The definition of God as ‘an infinite sphere, whose centre is everywhere and whose circumference is nowhere‘ has its roots in the Liber XXIV philosophorum, a Latin booklet by an anonymous author, which consists of 24 commented definitions of what God is.Marc Richir, And God became Space
The rhombicosidodecahedral connector node of the Zometool construction set may serve as a fitting center of that “infinite sphere, whose centre is everywhere and whose circumference is nowhere” which Pythagoreans call God, and may be associated with Agni Yoga’s Practice of the Presence, that is, bearing witness to the omnipresence of God.
The icosahedral group is the group of symmetries of the icosahedron and dodecahedron having order 120.https://mathworld.wolfram.com/IcosahedralGroup.html
There are five —and only five— convex, regular polyhedra in three-dimensional Euclidean space. These are:
- the tetrahedron (4 sides, its own conjugate),
- the hexahedron (6 sides, cube) and its conjugate the octahedron (8 sides), and
- the dodecahedron (12 sides) and its conjugate the icosahedron (20 sides).
These archetypal patterns, known as the five Platonic solids, are ubiquitous in Nature, as well as in the pentatonic musical scale and the number of petals in the chakras along the spine:
- root: 4 petals
- sacral: 6 petals
- solar plexus: 10 petals (a tetrahedron inscribed in a cube or joining two convex pentagons)
- heart: 12 petals
- throat: 16 petals (2 orthogonal octagons or a tetrahedron inscribed in a dodecahedron)
The 5 Platonic solids are further differentiated into the 13 Archimedean solids. These are convex polyhedra that have a similar arrangement of regular, convex polygons of two or more different types, many of which are produced by truncation of the Platonic solids.
The truncated icosidodecahedron or “great rhombicosidodecahedron” (30 squares, 20 hexagons, 12 decagons) contains the highest volume of all the Archimedean solids, occupying the largest percentage (90%) of the volume of a sphere in which it is inscribed. Expansion of either a dodecahedron or an icosahedron creates a rhombicosidodecahedron.
A rectified, truncated icosidodecahedron with golden rectangles is used as vertex element of the construction set Zometool.
The Zometool kits for making geodesic domes and other polyhedra use slotted balls as connectors. The balls are “expanded” rhombicosidodecahedra, with the squares replaced by rectangles. The expansion is chosen so that the resulting rectangles are golden rectangles.https://www.wikiwand.com/en/Rhombicosidodecahedron
3-D Projections (shadows) of 4-D Realities
There are six regular, convex polytopes in the fourth dimension. The two-dimensional analogue of a 4-polytope is a polygon, and the three-dimensional analogue is a polyhedron. In 5 or more dimensions, there are only three regular polytopes.
The convex, regular 4-polytopes are the four-dimensional analogues of the Platonic solids. The most familiar 4-polytope is the tesseract or hypercube, the 4D analogue of the cube.
Five of these polytopes correspond to the Platonic solids. In the same way that a 3D object can be flattened (projected) in 2D, a 4D object can be projected (shadowed) into 3D and 2D “shadows” (as in Plato’s Cave).
4-D Space or 4-D Consciousness?
To visualize 4D polytopes we need to imagine the cyclical displacement of 3D cells (polyhedra) joined by their polygons. Any displacement involves the notion of time, a dimension of consciousness.
Space can exist without consciousness. However, time is the outcome of consciousness. There can be no time without an observer.
“With clocks we measure the numerical order of motion in 3D space,” Sorli told Phys.org. “Time is ‘separated’ from space in a sense that time is not a fourth dimension of space. Instead, time as a numerical order of change exists in a 3D space.https://phys.org/news/2012-04-physicists-abolish-fourth-dimension-space.html
The dimensional analogy has been addressed elsewhere. In short, it states that a higher dimension is interpreted as time (displacement) by lower dimensions.
Quotes on 4D Consciousness
Letter on Occult Meditation, p. 75: Coordination between the centres becomes apparent; rotation is intensified, and the centres themselves change in appearance, becoming unfolded, and the rotary movement becomes fourth-dimensional, turning inward upon itself. The centres are then radiating nuclei of light, and the corresponding four lower head centres are equally alive.
A Treatise on Cosmic Fire, p. 161: All these material forms (existent in etheric matter which is the true matter of all forms) are primarily undifferentiated ovoids; they then become actively rotating or manifest latent heat; next they manifest duality or latent and radiatory fire; the expression of these two results in fourth dimensional action or the wheel or rotary form turning upon itself.
Esoteric Astrology: The entire question of the revolving wheel with its dual action and its dual effect upon consciousness (and therefore the whole problem of the three decans and their rulers in each sign of the zodiac) must remain a difficult and abstruse problem until [Page 370] such time as astrologers have developed a four-dimensional consciousness and know the true meaning of the Biblical phrase: the “wheel turning upon itself.” In reality, the wheel does not turn like a wheel in a car either forward or the reverse. It turns every way and both ways simultaneously. This fact is, as yet, an impossible one for the human consciousness to grasp. The complexity involved in the progress through the decans—conditioning also the rulers—is to be found to be based upon this multiple action of the wheel. The wheel, therefore, moves not only clockwise, but both ways at once and also at right angles to itself.
A Treatise on White Magic: Each centre or chakra is composed of three concentric interblending whorls or wheels which in the spiritual man upon the probationary path move slowly in one direction, but gradually quicken their activity as he nears the portal of the Path of Initiation. On initiation, the centre of the chakra (a point of latent fire) is touched, and the rotation becomes intensified, and the activity, fourth dimensional.
It is difficult to express these ideas in words that can be comprehended by the uninitiated, but the effect could be described as a changing from a measured turn to one of a scintillating radiation, a ‘wheel [Page 363] turning upon itself’, as the ancient Scriptures express it. Hence, when by purification, conformity to rule, and an aspiration that brooks no hindrance and that ceases not for pain, the aspirant has caused his centres to pulsate and to rotate, then—and only then—can the Master lead him into the Presence of the Hierophant.
The Initiator then, with full knowledge of the disciple’s ray and of his sub-ray, both egoic and personal, and recognising any karma that still may cling, touches the centre or centres which are in line for vivification, and the hidden fire will then rush up and become focalised. Remember always that in the vivification of a centre there is always a corresponding vitalisation of the analogous head centre, till eventually the seven centres in the body and the seven centres in the head rotate in unison. Remember also that just as the four minor rays pass into the three major rays, so the four minor centres carry on the correspondence and pass into pralaya, finding their focal point in the throat centre. Thus you will have the three centres—head, heart and throat—carrying the inner fire, with the three major head centres vibrating in unison also.
The world viewed from another perspective