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Three Sacred Solids - A Trinity of Triangles!

In the book All Things Are Numbers there is a descriptive model used that employs some basic shapes of geometry to describe a path of evolving ideas. This idea is repeated again in the More on Tarot section of this site. This Geometric model is also combined with verbal progression called the Dimensional model. Together they describe a series of shapes that are later used to give character to individual cards in a tarot deck. For example: the idea of a point moving out as a Ray conveys the idea of motion, which we see illustrated adequately in popular tarot decks like the Rider/Waite/Smith 2 of Pentacles. In the progression presented in the book All Things Are Numbers, only one example of evolving shapes is given, but there are other ways to describe the same progression, using slightly different shapes. The table below shows some of those variations.

In the book All Things Are Numbers, I chose to feature the Point to Sphere Linear progression, because it seemed easiest to follow. But... while drawing The Seasonal Tarot, I combined the linear progression with a slightly different progression. In all the decks on this site, the Minor Suits relate to each other in binary ways. Addition/Diamond/Coin and Subtraction/Club/Stave are supposed to be more analytical and linear, and the suit of Multiplication/Heart/Cup and Division/Spade/Sword are suppose to be more synthetical and circular. Because of this, I added a circular progression to The Seasonal Tarot, to illustrate that difference between suits. Both the linear and circular shapes follow the same path dictated by The Dimensional Progression model. In that progression, the three dimension of reality are found between the numbers 3 and 7. In both cases, a trinity of opening ideas leads to a First Form - a line. In both cases, a closing trinity of ideas leads to an explosion of that content into All Dimensional radiance. In both cases a two dimensional shape becomes a three dimensional shape, as it passes the midpoint of the number line; a square becomes a box and a circle becomes a can. As the box is filled in, it becomes a cube. As the can is filled in, it becomes a cylinder.

The hand of God at work! ? In The Numerical Tarot, the suits of Addition/Diamond/Coin and Sutraction/Club/Stave follow the path of the Square... while the suits of Multiplication/Heart/Cup and Division/Spade/Sword follow the path of the Circles. The Trump cards are then left to express the path of the Triangle, which is fitting considering how "Godly" the triangle is compared to all other shapes.

In the Linear progressions mentioned above, a point moves out for an arbitrary distance. When it stops, it creates a line - a First Form. The rules then dictate that every point along that line should then move out in another direction the same amount to create the next shape of a square, which... when filled in, becomes a plane or grid. In the Circular progression the line pivots on the point to complete a circle, which... when filled in becomes a disk. In both the Linear and Circular progressions, found within The Seasonal Tarot, the sixth shape was made by moving all points perpendicular to the plane or disk. Thus, a square plane became a box and cube, while a circular disk became a can and cylinder. But... strictly speaking, I errored in following that path, by not sticking to the defined criteria of movement within the circular progression (hey, it was early in my exploration of ideas). If I had stuck to the plan, the disk of the circular progression would have also pivoted around a radius to create a torus. Oh well, maybe I'll draw another deck in the future that corrects that error. Or, I may explore a third alternative where... instead of moving all points the same amount perpendicularly, or circularly, I ask them to converge back to a point somewhere away from their two dimensional state. If this had been done to the linear or circular progressions, our square plane would have become a quadrilateral pyramid, and our circular disk would have become a round cone. As they are, these two progression, Linear and Circular, work well at conveying the analytical/linear vs. synthetical/circular nature of the Minor Suits of The Seasonal Tarot, and any other deck that might be drawn from the underlying structure being presented everywhere on this site. But there is another progression of shapes that is also worth examining, because of how it uses the converging idea to exalt the nature of threeness.

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The Triangular Progression

Actually... by exploring the alternative of asking the sixth shape of the Linear and Circular progressions to converge upon a point, rather than move all points perpendicularly or circularly, we have already exalted the nature of threeness! Because... whenever anything converges upon a point, there is going to be some form of triangulation that is evident. Triangulation is pretty common, because a triangle is the basic building block of the universe! A triangle is the most fundamental of shapes. Three connected sides is the fewest sides possible to create an enclosed, two dimensional space, thus making a triangle very important as a First (two dimensional) Form.

As described in the table shown above, in the triangular progression, a point moves out as a ray... to create a line. But instead of moving all the points of the line perpendicularly or circularly, we utilize the converging idea, and ask that the ends of the line converge upon a point that is as far away as the original line. This creates a vector, which... when filled in, becomes a triangle. When we repeat this criteria of convergence from the two dimensional base of our triangle, we create a tetrahedron, which... when filled in, becomes... a filled in tetrahedron! If we had followed the perpendicular criteria, our triangle could have evolved into a prism. Also a cool shape! But in this particular essay, we want to focus on the tetrahedron because of how it exalts the idea of threeness.

Creating a tetrahedron is as easy as 1,2,3! Or... 1 through 7!

A tetrahedron, like the two dimensional triangle that created it, is a fundamental shape - the most basic of the so-called regular solids - a First (three dimensional) Form. A tetrahedron doesn't have to be regular, but some consider regular solids - where faces and angles are all congruent, and all points equal distance from a center (I think) - to be the most exalted and sacred of geometric shapes. Seeing how this essay is devoted to exalting threeness, we will want our Triangular Progression to result in a beautifully crafted, "regular" tetrahedron. And so it does. But, what good is a tetrahedron, regular or otherwise? To answer that question we need to examine the rest of the so-called regular solids, of which our tetrahedron is only one.

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The Three Regular Solids

This essay will not go into a detailed discussion about the nature of regular solids. One of the reasons this essay will not go into a detailed discussion about the nature of regular solids is because I am no expert at all on the subject! In fact, I only have a passing interest in these things, as I do not see much about them that excites me. Interesting yes. Exciting... not so much. Thus, I leave exploration of regular solids up to each individual... leaving this essay to focus exclusively on how these solids might be found to exalt the nature of threeness, and how that exaltation can be applied to the underlying structure of tarot that is being presented everywhere on this site. To accomplish that goal we must first acquaint ourselves with the other regular solids of which our tetrahedron is only one.

Most people will tell you that there are in fact five regular solids: tetrahedron, hexahedron, octahedron, dodecahedron and icosahedron. With mathematical minds that love to calculate quantities, they will often supply a table listing the number of faces and sides, degree of angles etc. for each solid. Like the solids themselves, these statistics are interesting to note and compare. But not as exciting to me as the shapes themselves. In observing the shapes themselves, there was one key fact about regular solids, that intrigued me, and lead me to see a way to make practical use of these shapes. That key observation had to do with the inverse compatibility between the shapes of hexahedron and octahedron, and dodecahedron and icosahedron. It seems that, when compared to each other - hexahedron and octahedron for example - all the points of one correspond perfectly with the faces of the other. Meaning... that by flattening points into faces and pulling faces out into points a hexahedron will morph into an octahedron, and... likewise, a dodecahedron will morph into a icosahedron. Or... the other way around. To me, someone not that interested in looking at these solids for how their sides and faces and angles can be calculated, but for what they are as shapes, this fact meant that in their most abstract essence hexahedron and octahedron are the same, and likewise with dodecahedron and icosahedron. To me, this meant that there were, in essence, only three regular solids.

Because hexahedron and octahedron, and dodecahedron and icosahedron are morphs of each other, one could argue that they are in essence the same, and that there are actually only three regular solids! Because the triangle is the basic building block of the universe, one could also argue it most appropriate to feature the state of morphed solid that features triangles... leaving the non-triangled equivalent to be considered a simplified version of the triangular original.

In determining that there are - in essence - only three regular solids, a choice had to be made as to which state of the morphing pairs to show as a representation of those particular points and faces. To settle this, the exaltation of threeness dictated octahedron and icosahedron to win out over hexahedron and dodecahedron. Seeing how the triangle is the basic building block of the universe, it seemed only fitting to feature the shapes that utilize triangles of threeness over the ones that don't. Hence, there are in essence only three regular solids: tetrahedron, octahedron and icosahedron. As suggested in the table of solids shown above, a tetrahedron represents the idea of threeness, applied to a base of threeness. While the octahedron represents the idea of threeness, applied to a base of fourness. Leaving the icosahedron to represent the idea of threeness, applied to a base of fiveness. 1, 2, 3 to tetrahedron, then, 3, 4, 5 to icosahedron. No more regular solids after threeness applied to fiveness? Guess not. So what significance can their be to ending with 5?

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The Three Regular Bases

When discussing things like Sacred Geometry, many like to show how basic shapes can be constructed, with compass and straight edge, within the confines of a Viseca Pisces... believing those shapes to be even more sacred if found within the two overlapping circles that represent the Monad of Unity reflecting itself into our Universe as a progenitor of duality, existence and all that comes after that. Seeing how this essay is devoted to the Sacred Geometry of Threeness, using construction lines that have evolved from a Viseca Pisces seems appropriate. If so applied, it will be noted that, while a triangle and square are easily found within the lines and arcs of a Viseca Pisces form... a pentagon is not as easily found. But... by adding just one additional circle, we can make the task of finding a pentagon easy (see diagram below).

Three circles equal five sides! And threeness applied to fiveness equals an icosahedron! A freakin' amazing coincidence that should have our jaws dropping to the floor? No. Just fun to note.

A Monad reflecting itself creates duality. But duality alone does not create the shapes of reality. To create the enclosed shapes of reality, we need threeness! We need to move out or away from that one dimensional line to establish two dimensional shapes. Thus... by adding a third circle to our Viseca Pisces form, we exalt threeness, and find additional shapes of interest. Adding more circles would create more construction line possibilities, until many more shapes are found. But for the purposes of this essay, and the exaltation of threeness, three circles is enough to get us the three regular bases we need to create the three regular solids that we will use to describe the underlying structure of our tarot deck. The fact that three circles - one drawn upon a centerpoint that is formed by the other two - create a pentagon is a happy coincidence to that of applying threeness to fiveness to achieve the last of our regular solids, the icosahedron. But, again, what good are these bases and solids?

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A Trinity of Triangles

By reducing the number of regular solids to three, we exalt threeness. By choosing to feature the solids that use triangles, we exalt threeness again... with a trinity of triangles! By using a threefold modification of a Viseca Pisces to find three regular bases, we exalt threeness even further. By connecting the underlying structure of our tarot deck to these Three Sacred Solids, we exalt the threeness that is to be found therein.

Some tarot scholars suggest that the number of cards in the Major Trumps - 21, not counting the un-numbered Fool card - was derived from the number of possible outcomes that result from throwing two six sided dice. Elsewhere on this site I have my Major Trumps and Minor Suits ascribed to nine six sided dice (see Tarot on Dice elsewhere on this site). Six sided dice - or cubes - are hexahedrons... one of our regular solids. One day... while contemplating regular solids I got the notion to try and ascribe the cards of my Numerical Tarot to the sides of a icosahedron. I figured; there are 20 sides, and I have 18 Majors. If I use the two left over as zeros, then each triangle could represent a Major Trump. Additionally, I could see how the unique structure of my deck design lended itself perfectly to the idea of exalted threeness, as every card in my Major Trump sequence is considered to be the parent of two children in the Minor Suites. In other words, where there's one, there's two (see The Deck Operating System elsewhere on this site). Thus, by placing Majors and Minors on the triangular faces of an icosahedron, the exalted threeness of my deck design is accentuated.

After placing the Major Trumps and Minor Suits on the triangular faces of an icosahedron, I began to think about the 8 sides of the octahedron, and how four triangles - separated from four other triangles... by a square base, resembled the equidistantly symmetrical nature of a basic number line, and how the numbers of a basic number line are in fact the parents of the Major Trumps. Thus, it seemed logical to use an octahedron to describe a number line, and its offspring... Major Trumps. Then... working backwards even further, to the more basic, essential tetrahedron, it's four equal sides made me think of the Quaternary part of my Numerical Tarot deck. By extension, the four Quaternary cards are the parents of the 8 equidistant numbers of any number line. Thus, an evolution of form became apparent between these three regular solids that at least approximated the evolution of my deck design! As many consider these regular solids to be Sacred Geometries, I thought it would behoove me to utilize this coincidence in form to better exalt my choice of deck design.

Three Sacred Solids. A Trinity of Triangles.

In the icosahedron, the two triangles of zero are to be placed diametrically opposite each other as pole points, with the two rows of 9 Trumps, and 36 Minors, orbiting around the equator. In the octahedron, the zeros are not represented by faces but by the area beyond the two equidistant points of each pyramid, just as they are with any of the equidistantly designed number line examples given throughout this site. In the tetrahedron, there is a zero beyond each point, and the 5 inhabits the center, just as it does in all our other theoretical models of existence. This is the best way I can think of to utilize these Three Sacred Solids. As for anything else...

This is as far as I've gotten with my contemplation of regular solids. Like I said, they don't really excite me that much, but this one idea seemed kind of interesting, so I developed it. At the same time though, this idea also seems kind of nutty to me. In fact, it's pretty much the kind of thing I don't really like to see when I examine other people's ideas about tarot or sacred geometry. I have to admit, some of the connections and subsequent conclusions seem a bit arbitrary, illogical, forced or inconsistent and questionable at times. Like the fact that the first two solids don't have faces for zero, but the last one does. That could be taken as inconsistent. And the fact that the octahedron does not have enough faces to display the central Major Trumps of Wizard and Witch. That too could be viewed as an arbitrary concession made in order to force things to fit. On the other hand... when comparing a number line to the characters of chess, a similar omission of the 5s was made, as their neutral position was considered equatable to the black and white checkerboard of the game itself - the neutral battlefield (see The Invisible Body elsewhere on this site). As an octahedron also has a square base at its center, we could consider it to be equatable to that same strategic omission.

But... some may find strategic omissions to be a kluge. If so, feel free to trash all this. In the end, all I want is to display my card positions on dice, so that they can be randomized in ways other than shuffling cards. These Three Sacred Solids accomplish that goal. The rest is just for fun. Assign sacredness or stupidity to it as seems fitting. For those who think it is an abomination of Sacred Geometry to reduce five solids to three, and that the sacredness of threeness does not justify such actions, feel free to ignore all these ideas. For me, threeness is sacred. And besides, those other solids don't exactly disappear for ever. They are still there... hidden inside the others, as a mystery to be revealed. In my opinion, regular solids are much more appealing and much more compelling when thought of as a trinity of triangles.

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The Magic Tarot Ball (Majors) and, The Magic 80 Ball (Minors)

Elsewhere on this site there are three different versions of the Major and Minor cards of The Numerical Tarot put onto dice (see Tarot on Dice). These can be printed out and folded down into cubical dice. Here are two more print out and fold down gadgets that can be used to randomize the ideas found within The Numerical Tarot deck. I call the first one The Magic Tarot Ball. I call the other one The Magic 80 Ball. Is there anything really magic about them? Is there anything really magic about tarot? Maybe not. These were just made for fun. The Magic Tarot Ball is good as an aid for studying the unique parent/child arrangement of Majors and Minors found within The Numerical Tarot. The Magic 80 Ball is basically the same content as the Sane/Crazy Quilts seen in other places around this site (let me know if you find any errors!).

The Magic 80 Ball ! Have fun !!

The Magic Tarot Ball puts the 18 Major Trumps, 36 Minor Suits and 4 Quaternary cards onto the 20 faces or 60 triangular corners of an icosahedron, as discussed above. But... rather than make all 20 faces as one large piece to cut out and fold down, this model has been divided into two mandala-like diagrams that represent a north and south pole or positive and negative numbers, thereby duplicating the nature of the parallel 9s approach used by The Numerical Tarot deck. As with our own Earth, the two poles represent a swirling vortex, so the numbers are divided into three groups of three in such a way as to swirl around the pole points in opposite directions. When the two halves are put together - preferably by aligning like numbers across from each other - the two rows of 9 orbit the equator.

In constructing this model, it's probably best to print these out on the heaviest paper possible, because there is no internal support to keep them from being crushed the way there is in the cubical dice. Then score and fold. It might be a good idea to add some tabs to the outer edges. That could allow some of the edges of the two hemispheres to be taped inside rather than outside. This is as far as I've gotten with this thought. I'll try and come back to it later and improve it. I'd like to turn these into full blown mandalas. But I've got other things to get to first. So, for now, use as best as possible. Roll The Magic Tarot Ball across a table or floor and invent a way to identify which triangle corner to read. Probably whichever one is upright and readable. But maybe not. Maybe drop it from a height onto a needle or nail and read whichever corner gets punctured! Just be random, or there'll be no magic!

Here is a PDF of the Magic 80 Ball.

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EXTRA CREDIT

Just for fun, I've also created a fold-down gadget that uses the fourfold nature of the Tetrahedron! I put the four elements on it. Of course... in the system being presented everywhere on this site, my arrangement of the four elements is not exactly the same as the way people with an interest in Alchemy think of the four elements... so... fold this down and use it with caution!

Here too is a color version in RGB. And... a PDF of a color version in CMYK that might print better on most color printers. It won't look as good at the RGB on screen, but, such are the limitations of printing color! In addition... here is another version that positions the Hot, Cold, Wet, Dry etc. in a way that fits better with tradition. For a more complete examination of the elements and our theoretical model of existence, consult the essay The Quaternary and the Elements of Tarot. In that essay, there is also a vertical arrangement of both this alternate form as well as the traditional form of elements.