Introduction

This chapter will show how the concept expressed by the famous yin-yang symbol "taijitu" (shown above) can also be found in both the icosahedron and dodecahedron regular solids. In the simplest terms, the taijitu symbolizes that opposites contain an element of their contraries -the white teardrop contains an element (the black disk) of the black teardrop and vice versa to create interdependence. Outside of the taijitu itself and the I-Ching, expressions or interpretations of the yin-yang concept tend to be of an intuitive and subjective nature. What is going to be presented here is an objective geometric model, and while I will not claim this demonstration is as simple as the black and white of the taijitu, it is certainly clear in red, blue and green, as the reader will see! This Wikipedia article gives a more in depth explanation on yin-yang with links to the taijitu article as well: Wikipedia on yin-yang

Moreover, the chapter is also something of a test-case to show how the Platonic solids can be viewed from a perspective of capability.

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The icosahedron: The icosahedron's relationship to the taijitu will be shown first because it is a little simpler and therefore easier to visualize. The internet is rife with information on the icosahedron and the Platonic solids in general so a description of the icosahedron itself will be brief. The icosahedron is a three- dimensional solid constructed of twenty equilateral triangles for faces, thirty edges of equal length, and twelve vertexes. Because all its faces are equal it is known mathematically as a regular solid:

 It is also known as a “Platonic solid”, because it was one of five regular solids described in Plato’s Timaeous dialogue and Plato associated it with the element water.  The five regular solids –tetrahedron, octahedron, hexahedron or cube, dodecahedron and icosahedron- are all intimately connected. The descriptions of tetrahedron, octahedron, cube and dodecahedron and their relationships will be described afterward. Everything applied to the icosahedron can be equally applied to the dodecahedron; however, the dodecahedron’s relationship to the taijitu will come later as well.

Step 1: The three-dimensional taijitu model starts out by connecting the icosahedron’s twenty faces so the connecting lines (in red) return to the starting face with each face only being crossed once. The resulting configuration will be termed a “Winding”:

 The Winding (and consequently the icosahedron as well) can be flattened-out in two dimensions:

Step 2: This Winding has a complementary mirror-opposite construction. The polarity symbols “+” and “-” will prefix and designate the opposite Windings.  The Winding described and pictured above in step one will be fully termed as “+Icosa Winding”, while “-Icosa Winding” will be shown below:

It is important to end step two by emphasizing that the creation of either a + or – Winding automatically brings about the creation of its opposite. For example, if +Winding is viewed as a line connecting the faces in time like an animation sequence, then its very first face-to-face step demands the possibility of a movement in the opposite direction. This in itself could be interpreted as the incorporation of yin-yang opposites; however, this chapter is going to demonstrate something more explicit.

The red +Winding and black -Winding flattened-out:

Step 3: The +Winding will be divided into four equally shaped parts termed as "Quarters". These Quarters are labeled numerically 1 through 4. Each of the five triangle faces constituting a Quarter will acquire a roman numeral label of I through V. Starting with "Quarter 1":

Below the mirrored Windings are shown as 3D and flattened-out views. -Winding reciprocates the I through V facial labeling for its Quarters. Note the location of the -Winding's Quarter 1 on the flattened-out image. This is because the Quarter's number is dictated by the Quarter its faces are opposite. Note also that because of the interior view relation of the three-dimensional model to the viewer -Winding's Quarter 1 appears to be the mirror of the exterior view flattened-out image.

Quarter 2 and again the mirror effect comes into play because of 3D interior versus flattened-out exterior points of view:

Quarter 3:

The Winding returns to the beginning with Quarter 4:

Step 4: A Winding crosses over the twenty icosahedron faces using only one of three possible paths. While these "paths" will be termed here as "Crossing-Paths", they are known primarily in mathematics as "geodesics". The goal now is to build a “complete” icosahedron  by utilizing all three possible Crossing-Path for each face so that each of the sixty total Crossing-Paths will be used only once:

The best way to accomplish this goal is to use three Windings of the same polarity in such a way that each Winding is aligned and identified with one of the X, Y and Z Cartesian-coordinate axis that define the respective side, front and top views. Each of the three Windings is also color-coded to match their respective assigned axis. Just as the axis are perpendicular to each other, so are Windings.   

It is important to emphasize that the green and blue +Windings are simply the configuration of the red +Winding revolved to the top and side positions respectively:

In the three images below the +Winding is shown in white with the -Winding now acquiring the appropriate color:

Again, the green and blue -Windings are simply the configuration of the red -Winding revolved to the top and side positions respectively:

Step 5: The II and V Faces of the four Quarters play a central and a seemingly controlling role for the Windings, because the three red, blue, and green +Windings all share or overlap the same physical icosahedron faces for their respective II and V faces:

None of the I, III and V faces have the synchronous feature of the II and IV faces. I, III and V faces are arranged just like that: I, III, and V. For example: red I face will be overlapped by a blue III face and a green V face.

Also the –Winding’s (fuchsia letters) II faces share same icosahedron face as the +Winding’s IV faces and vice versa:

Step 6: The flattened - and +Windings are shown together: 

The Quarter 1 of both + and -Windings are now extracted and their faces labeled I through V:

An operation is now performed on the extracted Quarters 1 for purposes of clarity. The Quarters are disassembled and their faces are rotated as needed so that the vertex opposite the red Winding line is always uppermost:

I, III, IV and V and then a reversal of red, green and then blue for its face II. This holds true for all four Quarters.

Compare below the clockwise color order of +Winding face IV on the right with the clockwise color order of -Winding face II on the left and vice versa:

The clockwise rotation predominate order of red, green and blue for the +Winding represents the white teardrop of the taijitu:

That “black disk” minority becomes the majority color order for the -Winding, which in turn represents the black teardrop:

-Winding’s Face II then is the white disk for the new minority order of red, blue and green

original majority color order of the +Winding and the white tear-drop.

 Consequently, the three red, green and blue Windings of a given polarity incorporate the color order rotation of the opposite polarity Windings directly into their resulting configuration! So the taijitu is incorporated explicitly into the structure of the icosahedron.

Other Platonic (regular) and semi-regular solids: Next, images and descriptions of three other regular/Platonic solids with their relationships to the B and D faces of the icosahedron and a semi–regular solid the cuboctahedron; then a description of the dodecahedron and its relationship to the taijitu.

1.      The first member of the regular solid family is the hexahedron or cube, which as can be easily seen is the simple box with six equal square faces, twelve edges and eight vertexes and signified the element earth in the Platonic dialogue Timaeus. The eight vertexes of a cube are the center points of the eight icosahedron faces that constitute the II and IV faces of the four Quarters:

2.      The second regular solid is the tetrahedron. It is the simplest of the five solids with four equal equilateral triangles for faces, six equal length edges, and four vertexes and signified the element fire in the Timaeus. Its edges can be six diagonals of a cube with the four vertexes becoming four of the cube's eight vertexes:

Now the tetrahedron is in what is called a “dual relationship” with itself. This means that if the blue-edged tetrahedron is copied and inverted  with the edges turned green, this new tetrahedron’s vertexes will now be the center points of the blue-edge original’s faces and vice versa. The beige-colored cylinders in the images below help to demonstrate this vertex to face-center relationship:

The other tetrahedral result of this dual relationship is that the new green tetrahedron's vertexes complement the blue's by completing the rest of the cube's vertexes. This in turn means that the tetrahedrons' vertexes also define the center points of the II and IV faces of the +Winding. In the image below on the right note how the green tetrahedron defines the center points of the IV faces while the blue defines the II faces!

Conversely in terms of the mirror opposite -Winding, the blue tetrahedron defines the IV faces while the green defines the II faces: 

These relationships between the tetrahedrons and icosahedrons allows the creation of a color-rotation axiom:

"Since one of the tetrahedrons has an alignment with a set of II faces from Windings of one polarity along with an alignment with the IV faces of Windings of the opposite polarity and since those II and IV faces will have the same color-order rotation, then that tetrahedron can be identified with the polarity of the set of Windings where that color-order rotation is the majority."

3. The third regular solid is the octahedron. It consists of eight equal sized equilateral triangles, twelve edges of equal length and six vertexes. It is the dual-solid of the cube, that is its six vertexes which are intersected in pairs by the X, Y and Z Cartesian axes are also the center points of the six faces of the cube. The cube reciprocates with its eight vertexes as the center-points of the tetrahedron’s faces:

Consequently the octahedron’s faces also coincide with the I and IV Quarter faces of both the +Winding and -Winding:

The face-vertex relationship between the octahedron and its sibling cube can produce another useful and interesting three-dimensional solid called a cuboctahedron. The cuboctahedron consists of eight square faces and six equilateral triangle faces, twelve vertexes equi-distant from the center and twenty four edges of equal length.

As its name implies the cuboctahedron can really be understood as a synthesis between a cube’s six square faces and an octahedron’s eight   triangular faces. The octahedron has been enlarged to clarify:

Since the vertexes of the cube are the center points of both the triangle faces of the cuboctahedron and the II and IV Quarter faces of both the +Winding and -Winding, then the triangle faces of the cuboctahedron can be said to correspond to the II and IV Quarter faces:

While the icosahedron has twenty faces and the cuboctahedron only has fourteen, the cuboctahedron can be potentially changed to have twenty faces by subdividing the rectangular faces along their diagonals respective to the X,Y and Z axis to create new triangle faces. In the image below the only the three front rectangle faces of the cuboctahedron and their corresponding icosahedron faces are shown for simplicity and clarity.

The final Platonic solid is the dodecahedron. It consists of twelve equal-sized pentagons for faces, twenty vertexes and thirty edges of equal length. In the both the Platonic dialogue it was given the distinction of being an image of the universe itself.  At Timaeus 55 C after just describing the other four solids as respectively the fire, water, air and earth elements Plato writes: “There was yet a fifth combination which God used in the delineation of the universe with figures of animals.” (B. Jowett translation found in Plato: The Collected Dialogues, edited by Edith Hamilton and Huntington Cairns, Bollingen Series LXXI, Princeton, 1989.  A truncated, but relevant version of that translation can be found by going here: Timaeus Dialogue and using the “Find” feature in your browser for the relevant quote.) The phrase “with figures of animals” is assumed to mean the twelve figures of the zodiac –one figure for each of the dodecahedron’s twelve pentagon sides.

At Phaedo 110 there is an extended description of a “true heaven and a “true earth” –a higher reality beyond our current state that is further described at 110 B:  “The tale, my friend, he said, is as follows:—In the first place, the earth, when looked at from above, is in appearance streaked like one of those balls which have leather coverings in twelve pieces, and is decked with various colors, of which the colors used by painters on earth are in a manner samples. But there the whole earth is made up of them, and they are brighter far and clearer than ours…” (B. Jowett translation that can be found here: Phaedo Dialogue and by using the “Find” feature in your browser for the relevant quote.)

Screen shots of the dodecahedron:

It is in a dual relationship with the icosahedron, this means that the vertexes of one are the center-points of the faces of the other: twenty vertexes of the dodecahedron are the center points of the twenty faces of the icosahedron and reciprocally the twelve vertexes of the icosahedron are the center points of the twelve pentagons of the dodecahedron. The one solid describes the other. Below are six images showing this dual relationship in respective front, perspective and side views. The dodecahedron’s edges have been exaggerated for clarity.

The similar views with the icosahedron as a frame only:

Because the icosahedron and dodecahedron have this dual relationship, the taijitu image can be duplicated in the dodecahedron. Twenty of the dodecahedron’s edges are used to connect its twenty vertexes to create a +Winding dodecahedron style:

Since the vertexes of the dodecahedron Windings are the center points of the icosahedron faces these vertexes will be termed as “Center-Vertex”.  The resulting Winding can be divided into Quarters with the five Center-Vertexes of each Quarter labeled I-V accordingly. In the image below the Quarters are shown in shades of red.

Of course the dodecahedron +Winding has a reciprocal –Winding as well (shown in black and slightly offset):

Below the dodecahedron Windings are flattened-out, but are shown in the context of the icosahedron’s triangle faces. The respective Quarter 1 Center-vertexes have been labeled.

Ophite Gnostic sects produced cosmological depictions that drew off both the Bible and the cosmology of Plato. Known as “Ophite Diagrams”, these drawings revised the biblical sea monster Leviathan into an Ouroborus that surrounded the cosmos. Since Plato wrote descriptions of the cosmos as the form of the dodecahedron, so with the Ophite diagrams in mind then a dodecahedron Winding can be understood as "Leviathan". The icosahedron version of the Winding was previously identified with the Ouroborus in the alchemist’s flask based on both the return to its beginning and serpentine design. Since the dodecahedron and icosahedron are in a dual relationship then the Hermetic maxim “As above, so below” takes center stage here and the alchemical Ouroborus undulating in the alchemical flask IS Leviathan. For more information on the Ophite diagrams here are two helpful sites: Wikipedia and Gnosis.org.

An Ophite Diagram:

It takes a minimum of three Windings of the same polarity and respectively oriented on the X, Y and Z axis to rebuild a complete dodecahedron. (“Complete” being defined as: “all edges crossed an equal number of times” with the minimal “equal number” being two.) Notice in the three images below how each dodecahedron edge is crossed only two times:

The two Quarter 1 of the + and – Windings are shown flattened-out below with the three respective colors connecting the Center-vertexes. Again, an icosahedron triangle-face context is used as a visual aid, but it should be remembered we are focusing on the Center-vertex of each face:

Similarly these Center-vertexes are disassembled and rotated to match the rotation of Center-vertex “D” so that the blue/green combination is perpendicular and on the bottom:   

Again, a clock-wise rotation is used to study the color-order combinations of the above image. It can now be seen that the dodecahedron +Winding has a green/red  -> red/blue  -> blue/green  combinations sequence for its Center-vertex II, while the majority of Center-vertexes I, III, IV and V have a sequence of red/blue -> green/red -> blue/green. Reciprocally, dodecahedron -Winding has a clock-wise color order sequence of green/red -> red/blue -> blue/green combinations for the majority of Center-vertexes I, III, IV and V with a reversed  red/blue ->green/red -> blue/green for its Center-vertex II which returns to the majority combination sequence of the +Winding.

Compare below the clockwise color-order of dodecahedron +Winding Center-vertex III on the right with the clockwise color order of -Winding Center-vertex II on the left and vice versa:

The clockwise rotation of the color combinations for the +Winding majority Center-vertexes I, III, IV and V   represents the taijitu’s white tear drop:

That “black disk” minority becomes the majority color combination for the -Winding, which in turn represents the black teardrop:

Which in turn is the return to the original majority color combination order of the +Winding and the white teardrop.

Thus the taijitu as a far-eastern symbol of the cosmos finds itself at home in a western cosmological symbol of Platonic descent and vice versa!