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, D_3>, D₄ = length, width, height (spatial dimensions)
D_1/4, D_1/3, D_1/2, D_2/3, D_3/4, D_4/3, D_3/2 = 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
i⁴ = 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 D_zero, 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 = D_1/4.

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

Mathematics is the language of the Universe.
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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.

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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 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 in his Timaeus ca. 350 BC. 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 (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

(1)

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

			(2)
			(3)
			(4)
			(5)
			(6)
			(7)

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

solid	r		R
cube
dodecahedron
icosahedron
octahedron
tetrahedron

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
dodecahedron
icosahedron
octahedron
tetrahedron

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

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© 1999 CRC Press LLC, © 1999-2004 Wolfram Research, Inc.

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