

Eric W. Weisstein. "Platonic Solid." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/PlatonicSolid.html

mathematica calculationcenter series explorer


Platonic And Archimedean Solids - Faces, Edges, Areas, Vertices, Angles, Volumes, Sphere Ratios

Polyhedra Elements
Faces Dihedral angles between polygons with this many sides
P = Platonic
A = Archimedean
# of triangles : 3 # of squares : 4 # of pentagons : 5 # of hexagons : 6 # of octagons : 8 # of decagons : 10 total # of faces total # edges total # vertices (a.k.a. apices) total surface area (edge length = 1) total surface area (circumradius = 1) total surface area (inradius = 1) Volume *
3-3 4-4 5-5 3-4 3-5 3-6> 3-8 3-10 4-5 4-6 4-8 4-10 5-6 6-6 6-8 6-10 8-8 10-10 Central Angle R/e =
edge length
e/R =
edge length/
r/e =
edge length
e/r =
edge length/
rho/e =
Edge Length
e/rho =
Edge Length/
R/r =
r/R =
R/rho =
rho/R =
r/rho =
rho/r =
P 1 Tetrahedron tetrahedral 4

4 6 4 1.732050808 2.828427125 8.485281374 0.117851130 70° 32'

109° 28' 0.612372435696 1.632993161855 0.204124145232 4.898979485566 0.353553390593 2.828427124746 3.000000000000 0.333333333333 1.732050807569 0.577350269190 0.577350269190 1.732050807569
P 2 Hexahedron (cube) 2,3,4-fold

6 12 8 6.000000000 6.928203230 12.000000000 1.000000000

70° 32' 0.866025403784 1.154700538379 0.500000000000 2.000000000000 0.707106781187 1.414213562373 1.732050807569 0.577350269190 1.224744871392 0.816496580928 0.707106781187 1.414213562373
P 3 Octahedron 2,3,4-fold 8

8 12 6 3.464101615 4.898979485 8.485281374 0.471404521 109° 28'

90° 0.707106781187 1.414213562373 0.408248290464 2.449489742783 0.500000000000 2.000000000000 1.732050807569 0.577350269190 1.414213562373 0.707106781187 0.816496580928 1.224744871392
P 4 Dodecahedron 2,3,5-fold


12 30 20 20.645728806 14.733704195 18.541019661 7.663118961

116° 34'

41° 49' 1.401258538444 0.713644179546 1.113516364412 0.898055953159 1.309016994375 0.763932022500 1.258408572365 0.794654472292 1.070466269319 0.934172358963 0.850650808352 1.175570504585
P 5 Icosahedron 2,3,5-fold 20

20 30 12 8.660254038 9.105929973 11.458980337 2.181694991 138° 11'

63° 26' 0.951056516295 1.051462224238 0.755761314076 1.323169076499 0.809016994375 1.236067977500 1.258408572365 0.794654472292 1.175570504585 0.850650808352 0.934172358963 1.070466269319
A 1 Truncated Octahedron (Mecon) 2,3,4-fold

14 36 24 26.784609689 16.940074571 18.822305079 11.31370850

125° 16'

109° 28'

36° 52' 1.581138830084 0.632455532034 1.423024947076 0.702728368926 1.500000000000 0.666666666667 1.111111111111 0.900000000000 1.054092553389 0.948683298051 0.948683298051 1.054092553389
A 2 Cuboctahedron (Dymaxion) 2,3,4-fold 8 6

14 24 12 9.464101615 9.464101615 12.618802153 2.357022604

125° 16'

60° 1.000000000000 1.000000000000 0.750000000000 1.333333333333 0.866025403784 1.154700538379 1.333333333333 0.750000000000 1.154700538379 0.866025403784 0.866025403784 1.154700538379
A 3 Truncated Cuboctahedron 2,3,4-fold
8 6
26 72 48 61.755172435 26.646048347 27.946789492 41.79898987

144° 44' 135°

125° 16'

24° 55' 2.317610912893 0.431478810545 2.209741210257 0.452541680156 2.263033438454 0.441884765381 1.048815536469 0.953456509013 1.024116954488 0.976450976247 0.976450976247 1.024116954488
A 4 Snub Cube 2,3,4-fold 32 6

38 60 24 19.856406460 14.777263402 17.152165352 7.889477400 153° 14' 142° 59'

43° 41' 1.343713373745 0.744206331156 1.157661790956 0.863810145426 1.247223167994 0.801781129201 1.160713244786 0.861539234167 1.077364026124 0.928191377986 0.928191377986 1.077364026124
A 5 (Small) Rhombicuboctahedron 2,3,4-fold 8 18

26 48 24 21.464101615 15.342829357 17.589734695 8.714045208 144° 44' 135°

41° 53' 1.398966325966 0.714813488673 1.220262953798 0.819495500448 1.306562964876 0.765366864730 1.146446609407 0.872260419103 1.070722470768 0.933948831094 0.933948831094 1.070722470768
A 6 Truncated Cube 2,3,4-fold 8

14 36 24 32.434664361 18.233771763 19.797982086 13.59966329

125° 16'

32° 39' 1.778823645664 0.562169275430 1.638281326807 0.610395774912 1.707106781187 0.585786437627 1.085786437627 0.920991426441 1.042010766560 0.959682982261 0.959682982261 1.042010766560
A 7 Truncated Icosahedron (soccer ball) 2,3,5-fold

12 20

32 90 60 72.607253029 29.300527163 30.544061106 55.28773076

142° 37' 138° 11'

23° 17' 2.478018659068 0.403548212335 2.377131605984 0.420675068003 2.427050983125 0.412022659167 1.042440667917 0.959287210080 1.020999837373 0.979432085486 0.979432085486 1.020999837373
A 8 Icosidodecahedron 2,3,5-fold 20

32 60 30 29.305982843 18.112093471 20.024238056 13.83552594

142° 37'

36° 1.618033988750 0.618033988750 1.463525491562 0.683281573000 1.538841768588 0.649839392466 1.105572809000 0.904508497187 1.051462224238 0.951056516295 0.951056516295 1.051462224238
A 9 Truncated Icosidodecahedron 2,3,5-fold
12 62 180 120 174.292030327 45.837440154 46.643971373 206.8033989

159° 6'
148° 17'

142° 37'

15° 6' 3.802394499851 0.262992175073 3.736646456083 0.267619645517 3.769377127922 0.265295821050 1.017595468167 0.982708778963 1.008759370795 0.991316689541 0.991316689541 1.008759370795
A 10 Snub Dodecahedron 2,3,5-fold 80

92 150 60 55.286744956 25.645137056 27.103030805 37.61664996 164° 11'

152° 56'
164° 11'
152° 56'

26° 49' 2.155837375116 0.463856880645 2.039873154954 0.490226560201 2.097053835252 0.476859479327 1.056848740756 0.946209198569 1.028031488212 0.972732850566 0.972732850566 1.028031488212
A 11 (Small) Rhombicosidodecahedron 2,3,5-fold 20 30 12

62 120 60 59.305982843 26.559470348 27.961449293 41.61532378

159° 6'

148° 17'

25° 52' 2.232950509416 0.447837959589 2.120991019518 0.471477715274 2.176250899483 0.459505841095 1.052786404500 0.949860290488 1.026053801952 0.974607762378 0.974607762378 1.026053801952
A 12 Truncated Dodecahedron 2,3,5-fold 20

12 32 90 60 100.990760142 34.009932348 35.002328800 85.03966456

142° 37'

116° 34'

116° 34' 19° 24' 2.969449015863 0.336762811773 2.885258312920 0.346589418189 2.927050983125 0.341640786500 1.029179606750 0.971647702152 1.014484897251 0.985721919281 0.985721919281 1.014484897251
A 13 Truncated Tetrahedron tetrahedral 4


8 18 12 12.124355652 10.339685242 12.637393073 2.710575995

109° 28'

70° 32'

50° 28' 1.172603939956 0.852802865422 0.959403223600 1.042314613294 1.060660171780 0.942809041582 1.222222222222 0.818181818182 1.105541596785 0.904534033733 0.904534033733 1.105541596785
face angles for corresponding face components
60° 90° 108° 120° 135° 144° * in multiples of edge length cubed

Impossible Correspondence Index

Copyright. Robert Grace. 2004