Introduction

The introduction of Chapter 1 described the Platonic solids as having capability. This chapter will now "lay down the gauntlet" and begin to describe the Platonic (regular) and semi-regular solids in terms of being a computer. 

Compute what?

It will compute the selection of tarot cards and it will select them in such a way that each card will in some way be dependent on the other cards chosen for its selection. This will be accomplished by using the numerical quantities associated with the symbolism of the tarot. The traditional 78 card tarot deck is a mongrel of history with an unclear origin, but since the late middle ages the cards have been bred with the Hermetic symbolism of both astrology and alchemy into a divination system. The following quantities from that symbolism forms the basis of this work:

1. The number four expressed through the four elements fire, air, earth and water of astrological signs of the major arcana. These elements are also symbolized in the four-by-four suit-matrix of the sixteen court cards and the four suits of the forty small cards.

2. Numbers three, seven, and twelve via the three principals of mercury, sulphur and salt (neutral, active and passive amongst other designations), seven metal/planets and twelve astrological signs which together add up to the twenty-two total of the major arcana cards. 

3. Number ten, the cyclical rebirth of unity expressed in the Pythagorean Tetraktys   and found in the total number of small cards ace through ten of one suit.

The Platonic or regular solids are a treasure trove of this quantified astrological/alchemical symbolism too. They are replete with geometrical structures incorporating three, four, ten, twelve and seven. Besides embodying these numerical combinations, they also have a self-referencing capability because their geometric structures can describe and construct one another and consequently themselves. The ability of the tetrahedron to define an inverted version of itself because its vertexes are the center points of the new faces as shown in Chapter 1 suffices as a simple example at the moment.

Psychological  and random elements will also be incorporated into this system as well. The psychological aspect will enter through the reader's (or querent's) interpretation of sigils and other two-dimensional images obtained from the geometric structures. Randomness comes into play in a fashion similar to the well-known use of these solids as dice in role-playing games. However unlike the randomness of cards simply pulled from a shuffled stack, this randomness is framed within the context of the psychological and numerical elements as well as elements from past selections.

Developing the "hardware" and "software" i.e. the respective geometric structures and sigils for psychological interpretation to run this computer is the current focus. Enough different geometric structures and sigils need to be generated from the solids to represent the individual symbolism of the various tarot cards. This chapter will focus on combinations and reflections of "twelveness" and "fourness" within the icosahedron. Aspects of twelveness will be handled first which will then transition to fourness.

A caveat: the information given on this page should be viewed as preliminary sketch and therefore potentially subject to change!

Chapter 2

Now the title of this chapter is a play off of an alternate title of "The Eighth Reveals the Ninth" for the Hermetic tract "The Discourse on the Eighth and Ninth" and part of the Nag Hammadi Gnostic Library:

http://www.gnosis.org/naghamm/discourse.html

In the tract the seeker travels to the eighth sphere which in the ancient cosmology is the stars including the zodiac above the seven planet spheres. There he discovers the ninth sphere which is the universal mind, but also can be understood as the world soul, or Anthropos (cosmic man) or Aion (depending on the tradition followed). This particular chapter works backwards and descends rather than ascends because the Winding configuration developed in Chapter 1 was likened to the world-encircling serpent Leviathan which in turn is traditionally associated with the world soul, so here the ninth sphere has actually been found first. Since the eighth sphere holds the twelve signs of the zodiac, it will be shown how the original Winding of Chapter 1 will reveal twelve other Windings that connect the icosahedron's twenty faces in a totally linear sequence.

The first piece of "geometric hardware" for this computer will now be demoed. It will be remembered from Chapter 1 that the Winding concept exists as + and - mirror opposites, so the combined face crossings of both the + and -Windings totals forty. Forty is the number of small cards of the tarot decks minor arcana (as separate from the court cards), so the Windings can also act as a literal Royal Road carrying two suites of ace through ten cards for each Winding to designate the traversed faces. This coincidence of forty carries even more weight when the concept of opposites is taken into account with opposing elements of fire versus water and air versus earth syncing with the + and - of the Windings.

These twelve new Windings will now be described and derived from the Winding introduced in Chapter 1, but first some superficial color changes must be made to that Winding of by changing the color to white with the icosahedron's edges to black. Also "Winding" will be suffixed with "_0" to designate that it is now not the only one:

The "+" prefix of course means that this page is set in the context of the positive version of the Winding. However information applied to it also equally applies to the negative version as well!

The new Windings also linearly connect all twenty faces of the icosahedron, but the difference between them and Winding_0 is they do not return to their beginning face. They also have left and right handed mirror opposites, so the "+" and "-" polarity prefixes will be retained. The total thirteen Winding configurations appear to be the total number of ways that all twenty icosahedron faces can be linearly connected, however no mathematical proof of this can be offered. They are simply the product of hours of trial and error work that could not produce any more configurations.

A correlation can be made between each of these new Windings and each of the icosahedron's twelve vertexes with Winding_0 representing the center point. Hypothetically then it also becomes possible to associate each Winding with a zodiac sign. However such an association can be potentially modified by a four cubed -sixty four- matrix of possibilities as will be shown later.

Because of the limitations of a 2-dimensional format of the web-page and the complexity of the subject matter, the work will be demonstrated "backwards" by showing an example set of correlations first and then how they were achieved. Of course the best way to study and compare the new Windings is by three dimensional models -either with actual wire and string models or computer generated 3D- but since the reader is limited to the two dimensions of the web page then both multiple and redundant images augmented with animations become necessary.

First, the twelve new Windings will be presented like an Ouroborus undulating in its black-edged icosahedron retort through a rainbow of colors:

The same twelve +Windings without the icosahedron frame:

Each new Winding has a corresponding mirror image just like original Winding_0 has. Besides the reversed handedness aspect, the -Windings have also been rotated 180 degrees on the X axis so that the relationships of diametrically opposite vertexes are achieved:

Next are a series of paired interactive images for the reader to compare Windings by clicking the buttons below the images.

Images 1:  Slow-motion animations of the +Windings that show a full 360 degree rotation around the Z axis.  

Images 2: Front views of the thirteen +Windings at a slight angle shown with and without the icosahedron. Changing these images will also change Images 1.   

Images 3: Mirror opposite -Windings from the same viewpoint as Images 2 also shown with and without the icosahedron. 

Images 4: Side, front and top views of the +Windings. 

Images 5: Perspective views of +Windings with and without the icosahedron. 
 

Next the Windings will be shown flattened out. A lot of the differences between the Windings are simply the direction reversal of loops such as Windings 4, 5, and 6. Some Windings are hybrids of two other Windings. Winding_5 is a hybrid of Windings 4 and 6. Winding_7 is a hybrid of Windings 6 and 8. Winding_11 is a hybrid of Windings 10 and 12.

In order to demonstrate the correspondence between a particular Winding and a single vertex certain geometrical aspects have to be defined and given terms. The following terminology is strictly my own invention.

 A. "Cross Vector" -A line that bisects both the Winding's Crossing Path of a Face and the Face edge parallel to the Crossing Path.

B. "Cross Point" -A Face vertex that is the end point of the Cross Vector with the other end point being the middle of the opposite edge.

C. "Cross Face" -One of the eighteen Faces fully crossed by a Winding. Each Cross Face has a corresponding Cross Vector and Cross Point.

D. "Terminal Face" -The one of two Faces where the Winding ends. An End Face is not fully crossed by a Winding.

E. "Terminal Edge" -The last edge crossed by the Winding's Crossing Path as it enters the End Face.

F. "Terminal Vector" -A line that connects the mid-point of a Terminal Edge with the opposing vertex ("Terminal Point" below).

G. "Terminal Point" -The opposing vertex of the Terminal Edge and one of the end points of the Terminal Vector.

H. "Cross Terminal Point" -The Cross Point of the Cross Face sharing the Terminal Edge with the End Face.

J. "Cross Terminal Vector" -A Cross Vector on a Terminal Face that ends at a Cross Terminal Point (H above).

K. "Midpoint Edge" -The icosahedron's edge that when crossed by a Winding and divides the Winding in equal size parts of nine Cross Faces and one Terminal Face each.

+Winding_12 will be used as an example:

The correspondence between a particular Winding and vertex was established using three rules or criteria that synchronize and work together.

Rule 1:

A unification of a system of Cross Vectors and axis termed "Cross Vector Alignment Pattern Set" (or "CVAP Set" for short and "Alignment Axis") is the basis for Rule1. A CVAP Set consists of six Cross Vectors of certain Crossing Paths from ALL of the four Quarters (per Chapter 1). These particular Cross Vectors were singled out because they are identical in the configurations of all thirteen Windings. The CVAP Set is a structure they all have in common and therefore unifies them. The "Alignment Axis" is a line that connects the pair of opposite vertexes that the six Cross Vectors are aligned to. It is really an integral part of the CVAP Set. 

There are four possible CVAP Sets (1, 2, 3 and 4) for Winding_0.  CVAP Set 1 will be examined in detail and uses the following Crossing Paths from all four Quarters with Quarter 1 as the start:

Quarter 1: IV and V

Quarter 2: V

Quarter 3: II

Quarter 4: II and I

It must be those specific Crossing Paths within that sequence. Even though Windings 1 through 12 change the shape of various Quarters, the Crossing Paths listed above will still exist in the same relative configuration. The image below shows the "CVAP Set 1" for both + and - Winding_0:

Side view perspectives with the Alignment Axis near vertical for +Windings 1 through 12:

Using this new vertical perspective the two Winding images below can be changed using the buttons underneath to compare the different structures:

Rule 2:

All of the new twelve Windings are positioned to nest (follow and align) with the original Winding_0 as best their design differences allow them. Besides the Quarter sub-division already described above and in Chapter 1, Winding_0 is constituted of two basic elements that control the nesting of the other twelve Windings. These are termed and shown below as "S Curves" and "Great Arcs".

The other twelve Windings are shown with Winding_0 offset in the sets of images below. They can be compared six at a time by using the buttons below:

Rule 3:

The twelve Windings also have to nest each other as best their design differences allow them to.

The implementation of these three rules resulted in four distinctive Winding groupings that will be termed "Nesting Groups":

1. Great Arc Midpoint Edge Nesting Group:

It is made up of Windings 4, 5, 6, 7, 8 and 9. These six Windings nest together in such a manner that their Midpoint Edges coincide on the same edge located on the one of the Great Arcs of Winding_0. This edge also is located in the center of their CVAP Set. The Windings' relationship differences are a series of loop direction reversals. Winding_5 is a hybrid of Windings 4 and 6. Winding_7 is a hybrid of Windings 6 and 8. Winding_9 is a modification of Winding_8.

The Great Arc Midpoint Edge Nesting Group is shown below using a downward looking perspective. The Midpoint Edge is shown in using the adjoining faces outlined in black:

  2. S Curve Midpoint Edge Nesting Group made up of Windings 10, 11, and 12.

These three Windings nest together in such a manner that their Midpoint Edge bisects Winding_0's and their nesting S Curves. Again, the Windings' relationship differences are based on a loop direction reversal.  Winding_11 is a hybrid of 10 and 11 .The Midpoint Edge is again shown in the context of its adjoining faces with the view switched back to front perspective: 

3. Winding_1 Nesting Group:

Winding_1 defines its own group. It nests with Winding_0 in such a way that it shares both its S Curves and Great Arcs. Its Midpoint Edge however does not lie on either its S Curves or Great Arcs:

4. Asymmetrical Nesting Group is comprised of Windings 2 and 3:

Both these Windings are asymmetrical in nature and differ from the other asymmetrical Windings 5, 7 and 11 by not being simply hybrids of symmetrical Windings. They differ from the other Windings in that their Midpoint Edges play no part in their nesting. However their CVAP Sets do line up with the other Windings.

It should be added that within these four Nesting Groups it is also possible to find many sub-nesting groups, but these main four are all that are required do establish a twelve Winding to twelve vertexes relationship. Next it will be shown how +Windings 1 through 12 of CVAP Set 1 and the four Nesting Groups can be correlated with a particular vertex. It took a very long time to achieve this correlation, because two major mistakes were being made during research:

1. The certainty that Terminal Vectors were a "pointers" per se to the correct vertexes (the Terminal Points) because of the simple duality division of twenty four Terminal Vectors for twelve Windings. This turned out to be not the case as will be explained shortly. While sometimes a Winding is correlated with a Terminal Point (using a Terminal Vector), most of the time it is the Cross Terminal Vector and its Cross Point that does the correlation.

2. Too much on the Windings' twenty faces count. Once focus changed to the "twelveness" of the vertexes the solution began to emerge.

This "twelveness solution" is to convert each Winding to a short-hand version of itself that will be termed "Icosa Short Hand" or the acronym "ISH" which will prefix "Winding". An "ISH_Winding" simply follows twelve of the icosahedron's twenty edges to sequentially connect the Winding's Cross Points so that all twelve vertexes are crossed only once. Sometimes it finishes on the Winding's Terminal Points, but most times not.

Here are all Thirteen ISH_Windings superimposed on their respective flattened +Windings:

The thirteen 3D +Windings with their ISH_Windings for CVAP Set 1 are shown below in using a pure front view. The sphere at an end of each ISH_Winding designates that the Winding is correlated with that particular vertex, on the other hand the cylinder indicates that the Winding is NOT correlated that vertex. Why these correlations were established will be described further below in detail using a game approach.  

Again, the same 3D front views but with the now color-coded ISH_Windings by themselves:

The reader can compare and change the two sets of front and isometric images below:

How an ISH_Winding is correlated to a particular vertex (designated with a sphere) is accomplished through a game. This game first requires a labeling of the twelve vertexes using neutral ISH_Winding_0 for a progression of lower-case letters:

Next, this game has the following rules:

A. If an ISH_Winding is the only ISH_Winding terminating on a particular vertex it will "Own" that vertex. The terms "Own" and "Ownership" denote the Winding IS correlated with the vertex.

B.  An ISH_Winding can only Own one of vertexes of the two it terminates at. The other terminating vertex is therefore described as being "Leased" by it.

C. If a group of ISH_Windings R, S, T and V all terminate at the same vertex and R, S, and T already Own other vertexes, but V does not, then V will Own that vertex by default while R, S and T will Lease it.

This game is applied to only one of the four CVAP Sets at a time. There are sixteen possible configurations for this game with in each CVAP Set that makes for a complete total of sixty four! When the opposite handed -Windings are taken into account the permutations double to one hundred and twenty-eight. This example it is played in CVAP Set 1. Configuration one will now be demonstrated step-by-step.  The other fifth teen will be shown and compared later. Some of the Winding relationships that occur within these sixteen possibilities never change and are therefore termed as "Relatively Static". Others that are subject to change and create the uniqueness of each outcome will be termed "Variable". The description of this particular configuration will start with the ISH_Windings to which rule A applies. These ISH_Windings are 2, 3, 1, and 9, which in turn are paired off as 2, 3 and 1, 9. A river analogy will be applied here with each ISH_Winding "headwater/source" being the Owned/sphere vertex and flowing to the "confluence" of its Leased/cylinder vertex. The river starts with pairings 2 and 3 being tributaries to a river sequence of 4 ->5 ->  6 -> 7 -> 8 that is the majority of Windings from the Great Arc Midpoint Edge Nesting Group, while 1 and 9 are tributaries to a second smaller river sequence that is S Curve Midpoint Edge Nesting Group of 12 -> 11 ->10. These two rivers converge at 8 and 10 which in turn actually converge on each other as a loop or -continuing the water theme- the currents of a whirlpool.

In the image below each river will be traced from its tributaries' sources by following the black arrows:

Starting with ISH_Windings 2 and 3, none of the other nine ISH_Windings terminate on vertexes c and b. Because of rule A, 2 and 3 Own vertexes c and b respectively.  

ISH_Winding 2 <- vertex c

ISH_Winding 3 <-vertex b

Further, because of rule B then ISH_Winding 3 must default Ownership of its other terminating vertex (l) to ISH_Winding_4 since 4 is the only other Winding terminating at vertex l.

ISH_Winding 4 <- vertex l <- ISH_Winding 3

In turn, both 2 and 4 must Lease and give up Ownerships of vertex j to ISH_Winding 5 for the same reason. 

ISH_Winding 5 <- vertex j <- ISH_Windings 2 and 4

A linear sequence of vertex Ownership and consequent Leasing is then followed.

ISH_Winding 6 <- vertex a <- ISH_Winding 5

ISH_Winding 7 <- vertex h <- ISH_Winding 6

ISH_Windings 8 and 10 terminate on the same vertexes for both ends, so they windup exchanging vertexes:

ISH_Winding 8 <- vertex i <- ISH_Windings 7 and 10

ISH_Winding 10 <-vertex f <- ISH_Winding 8

Next tributaries ISH_Windings 1 and 9 to a new river will be shown in the below image and following description: 

Again, both ISH_Windings 1 and 9 Own vertexes c and b respectively because of rule A. Together they also Lease vertex k and default Ownership of that vertex (following rules B and C) to ISH_Winding 12.

ISH_Winding 1 <- vertex c

ISH_Winding 9 <- vertex b

ISH_Winding 12 <- vertex k <- ISH_Windings 1, 9

With 12 a one river of Ownership and Leasing sequence commences:

ISH_Winding 11 <- vertex d <- ISH_Winding 12

ISH_Winding 10 <- vertex f <- ISH_Winding 11 

ISH_Winding 8 is met up with again at vertexes f and i:

ISH_Winding 10 <- vertex f <- ISH_Winding 8

ISH_Winding 8 <- vertex i <- ISH_Winding 10

The twelve "Owning" spheres on their vertexes using +Winding_0:

When two Windings share the same Terminal Vertex that relationship will be called a "Node". There are two types of Nodes. The first type is a "Conjunct Node". "Conjunctive" means that the Windings have the same Terminal Edge, Terminal Face, and Terminal Vector. The ISH_Windings of Conjunctive Nodes will share the same edge that leads to the Terminal Vertex. In the example image below +ISH_Windings 8 and 10 create a Conjunctive Node at vertex F and share the edge f-g:

The second type is a "Disjunct Node". “Disjunct” means that only thing the Windings have in common is the Terminal Vertex. They have different Terminal Faces and Edges. ISH_Windings reach the same Vertexes via different edges as shown in the following image demonstrating vertex j:

Because a vertex can have more than two Windings terminating at it, it can have sets of both Conjunctive and Disjunct Nodes. In fact a Winding can be in a Disjunct Node with one Winding at a Terminal Vertex and at the same time be in a Conjunctive Node with another Winding at the same vertex. However only vertexes j and i carry sets of Disjunct Nodes. In the case of vertex j the Windings 4 and 5 terminate there in a Variable Conjunctive Node, while Winding_2 is in a Disjunctive Node there with both of them. The Disjunctive Node of vertex i is a Relatively Static one using Windings 8 and 10. Any possible Conjunctive Nodes these two have with other Windings (7 and 11 respectively) are Variable.

The concepts of Conjunctive and Disjunctive Nodes are important from the standpoint of differentiating and (as shall be seen later) interpreting the different structures. "Disjunctive" implies separate and a dichotomy therefore and less unified, while "Conjunctive" implies together and more unified.

Now to show the sixteen possible configurations of the twelve Windings coordinated with the twelve vertexes two images will be used in combination.

1. A 3D screen shot showing the twelve spheres on their Winding_0 vertexes with a corresponding line connecting to the Leased Terminal Vertex like the example below for Configuration One:

2. A schematic drawing will be utilized as a simplifying guide like the example on the below that accompanies the above 3D image. A combination of matching color-coding and lower-case vertex labeling on the schematic will help clarify 3D screen shots:

One aspect of the 3D image not reflected in the schematic are the distances between the two Terminal Vertexes of each Winding. The Winding symbol in the schematic only reflects the total twenty triangle faces used by each Winding.

Now a key to understanding the schematic:

The schematic and 3D images are then shown in a composite image:

Next, the five basic building-blocks that consist of one Relatively Static and four Variables which make up each configuration will be demonstrated.

State 1 -the Relatively Static Windings: Each configuration always has the same four Windings which are Relatively Static in nature because the vertexes they Own never change with in the confines of the configuration's CVAP Set: 

State 2: The Variables Windings 8 and 10 have the same Terminal Vertexes at both ends with vertex i as a Disjunctive Node and vertex f as a Conjunctive Node. Therefore the Ownership and Leasing of their mutual Terminal Vertexes can be swapped between them making two different configurations possible. It is very important to note the impact these Variables have on the schematic diagrams because sometimes the ONLY difference between some of them is the reversal of Terminal Vertex Ownership between Windings 8 and 10!

State 3: Variable Winding_11 is a synthesis of Winding 10 and Winding 12 and asymmetrical. If it is rotated 180 degrees on the Y axis it will cause Ownership of the vertexes k and d to swap between itself and Winding_12. Also, if Winding_11 Owns vertex d, it will Lease vertex f in a Conjunctive Node with Winding_10 and Winding_8.

On the other hand if it Owns vertex k it will Lease vertex i (diametrically opposite from vertex f) in another Conjunctive Node with Winding_10, but a Disjunctive Node in relation to Winding_8. Notice too in the below image how the positions of relatively static Windings 1 and 9 are actually changed visually through the change of Ownership of vertex k.

State 4: Variable Winding_5 is also asymmetrical as a synthesis of Winding_4 and Winding_6. When it is rotated 180 degrees on the axis shown it swaps Ownership and Lease of vertexes j and l with Winding_4. The rotation can also alter the Ownership and Lease vertexes h and a by Winding_6. More on this swap further on.

State 5 -Part 1: Variable Winding_7 is asymmetrical as well, being a synthesis of Windings 6 and 8. It is shown below in the context of a change of Ownership of vertex i between Windings 8 and 10. While it is in a Conjunctive Node with Winding_8 at vertex i, it is also in a Disjunctive Node with Winding_10 at the same vertex. This is a similar relationship to the one described with Winding_11 at State 3 above.

State 5 -Part 2: Variable Winding_7 can be revolved 180 degrees on the same axis as Winding_5 described in State 4. This will cause a change of Ownership and Lease of vertexes h and a in Conjunctive Nodes with Winding_6. It also causes Winding_7 to Lease vertex f in a Conjunctive Node with both Windings 8 and 10:

State 5 -Part 3: Here we step back a little to State 4 and show the interaction and impact of Windings 5 and 7 on Winding_6. In Steps I and IV shown below Windings 5 and 7 are separated but connected by Winding_6, but in Steps II and III a 180 degree revolution of either Winding 5 or 7 joins the two together in a Conjunctive Node at either vertex a or h and removes Winding_6 from the main flow and turns it into an appendage or side stream thus altering the configuration:

Below are four separate schematic/3D configuration images belonging to CVAP Set 1. These can be changed by clicking the buttons below them so that all sixteen permutations can be compared:

The CVAP Set 1 schematics 1 through 16 (but without the vertex labeling) are shown below. It's interesting to attempt to correlate the schematics with the 4 x 4 suit/element matrix of the tarot court cards:

knight/fire of wands/fire, queen/water of wands/fire, prince/air of wands/fire, and princess/earth of wands/fire

knight/fire of cups/water, queen/water of cups/water, prince/air of cups/water, and princess/earth of cups/water

knight/fire of swords/air, queen/water of swords/air, prince/air of swords/air, and princess/earth of swords/air

knight/fire of discs/earth, queen/water of discs/earth, prince/air of discs/earth, and princess/earth of discs/earth

Any interpretation would be subjective and have an analogical approach similar to: "This planet represents Mars the god of war because it is red, since red is the color of a flushed angry face and blood", but would use the schematics' design elements relationships and their dynamics to frame the analogy. Admittedly that given the similarity of the schematics, the interpretations would require some time and study. 

It is possible to develop a simple interpretation just based on the numerical sequence above. However, another aesthetic interpretation is shown below. The whys and therefores of which schematic is a particular element become obvious with a little examination, but these interpretations are not written in stone. 

CVAP Sets 2, 3 and 4 also generate their own individual sets of sixteen schematics. These are developed from the perspective of the CVAP Set 1, but modified to convey the various rotations and mirroring of the particular CVAP Set relative to CVAP Set 1. It's important to remember that the vertex lower-case labeling of the ISH_Winding_0 remains the same for all four CVAP sets! This makes for sixty-four total schematics for CVAP Sets 1, 2, 3 and 4. Also, the CVAP Sets of the -Windings will generate mirror opposite sets. This means these will actually duplicate the sets shown here, but they will correspond to different configurations! For instance the CVAP Sets 1 and 2 will simply be interchanged as will Sets 3 and 4.

It's tempting to attempt a correlation between these and the sixty-four hexagrams of the I-Ching, especially with the existence of the taijitu within the icosahedron/dodecahedron demonstrated in Chapter 1, but that is to be set aside for another day.  Like the element interpretation above there is nothing really is set in stone in regards to which. Also, the reader is free to interpret their own elements within the four by four matrices.

CVAP Set 2 is a left-right mirroring of CVAP Set 1 with the additional of the angles expressing the Disjunctive Nodes set in the opposite direction from those in CVAP Set 1. The latter modification was done to create the same clockwise/counter-clockwise reversal that compares to when a helix is viewed from opposing sides.

The sixteen schematics for CVAP Set 2:

Next, CVAP Set 3 is simply CVAP Set 2 inverted creating a rather provocative set of stylized human-esque stick figure images to compare with the other sets:

Finally, CVAP Set 4 which is an inverted CVAP Set 1:

Having now transitioned fully from the twelveness of the Windings of the "Eighth Sphere" to their potential correspondence with the icosahedron's twelve vertexes through a four x four x four matrix; "fourness" will now be developed symbolically in the aforementioned matrix and other aspects as well by developing a series of  element sigils for fire, earth, air and water. Even a simple wire-frame view looking directly down on an icosahedron's face reveals the "Seal of Solomon" or "Star of David" which is the well-known source of the traditional element symbols for air, water, fire and earth (headlining this chapter using Islamic color-coding):

This "Star of David" viewpoint will be the constant frame of reference in the interpretation of these sigils.

First, element sigils will be developed for CVAP Sets 1-4. My method of obtaining sigils is by connecting the four Owned Vertexes and the four Leased Vertexes of the Relative Static Windings 1, 2, 3, and 9 into a pair of separate figures. The Owned Vertexes creating a coplanar four sided figure with the Leased Vertexes being a planar triangle. These resulting pairs of figures are to be considered as a unit and work together as a sigil. There is a four step process in for creating these particular sigils:

Step 1. is the initial creation of the figures using a convenient front angle perspective. The vertexes are labeled in lower-case and given color-coded swatches for Windings 1, 2, 3, and 9. Square swatches are for the Leased Vertexes and the Circles for the Owned.

Step 2. switches to the Star of David viewpoint looking downward on the center of a icosahedron triangle face. The chosen face is Face III, Quarter 1 of Winding_0. The color swatches are retained to compare to Step 1 and give a sense of orientation.

In Step 3. the pair of figures are treated as two dimensional silhouettes as if the Star of David viewpoint showed their true shape. They are filled in with solid black to emphasize this transformation.

Steps 1-3 are shown in the following four images for CVAP Sets 1 through 4. Step 4 is reserved for the finish.

Elemental Sigil for CVAP Set 1 (CVAP Set 1 being the on-going example above):

Elemental Sigil for CVAP Set 2 (CVAP Set 1 rotated 180 on the X axis):

Elemental Sigil for CVAP Set 3 (CVAP Set 1 rotated 180 on the Z axis):

Elemental Sigil for CVAP Set 4 (CVAP Set 1 rotated on the Y axis):

Step 4. is the interpretation and finish. The four silhouette pairs are then compared and accessed to which best represents fire, earth, air and water. Below is an interpretation using the Islamic color coding system to signify earth, water, fire and air. The simple triangle shape of each sigil is meant to function similar to how a diacritical mark will change pronunciation in a language with the quadrangle silhouette-shape being the letter itself:

Now quaternaries of element sigils will be developed for aspects of the original Winding_0.

+Winding_0's configuration returns to its beginning. However there are six basic ways it can accomplish this and these will be termed +Sub-Winding_0 and suffixed _A through _F. They are shown below flattened-out with their Terminal Faces correlated to the Quarter Faces of the 3D view:

The quantity of six is only superficial though because +Sub-Windings 0_A and 0_F can only occur at junctures of pairs of I and V opposite faces respectively and there are only four total of such junctures within the structure of  +Winding_0. These junctures are the boundaries of Quarters 1 through 4 as first shown in Chapter 1:

First, a set of element sigils from the Star of David viewpoint for these two pairs of Sub-Windings 0_A and 0_F will be created.  In the image below +Winding_0 has been changed to peach color matched with its analog ISH_Winding_0 in white. The four junctures that are the boundaries of the four Quarters are reiterated:

Next, in the image below Steps 1 and 2 outline the respective pairs of Terminal Faces for Sub-Windings 0_A and 0_F. Then Step 3 abstracts and colors the Terminal Faces keeping the edges of ISH_Winding_0 bordering the Terminal Faces as diacritical marks for interpretation of the four elements. Since those Terminal Face edges function so critically in the interpretation they are given a bright color, while the surrounding Terminal Face is darkened.

Another set of sixteen sigils can be obtained from the four +Sub-Windings 0_B, 0_C, 0_D and 0_E that exist for each Quarter. Quarter 1 is demonstrated  below as an example using the Star of David viewpoint again as a point of reference:

The sets of four sigils for Quarters 2, 3, and 4 are constructed using the same procedure. Notice the order direction reversal of _B through _E in the context of different Quarters:

Finally, a set of sigils representing the four Quarters themselves is created. The creation of these sigils incorporates the three possible Cross Points for each Quarter. However the process really just follows the vertexes of ISH_Winding_0:

The ISH_Winding_0 is then traced in black from where each Quarter of Winding_0 starts and finishes, incorporating the three Cross Points (vertexes):

These are then extracted and the three Cross Points are connected as a solid triangle and colored according to a personal and subjective interpretation of the four elements. It will be noticed each of the sigil has a pair of "legs" that connect the triangle with the where the Quarter started and finished crossing the ISH_Winding. These legs function again as diacritical marks to aid interpretation.

Here is a sample of the reasoning for the interpretation of the elements in the image below: Quarter 1 sigil was given the element air and Quarter 2 sigil the element earth because they interpenetrate each other and because one of their legs points up and the other down. Air is the upward pointing triangle and earth the downward following the traditional symbols. Quarter 4 and 3 sigils are fire and water because they are distinctly separate and both of their legs point either up or down respectively:

Below on the right are the element sigils for all sixteen Sub-Windings 0_B through 0_E grouped under the four Quarter sigils on the left. Potentially the four sigils of Sub-Windings _0_A and _0_F could be used instead. At the moment however there lacks a way to directly associate either Sub-Windings _0_A or _0_F with a particular Quarter! It is important to mention that the size difference of the sigils is because Quarter 1 on top is closer to the screen-shot camera then Quarter 2 on the bottom with Quarters 3 and 4 wedged in between -the consequences of using a 3D model! All these sigils are based on subjective perceptions, so the size difference is a phenomenon as much as the angles that change the shape of the four Quarters are changed by their angle relative to the viewer.

A summary sketch of the developed element sigils for the tarot computation:

1. The four-by-four-by-four matrix controlling the vertex to Winding correspondence allows an element (from either the ace-ten suite or aspect of zodiac sign) to influence the court card selection which in turn can influence the astrological twelve-sign major arcana selection. Since each Winding is constructed of twenty faces/vertexes accommodating two small card suits in sequence, then its selection can influence the obtaining of a particular small card from ace through ten.

2. Sub-Windings of Winding_0 produce yet two more different approaches:

  a. Sixteen total Sub-Windings_0_B through _0_E for selecting court cards. This allows for a different approach from the one described above and  different approaches make for a more flexible system.

  b. Four sigils for Sub-Windings _0_A and _0_F. These can select an element for the small cards suit or the elemental aspect of zodiac sign major arcana card.

3. Sigils for the four Quarters of Winding_0 form four "super suits" that encompasses the sigils of Sub-Windings 0_B through 0_E as demonstrated in the last image above.  The relationship is just like four court-cards to the parent suit. 

All these manifold divisions of four reveal the icosahedron as a "hyper-mandala" that leaps the wholeness-through-quaternary concept of the Jungian Mandala beyond the sum of its parts. Leaping beyond the sum of the geometry, randomness, elements and tarot per se is the ultimate goal of this system too.

Continue to Chapter 3: Conversely, the Twelve also Reveals the Thirteenth -or- the Strange Congruence

or read the Appendixes.

Appendix 1: Dodecahedron Windings

Presented here are the direct dodecahedron Winding analogs of the icosahedron's Windings so "Winding" will be prefixed by "Dodeca_".

First an image of all thirteen dodecahedron +Windings:

Next below are images of the Dodeca_Windings that can be changed by the user for comparison. If the image of a particular Dodeca_Winding is already shown, then the Icosa_Winding analog will be shown.

Appendix 2: Sphere Packing of the Cuboctahedron

It should be mentioned that there is another well-known twelve and thirteenth relationship involving the cuboctahedron ( shown in Chapter 1 ). This relationship involves a mathematical subject called "sphere packing".

In the case of the cuboctahedron, if spheres with a radius equal to one-half the length of the cuboctahedron's edges are centered on the cuboctahedron's vertexes, then there will be in the center of the cuboctahedron a space of equal in size to one of the surrounding spheres. This concept has an easily appreciable 2D analog! If you take six equal diameter discs and arrange them in a circle so their edges touch, then those six discs create an exact space in the center which measures the diameter of seventh disc equal to the other six:

Image 1 below are views of the cuboctahedron's triangle and square faces with spheres beginning to be packed on to the vertexes. Notice how the spheres only touch but do not intersect and how they divide the cuboctahedron's edges in half:

Image 2 below are views of the cuboctahedron with the three sets of green, red and blue spheres of four each:

Image 3 below are front, side and top cut-away views showing the inside of the cuboctahedron and the center white sphere that is of equal size to the surrounding spheres. Notice how the center sphere touches but does not intersect them:

Image 4 below shows two final views of all thirteen spheres without the cuboctahedron:

Continue to Chapter 3: Conversely, the Twelve also Reveals the Thirteenth -or- the Strange Congruence