Three – Dimensional Geometry
Notes by Steve Bass
written 1995 revised 2024
References
One, Two and Three.
At the beginning of each area of study we return to the Monadic One. We do this as a discipline of relatedness to aid in the search for Unity. In the previous sections on plane Geometry we saw that the Monad is represented initially as the paradoxical point, then as the two dimensional circle. But, as noted in Timaeus [53d], “…the form of a body, in every case possesses depth also. Further, it is absolutely necessary that depth should be bounded by a plane surface; …”
Material, sensible objects exist in three dimensions. The mathematical forms discussed here exist only in the realm of Mind and are re-presented as drawn images. But even a drawing of a dot or line is three dimensional in that it is a graphite mark on a piece of paper. Under sufficient magnification it is seen that the graphite has thickness and that the ‘line’ is a pattern of dark material deposited on a textured surface. As symbol, however, it can be used to represent a two dimensional idea.
Point – line – plane generation of the sphere.
The One, the initial, non-dimensional Monadic position represented by the dot, in order to manifest, must divide, or reflect from within itself to become the Dyad, The point may be said to simply ‘inflate’, like a balloon. The center remaining passive and its surface being thought of as active. But in this manner the sphere remains undifferentiated, figure 1a. The Dyad may also be represented by the line, figure 1b. The polar, nature of the Dyad differentiates the line into active and passive ends; one end point remains still and the other moves to create the line. These qualities, by their inter-relation cause the line to rotate and create the circle. In the 2-D Geometry session, the rotation was restricted to a single plane only. But if the active end of the line, having created the circle by rotation, is allowed to move in three dimensions it will trace out a sphere.
The sphere, in these ways, is a sequential projection of the point as it moves into materiality. The point, then, may be represented by the sphere; as it is by the circle and the dot.
Nature of the sphere.
The sphere may be considered the primary solid figure. It is Monadic in that it contains the potential for all other solids. All parts of its surface are the same distance from the center. The same symbolism of harmonic inclusiveness and equality ascribed previously to the circle also applies to the sphere.
In Timaeus [33-34] we read, concerning the fabrication of the universe, “Now for that living creature [the universe] which is designed to embrace within itself all living creatures the fitting shape will be that which comprises within itself all the shapes there are; wherefore he
[the Demiurge] wrought it into a round, in the shape of a sphere, equidistant in all directions from the center to the extremities, which of all shapes is the most perfect and the most self-similar, since he deemed that the similar is infinitely fairer than the dissimilar. And on the outside round about, it was all made smooth with great exactness…” In an unpublished comment to this paper Critchlow writes, “Here we can see how appropriate the English word ‘fair’ is as it simultaneously means ‘just’ and ‘beautiful’.”
Speaking of the sphere mathematically, Critchlow writes [OIS, p4], “Any form, however random, if completely rotated on its center of gravity eventually describes a sphere at its extremities, hence the sphere is the most economical form that is non exclusive. The sphere thus seems the most suitable form to give to the ‘point’ as it has complete rotational symmetry and is least ‘biased’.” Because of this, he chooses to call this figure the ‘spherepoint’. Materially, the sphere encloses the greatest volume with the least surface area; thereby accounting for material forms observed across a range of scales from stars and planets, to soap bubbles or cells.
The sphere, the 6 directions and the 7 movements.
Looking again at the planer generation of the sphere, the initial circle, if it is to achieve a third dimension and create a sphere, must rotate about a diameter. As the circle enters the third dimension it automatically creates a line of intersection with the original plane of the circle. This is the ‘polar’ axis. This first axis is shown in vertical position in figure 1c.
Critchlow calls this first axis [SSA, p19],“Axiality or the balance of motion through spin. The complimentary of unity. The axis of consciousness, sometimes known as the ray of creation. Identification of qualitative axis of being. The axis of stillness (eternity) through the motion of time. Two points in the limit sphere.” This first vertical axis can also be thought of as the heaven – earth axis and represents the potential for up and down spatial movement. Critchlow [SSA, p18] calls it the ‘Ontological’ axis, thus relating it to the process of coming into being.
We may continue the rotation of the original circle until it is 90o to its first direction, thus expressing the minimum condition of the double Dyad – the right angle cross, figure 1c. But we still have only one axis of motion, one Dyadic pair of directions, up and down. Since we wish to describe three dimensions we may cause a third plane emerge, creating two new axes where it penetrates the earlier two planes. These three double Dyads are balanced equidistant between each other on the surface of the sphere,.
Each of the three axes are at right angles to each other. Together they represent the six primary directions of movement; up – down, forward – backward, right – left. The original point allows for a seventh movement, rotation, in which the original point remains still and unmoved.
Timaeus [34], in discussing the spherical nature of the world soul says, “For movement he assigned unto it that which is proper to its body, namely, that one of the seven motions which especially belongs to reason and intelligence; wherefore he spun it round uniformly in the same spot and within itself and made it move revolving in a circle; and all the other six motions he took away and fashioned it free from their aberrations”
At a later stage of creation, that of the descent of the individual soul into materiality, Timaeus [43] adds, “that its [the souls’] progress was disorderly and irrational, since it partook of all the six motions: for it progressed forwards and backwards, and again to right and to left, and upwards and downwards, wandering every way in all the six directions.”
Critchlow calls these three axes [SSA, p19], “The complementary axes or ‘boundaries of creation. …The three dimensional cross. The six directions or paths to and from the center and the six motions according to Plato.” Later, he adds, regarding the axes of the three dimensional cross [SSA, p20], “These have symbolic associations with human orientation as the determinatives of direction, i.e., the ontological axis is representative of ‘up’ and ‘down’; the axis almost in the plane of the paper [longitudinal] represents ‘right’ and ‘left’. The third (equatorial) axis represents ‘front’ and ‘back’.”
Groupings of spheres.
The sphere, as Monad, may divide itself from within itself, or multiply by reflection or re-production of itself. Any first division initiates a process of relational development of the centers from point to line; and, when the length of the movement of the reflection equals the spheres’ diameter, this may be seen as analogous to the movement from One to Two, figure 2a-b. We now have two spheres in contact at one point. Both spheres are free to rotate in any direction, but if one sphere rotates clockwise, the other will rotate counterclockwise [assuming that they rotate without any external resistance or friction]. To achieve 2-dimensional or planer stability, a third sphere needs to be added, figure 2c. Now we have three spheres with three points of contact. The only possible rotation is vortical, that is, toward or away from their common center. This assumes that the spheres are not arranged along a line but are in closest relationship, or ‘close packed’. Close packing is said to occur when the center to center distance of each sphere is minimized.
The Tetrahedron.
To achieve three dimensional ‘solid’ form, a fourth sphere needs to be added, figure 2d. If the four spheres are close packed they have six points of contact and no rotation of the spheres is possible. If each of the four sphere centers is taken to represent a vertex, the six lines connecting the sphere centers represent edges and the four triangular shapes enclosed by the edges represent faces.
The solid so formed is called a tetrahedron, ‘tetra’ meaning four faced and ‘hedron’ meaning a regular solid. The tetrahedron is the minimal expression of regular form in 3 – dimensional space. It is the primary, or parent regular polyhedron. The tetrahedron occupies the least volume for a given surface area, the reverse of the sphere.
As with the regular 2 – dimensional grids, the regular solids can have duals. In dualing the solids, the center of each face of the parent figure becomes a vertex of the offspring figure, figure 3a. Since the tetrahedron has four faces and four vertices, it duals itself, a unique property among the regular solids.
Generation of the solids by internal division.
If we connect the center points of each edge of a tetrahedron, figure 3b, we find have a new figure made of four tetrahedrons, one at each vertex, with a remaining regular solid at the center composed of eight equilateral triangles with twelve edges and six vertices, known as the octahedron, figure 3c. If the octahedron is dualed, figure 3d, that is if the center of each face of the octahedron becomes a vertex, the figure produced is the cube, with eight vertices, six square faces and twelve edges. The dual of the cube is the octahedron.
Returning to the tetrahedron, if we further divide each edge into three, each face now may now be composed of nine triangles outlining hexagon A-F, figure 3e. This shape is the outline of another regular solid figure. To define the figure further within the hexagon we draw triangle BDF and then make a two part division of its sides to obtain triangle GHI. GHI is one face of the solid outlined by hexagon A-F. If each of the tetrahedron’s four faces is similarly treated, the figure defined is the icosahedron, having twenty equilateral triangular faces, twelve vertices and thirty edges. The dual of the icosahedron is the dodecahedron, figure 3f, which has twelve regular pentagonal faces, twenty vertices and thirty edges. The dodecahedron in turn, duals the icosahedron figure 3g.
The family of regular solids.
These five figures comprise the complete set of regular solids, figure 4a. They form a family-like group, related by number, shape and means of generation. Because of this ‘intimate’ relationship any of the regular solids may be derived or generated from any of the others, figure 4b.
One way of discriminating among the solids is by their type of axial symmetry. Symmetry is defined in this case as an orderly disposition of parts about a central axis. In this case, figure 4c, if we look at the tetrahedron vertex on, using the vertex as the center, the figure displays three fold symmetry. The octahedron has fourfold symmetry and the icosahedron has fivefold symmetry. The key numbers here are 3, 4, and 5 which recall the root systems, √2 (2 is itself the root of 4), √3, and √5; as well as the 3-4-5 triangle
Interior qualities of the solids.
The interior structure of the solids also has a 3, 4, 5 nature, as shown in figure 5a. If we take a section perpendicular to the axis of any vertex of the tetrahedron, cube or dodecahedron, figure 5a, the section will be an equilateral triangle; the octahedron’s vertex section will be square; and the icosahedron vertex section will be a pentagon. If a section is taken parallel to any edge, through the center and out the other side, figure 5b, the octahedron section will be a square and the icosahedron section will contain a Ø rectangle (1 : √5 – 1/2). The cube yields the √2 rectangle and the dodecahedron a rectangle of Ø plus a square (1 : 2.618).
Generation of the solids by division of a single sphere’s surface.
The path of generation of the family of regular solids we have followed is only one of many possible. Taking another path, we may also note that each vertex of each of the regular solids lies on the surface of a sphere. A section through a sphere which goes through the center creates a circle on the surface of the sphere known as a ‘great’ circle. It is a 3-D equivalent to a diameter of a circle, which might be called a ‘great’ line. The vertices and [spherical] edges of the solids will all fall on great circles, demonstrating an intimate relationship of the regular solids and the undifferentiated Unity of the sphere.
In the generation of the sphere by rotation of the circle, the two great circles that give rise to the first axis also contain the four vertices of the tetrahedron. These fall at the points on the circles where a root two rectangle may be inscribed, figure 6a.
About this division Critchlow notes, (SSA, p19), “The fourness of the spherical surface symbolizing the fourness of creation, earthly or solid, water or liquid, airy or gaseous, and fiery or radiant, outward expressions of the four states of consciousness in themselves symbolic of the four ‘worlds’ or ‘divine presences’. The six paths between the four points
symbolizing the six days of creation or in the ancient Chinese tradition the six lines of Ch’ien, the creative principle.”
The third great circle, at 90o to the first two, gives rise to the six directions, generates the octahedron whose vertices fall where the great circles cross one another, figure 6b.
By inscribing a Ø rectangle in each of the three great circles, the twelve vertices of the icosahedron are yielded, figure 6c.
The cube and dodecahedron follow as duals or they may also be formed directly, the
The cube and dodecahedron follow as duals or they may also be formed directly, th cube by two great circles inscribed with root two rectangles, as with the tetrahedron, but using all the eight intersection points, figure 6d; and for the dodecahedron, three great circles each inscribed with a 1:2.618 rectangle for twelve vertices and another two great circles that would inscribe a cube with eight vertices, figure 6e.
Generation of the solids by spherical groups.
We may return again to the path of generation of the solids by sphere-point construction; we had stopped these at four sphere-points, figure 7a, generating the tetrahedron. We now see that there are six points of contact (A – F) between the four spheres. Figure 7b.
If each of these points of contact is considered a sphere center, we then have a close packed arrangement of six spheres with twelve points of contact (a – l), figure 7c. Each of the sphere-points may be said to define one of the six directions and each one may become a vertex of the octahedron. If the twelve points of sphere contact in their turn become spheres and then vertices, the figure generated is the cuboctahedron with six square faces and eight equilateral angular faces, figure 7d.
This arrangement of twelve spheres will exactly surround a thirteenth sphere of the same size. But since there is no thirteenth sphere the arrangement is unstable, and its square faces must triangulate to achieve stability, figure 7e. The six square faces double into twelve triangular faces, which, when added to the previous eight give the twenty faces and twelve vertices of the icosahedron, figure 7f. The cube and the dodecahedron may be produced through dualing.
Semi Regular Solids.
The cuboctahedron in figure 7e requires mention of the semi-regular solids. Just as the three regular 2-D grids could, by the use of more than one figure become semi-regular; so the five regular solids, whose faces are all the same, can generate 13 semi-regular solids by having faces of more than one polygon, but still with only one kind of vertex. Figure 8a, from Critchlow (OIS, p 33), shows the entire family of semi regulars. These figures were explored by the philosopher Archimedes of Syracuse, in the third century BC and are known as the Archimedian solids. As Critchlow notes there are thirteen of them, analogous to the twelve spheres that can completely enclose a thirteenth.
When the ability of any of the solids to be transformed into any other one was discussed previously, this change was thought of as instantaneous. But it can also be thought of as a continuous process of transformation, which may be stopped at any point, resulting in intermediate figures. In these semi-regular solids, all the edge lengths are equal and all the vertices are equal, but the faces are comprised of two or more types of regular polygons. For example, starting with the cube, figure 8b.1, if each of the vertices is symmetrically sectioned, that is sectioned at right angles to the center of the circumsphere, each of the cube’s vertices will appear as an equilateral triangle and at a ‘certain point’, each face of the cube will be an octagon. This figure is called a truncated cube, figure 8b.2, its faces are regular octagons and triangles. Continuing the inward movement until the triangles meet at the center of the cube’s edges, the figure becomes a cuboctahedron, figure 8b.3. If the process is continued until the triangles become hexagons by intersecting each other, the truncated octahedron is produced, figure 8b.4. Continuing the inward movement until the faces of the cube are reduced to points, leaves the octahedron, dual of the cube, figure 8b.5.
A Cycle of the solids.
The generation of the five regular solids may be said to follow a Tetractyc reproductive cycle, as illustrated by Critchlow [OIS, p 21], starting with the monadic, ‘parental’ tetrahedron. We may set its edge length equal to 1. It may be re-presented as a sphere, and as such is the first or ‘point’ stage, figure 9.1. From the tetrahedron are generated by internal division the octahedron and the icosahedron. These are the senior members of their families of, 4 fold and 5 fold symmetry. Their spheres form a Dyadic polarity. This is the second or ‘line’ stage, figure 9.2. The three spheres together represent the Triadic or ‘planer’ stage, figure 9.3. Through dualing, the octahedron and the icosahedron generate the spheres of the cube and the dodecahedron. These now form two pairs of dyadic polarities which, when close packed in a crossing relationship, give rise to a new tetrahedron, figure 9.4-5, thus completing the cycle, in a Tetradic, fourth or ‘solid’ stage. The new tetrahedron has an edge length of 2, and we may thus describe it as being an octave lower than the parent body.
Plato and the regular solids.
The senior mathematical creation story of them all is the Timaeus of Plato. The character called Timaeus is said to come from Locris, a Pythagorean colony in southern Italy. In the dialog Timaeus tells us [53] that the solids are the material representations of the four qualitative elements, earth, water, air, and fire. Timaeus creates the solids starting with two kinds of triangles, the right isosceles and the half equilateral. These will be familiar to architects as the two triangles that are on their desks. The right isosceles is known as the 45o and the half equilateral is called the 30o-60o, figure 10a-b. Starting with the 30/60, six triangles are arranged to form an equilateral triangle. First four, then eight, finally twelve such equilateral triangles form the tetrahedron, the octahedron and the icosahedron, respectively. Four 45o triangles form a square and six squares form the cube.
To these four solids Timaeus [53] then assigns the four elements: “To earth let us give the cubic form; for of the four kinds earth is the most immobile and the most plastic body, and of necessity the body with the most stable bases must be pre-eminently of this character. Now of the triangles we originally assumed, the basis formed by equal sides is of its nature more stable than that formed by unequal sides; and of the plane surfaces which are compounded of these several triangles, the equilateral quadrangle, both in its parts and as a whole, has a more stable base than the equilateral triangle. Wherefore we are preserving the likely account when we assign this figure to earth, and of the remaining figures the least mobile to water, and the most mobile to fire, and the intermediate figure to air; and further when we assign the smallest body to fire, and the greatest to water, and the intermediate to air; and again the first in point of sharpness to fire, the second to air and the third to water. “
“As regards all these forms, that which has the fewest bases must necessarily be the most mobile, since it is in all ways the sharpest and most acute of all; and it must also be the lightest, since it is composed of the fewest number of parts; and the second comes second in point of these same qualities, and the third. Thus, in accord with the right account and the most probable, that solid which has taken the form of a pyramid shall be the element and seed of fire; the second in order of generation we shall affirm to be air, and the third water.”
There are four elements because, as Timaeus [31] says, “that which has come into existence must needs be of bodily form, visible and tangible; yet without fire nothing could ever become visible, nor tangible without some solidity, nor solid without earth. Hence, in beginning to construct the body of the All, the Divinity was making it of fire and earth. But it is not possible that two things alone should be conjoined without a third; for there must needs be some intermediary bond to connect the two. And the fairest of bonds is that which most perfectly unites into one both itself and the things which it binds together; and to effect this in the fairest manner is the natural property of proportion. For whenever the middle term of any three numbers…is such that the first term is to it as it is to the last term [a:b::b:c] …then the middle term becomes in turn the first and the last,…and the necessary consequence will be that all the terms are interchangeable, and being interchangeable they form a unity. [b2=ac] Now if the body of the All had come into existence as a plane surface, having no depth, one middle term would have sufficed to bind together both itself and its fellow terms; but now it is otherwise: for it behooved it to be of solid shape, and what brings solids into unison is never one middle term alone but always two. “
“Thus it was that in the midst between fire and earth the Divinity set water and air, and having bestowed on them as far as possible a like ratio one towards another – air being to water as fire is to air, and water being to earth as air is to water – was joined together and constructed a heaven visible and tangible. For these reasons and out of these materials, such in kind and four in number, the body of the Cosmos was harmonized by proportion and brought into existence. These conditions secured for it Amity, so that being united in identity with itself it became indissoluble by any agent other than That which had bound it together.”
The fifth solid, the dodecahedron, is mentioned by Timaeus only once [55], “And seeing that there still remained one other compound figure, the fifth, the Divinity used it for the Universe in the decoration thereof.” This last phrase is sometimes translated as “embroidering the constellations” and is possibly a reference to the twelve signs of the zodiac. It is also sometimes assigned to the element of eather, or containment, what Timaeus [52] has called ‘the nurse of becoming’ or space.
Heath [Euclid IV, 10, p 98] notes, concerning the dodecahedron and space, that, “Plutarch has in two passages spoken of the division of the faces of the dodecahedron into triangles, remarking in one place (Quaest. Platon. V. 1) that each of the twelve faces is made up of thirty elementary scalene triangles, so that, taken together, they give 360 such triangles and in another (De defectu oraculorum, c. 33) that the elementary triangle of the dodecahedron must be different from that of the tetrahedron, octahedron and icosahedron. Another writer of the second century, Alcinous, has, in his introduction to Plato (De doctrina Platonis, c. 11), spoken similarly of the 360 elements which are produced when every one of the pentagons is divided into 5 isosceles triangles, and each of the latter into 6 scalene triangles.”
This is shown in figure 10c. The arithmetic here is, 6 scalene triangles make 1 isosceles triangle; 5 isosceles triangles make one pentagonal face of the dodecahedron, 6 x 5 = 30; 12 pentagonal faces have 12 x 30 = 360 scalene triangles. The suggestion here is that the sun moves approximately 1 degree per day along the path of the zodiac. We may also see a progression here, from the threeness of the initial triangular components of the plane figures, to the fourness of the four elements, to the fiveness of the all containing dodecahedron.
Euclid and the solids
Euclid, in Book XIII of the ‘Elements’, propositions 13, 14, 15, 16, and 17, constructs the tetrahedron, octahedron, cube, icosahedron and dodecahedron respectively; that is, starting with the tetrahedron, the parent solid having three part symmetry, he goes first down the four part symmetry branch of the family with the octahedron and its dual, the cube, and then down the five part symmetry branch with the icosahedron and its dual, the dodecahedron. Euclid constructs each figure separately and then shows us their relationships in proposition 18, . He has been careful in each construction to point out the ratio of the edge of the figure to the diameter of its circumscribing sphere. Now Euclid starts with the diameter of a sphere and obtains the edge of each figure. The diameter of the sphere is treated like an octaval distance. The half point is used to obtain the side of the octahedron, figure 11a; the two thirds point to obtain sides of both the tetrahedron and the cube, figure 11b. The side of the dodecahedron is in phi relation to the side of the cube, figure 11c. The side of the icosahedron is found by constructing a triangle who’s base and height are in octaval (1:2) relation, with the base equal to half of the sphere diameter. A second triangle is constructed whose altitude is at the Ø/2 (or .809) point of the sphere diameter. One side of this the icosahedron’s side, as shown in figure 11d.
Cosmologies of the solids.
As we have seen, any one of the five regular solids can be generated from any other of the four, in any order, because all of their governing numbers are proportionally related. To illustrate this point we present two cosmological myths using the solids. The first follows the Platonic cosmology described by Timaeus above in a sequence of geometric constructions performed by Critchlow at his ‘KAIROS School of Sacred Architecture’, held in Crestone, Colorado, in 1987, and attended by your author. The second, is based on Robert Lawlor’s study of Indian and Chinese religion and is presented in his book, Sacred Geometry [SG, p 98].
Critchlow’s Platonic sequence begins, figure 12, with a circle who’s circumference is divided into ten equally spaced points, figure 12a. From each point, project lines to the five points symmetrically opposite. In figure 13a this is from point 1 to points 4, 5, 6, 7, and 8. This creates the outer decagon and the inner pentagon. Connecting them gives a face on view of a dodecahedron. A tetrahedron can be placed inside the dodecahedron by connecting every sixth vertex of the dodecahedron, figure 12b. The octahedron is placed within the tetrahedron by positioning its vertices at the midpoints of the sides of the tetrahedron, figure12b. The icosahedron is placed within the octahedron, figure 12c, by marking the points of intersection of lines from the center of the construction to points 2, 4, 6, 8, and 10 and the front faces of the octahedron. These are five inside vertices of the icosahedron. Using the distance from these points to the center as a radius, draw a circle. Where the circle crosses lines from outer points 1, 3, 5, 7, and 9, are the outer five visible vertices of the icosahedron. The center provides the eleventh vertex. The twelfth is directly behind the center. Lastly, the cube is placed within the icosahedron, figure 12d. Four of its vertices lie on the vertices of the icosahedron, opposite outer points 2, 4, 7, & 9. The other two visible vertices of the cube lie where its top and bottom plane cross lines from outer points 1 and 5.
The ten points are reminiscent of the tetractys. The tetractys is, in turn made up of four levels, or worlds. Some suggested correspondences are shown below:
Also note that the ten points in Critchlow’s construction, are arranged in a two dimensional circle but give rise to the three dimensional forms, symbolically establishing the primacy of the two dimensional realm over the three dimensional. The circle itself leads us from the two dimensional shape of its circumference, back along its one dimensional radius line, to the non-dimensional, paradoxical centre.
Lawlor [SG, p 98], provides a different arrangement of the solids, which he reports is based on Hindu symbolism, though he does not provide specific references. It begins, figure 13a, with an outer circle in which a hexagon is inscribed, and an inner circle who’s diameter is .618 (1/Ø) of the larger, also inscribed with a hexagon. These points, 1-12, mark the vertices of an icosahedron, figure 13b. By connecting the twelve controlling points as shown a dodecahedron is placed within the icosahedron, figure 13c. A cube is then placed within the dodecahedron by connecting the diagonals of the pentagonal faces, figure 13d. Two interpenetrating tetrahedrons are next placed so their edges are diagonals of the cube’s faces. The octahedron is placed within the tetrahedrons, its vertices at the midpoints of the tetrahedrons edges, figure 13d. Lastly, another icosahedron is placed within the octahedron, figure 13e.
In describing the symbolism of this construction Lawlor writes [SG, p 96], “…the Hindu tradition associates the icosahedron with the Purusha. Purusha is the seed image of Brahma, the supreme creator himself, and as such this image is the map or plan of the universe. The Purusha is analogous to the Cosmic Man, the Anthropocosm of the western esoteric tradition. … The Hindus envisioned the Purusha as unmanifest and untouched by creation just as in the drawing the icosahedron is untouched by the other forms. The dodecahedron was then seen to be Prakriti, the feminine power of creation and manifestation, the Universal Mother, the quintessence of the natural universe. This dodecahedron touches all the forms of creation within her silent, observing partner. The interlocking tetrahedron is then seen as the yin and the yang, for the tetrahedron is a volume of threeness and is therefore a primary symbol of a function accompanied by its reciprical. The result of this harmonic interaction of opposites gives the cube, symbolic of material existence, the four states of matter, earth, air, fire, water. Both the cube and the interlocking tetrahedron touch the dodecahedron. At the heart of this tetrahedron is the octahedron and as the cube is a formation of its extremities, the octahedron symbolizes the crystallization, the static perfection of matter. It is the diamond, the heart of the cosmic solid, the transformed and clarified lens of light, the double pyramid. The outer progression, extending into vaster and vaster realms, demarcates the same progression, the same genesis: icosahedron, the Purusha, generating the dodecahedron, the Prakriti, and within Prakriti the full play of manifested existence. The whole coagulation is begun by the secret seed which contracts the circle, the infinite, undifferentiated spirit, into the icosahedron. The seed is phi, the fire of spirit.
The transcendent principles, the icosahedron and the dodecahedron, Purusha and Prakriti, the primal duality, each have phi proportions. But when we arrive at the level of the natural world of oppositional dualities, yin and yang, and the cube of matter and its crystallization in the octahedron, it is the square root of 2 which is active. The square root of 2 is the way phi acts in nature. And from the octahedron, the purified state of matter, its crystallization into the mineral gem is reborn the icosahedron with its phi dimension, 1/phi2. …Being a square, it represents the Son; and being the side of the inner icosahedron it is the incarnation or exact image of the initial, generating icosahedron, the Father, Purusha, Anthropocosm.”
Lastlywe look at a construction publiched in the 15th century by the astronomer Kepler, figure 14. In this view the mythic god names or symbolic elements are replaced by the material bodies fo the planets. Since Kepler a pillar of te materialist, reductionalist view of the cosmos, the illustration is suggestive that a hidden pythagorean root may have fed the flower of modern science.
Music of the Solids
The number patterns formed by the edges, faces and vertices of the regular solids may be grouped in the chart below:
Each of the numbers in the chart 4, 6, 8, 12, 20, and 30 are related ‘musically’ in the sense that these numbers fall at positions of notes on the Pythagorean, or diatonic octaval scale, which will be derived in the Music chapter. There we see that the octave may be considered as a doubling of linear distance. Starting from any primary interval, 1, the first octave interval or distance is a doubling, from 1 to 2. A second octave is from 2 to 4, A third from 4 to 8, a fourth from 8 to 16, and a fifth from 16 to 32. 1, 2, 4, 8 is also the double sequence of the Lambda figure, and a geometric progression. The placement of the numbers of the solids in an octaval scale is shown on the chart below. Note that the numbers 20 and 30 are rounded off to the nearest whole number.
This method will also work on the other leg of the Lambda, with an interval of 1 to 3 as the first octave, 3 to 6 as the second, then 6 to 12, 12 to 24 and 24 to 48:
In this version, the tetrahedron is in the highest octave, the octahedron and cube are in the second octave, and the icosahedron and dodecahedron are in the lower two octaves. This comparison sets the stage for the next subject of the Quadrivium, Music, or Harmony.
Figure 15 is a sketch of the class exercise to create the solids using ping-pong balls’ colored white, red, yellow and blue.
End of 3-D Geometry Session.