https://www.youtube.com/watch?v=Jknrt4PpEAY ]]>

Now, not only is 26 the approximate number of weeks between the two solstices of a year (or, even more remarkably, between the two equinoxes), and the exact number of red cards in a pack of cards (or black ones, if the colour suits you better), it is also the number of letters in the alphabet. This is deeply significant.

Similarly, the number of points on a rhombicuboctahedron is 24, which, besides being twice the number of zodiac signs, is the exact number of letters in the Greek alphabet. And the number of edges on a rhombicuboctahedron is 48, the number of upper and lower case letters in the Greek alphabet. That’s more significant relationships.

All this explains why the rhombicuboctahedron was included in paintings, and why it is shown larger than the dodecahedron. It was to stress the importance of language and writing – more important than geometry and the rest of mathematics. English (and other languages that use the 26 letters) and Greek are the best languages, because their alphabets are encapsulated in the rhombicuboctahedron.

Hang on, you might say, the alphabet hasn’t always had 26 letters, and the Greek alphabet hasn’t always been 24 letters. It’s true, but don’t you think that someone like Leonardo da Vinci, acknowledged by many as a really smart cookie, couldn’t have had a deeper insight into how the alphabets would finally evolve into their perfect forms.

Another interesting feature of the rhombicuboctahedron is the number of triangular and square faces, 8 and 18 respectively. If you take an equilateral triangle positioned point down, and then remove the top edge, it is a V. If you take a square and remove its right edge, you have a good approximation to a C. Those letters, V and C, obviously stand for ‘vowel’ and ‘consonant’. Therefore, the alphabet contains 8 vowels and 18 consonants. The 8 vowels are A, E, I, O, R, U, W and Y. Besides A, E, I, O and U, most references will say that Y can be a vowel, but the rhombicuboctahedron confirms that Y is a vowel, and the same goes for R and W, which some more enlightened researchers have considered to be vowels.

There’s much more that can found in the rhombicuboctahedron. I have discovered a truly marvelous proof, but this comment box is too small to contain it.

]]>When you said the "Platonic Monad," (and the Demiourgos δημιουργός; architect of the Universe), and the "Invisible Sun," I immediately thought about Neo-Platonists and their "Three Suns," not the Gnostics.

Their influence in Geometry and building structure can also be found in Islamic arts.

Neoplatonism and Geometry in Islamic Art:

https://academic.oup.com/arthistory/article-abstract/21/1/135/7278429?redirectedFrom=PDF

]]>“The more I study the dodecahedra, the more I come to feel that the holes and the knobs were ultimately a function of their manufacture: to lighten their weight, decrease their cost, and to securely fix their sides together at the vertices. I believe that the original Greek dodecahedra were solid, as one would expect Platonic Solids to be.”

After reading this, it suddenly struck me that if the dodecahedra were indeed symbolic of the shape of the cosmos and the 12 houses of the zodiac, then what might amount to a more accurate representation of its shape would be a hollow dodecahedron. That is, the word “solid” in “The Platonic Solids” can be a red herring, for though the four-sided cube is associated with solid matter (earth), the octahedron is associated with atmosphere (air) — which we all reside in and under. In a similar way to the eight-sided octahedron, the twelve-sided dodecahedron was construed to represent the outer shape to the mostly empty space that the 12 constellations of the zodiac, the planets, and the earth inhabit “within.” This idea of mostly empty space within a dodecahedron might be why the Roman dodecahedra were deliberately fashioned as hollow. As I mentioned previously, the varying size holes could then account for the varying amounts of light that the sun provides during the 12 months of the year (per its 12 sides, which, as mentioned earlier, via Plato, are symbolic of the 12 constellations of the zodiac that the sun journeys through — and which face “within” towards its center).

And where does that light come from? To speculate a bit more, and to try to think outside the box (pun intended), the origin of the light might be derived from the Greek notion of the “Monad”, “The One,” the “unseen” deity who presides “above” everything — i.e., just above the open/holed surface of the material twelve-sided dodecahedron (the blueprint of the cosmos/everything). That is, the Monad resides at the highest level of what the Greeks and Gnostics call the super celestial sphere, the “pleroma,” (the region of light constituting the “fullness of the Godhead” and the realm of all of the Aeons — the pairs of male/female gods). This pleroma would, thus, constitute the mathematically modeled surface of the dodecahedron itself. As the ultimate source of the light of the pleroma, the “unseen” Monad would then seem to correlate to the varying size holes in the surface of the Roman dodecahedra (from which the varied light it creates enters) and/or to the unseen region that that surface is encased in (which then enters those holes in the surface).

But it’s the mathematical aspect of the Monad and its Aeons that just might help to understand the formation of the 20 points or outer “knobs” that rise above, or “outside,” the 30 lines of the surface/super celestial sphere of the Roman dodecahedra.

But before I discuss the knobs and the correlation of numbers, it’s important to note that the idea of a Monad comes out of Greek Philosophy (via Pythagoras, who was said to have travelled to and studied in Egypt), where

“the Monad is the Supreme Being, divinity or the totality of all things . . . According to Hippolytus, the worldview was inspired by the Pythagoreans, who called the first thing that came into existence the "monad", which begat (bore) the dyad (from the Greek word for two), which begat the numbers, which begat the point, begetting lines or finiteness, etc.” [Wiki]

It should also be noted that Pythagoras, who influenced Plato, is also credited with the invention of the Five Regular Solids, which includes the Dodecahedron. This idea of the Monad then went on to influence the later Christian Gnostics. According to the Gnostic text Apocryphon of John (c. 180):

“The Monad is a monarchy with nothing above it. It is he who exists as God and Father of everything, the invisible One who is above everything, who exists as incorruption, which is in the pure light into which no eye can look.” [Wiki]

Also,

“In many Gnostic systems, God is known as the Monad, the One. God is the high source of the pleroma, the region of light. The various emanations of God are called æons.” [Wiki]

But more importantly, the Gnostic systems appear to follow the mathematics of the Pythagoreans and the Five Solids. According to Marcus, the Gnostic Aeons are “numbers and sounds.” [Wiki]. That is, from the Monad, the “one,” came the Aeons — the dyads, the “two” (male and female pairings of the gods/goddesses, the “sygygies”) of the Ogdoad, the “eight” (which some believe might have its roots in the eight pairs of gods of the Egyptian Ogdoad) [Wiki], of the “fifteen” pairs of 30 male/female divine beings in some of the systems, such as Valentinus’. Thus, there is a 1, 2, 8, 15, 30 sequence of the formation of the Gnostic gods.

This unfolding mathematical sequence of the Gnostic system appears to approximate the unfolding sequence of the Greek “Monad” and the Five Platonic Solids: that is, the Greek solids “came into existence” via the “monad,” the “one,” which then “begat” the “dyad” (two), which then “begat the numbers,” which then “begat the point,” which then begat the “lines.” In the dodecahedron, those lines are 30. Thus, it’s interesting that the unfolding of the “Monad” and numbers that make up the 30 Gnostic Aeons (who are “numbers and sounds”) appear to correlate to the unfolding of the “Monad” and numbers that result in the 30 lines of the dodecahedron (which is fitting, as the 30 Aeons are the Ogdoad, the gods, and thus, would seem to naturally fit as the 30 lines of the one Platonic Solid that is the blueprint or structural model of the cosmos).

Furthermore, this unfolding of the “one” into “two” into “thirty” might then help explain why there are the 20 knobs on the “outside” of the Roman dodecahedra. That is, they appear to symbolize the “points” of the 20 vertices that unfold into the 30 lines that compose its form. Those points have zero dimension and are thus symbolic of “the invisible One who is above everything, who exists as incorruption, which is in the pure light into which no eye can look” (akin to the center point of god as a circle, whose center is everywhere and circumference nowhere). This could further help explain why those points/knobs are not seen on a normal dodecahedron. They are there for symbolic purposes in the same way that the varying size holes of the surface symbolize the varying amounts of light that shine in from the pleroma and Monad.

Lastly, it’s important to re-emphasize Professor Mair’s point about the Greek rock crystal dodecahedron found in a Greek Idaean cave. Though it’s hard to correlate the letters with the constellations of the zodiac [as the Greek numbering system used the letters as numbers, and, thus, the letters could simply be artifacts of the numbers used for the dice. Further, the fact that the sequence of numbers do not go to 12, and the fact that the numbers are separated from each other, further distances those numbers from a logical correlation to the zodiac and the alphazodiac], the strong possibility that it was used for divination/ritualistic purposes is critical, for that helps to establish a sequence of its religious use from Greece to Rome to the Roman territories. The fact that others have also been found in ritualistic or ritualistic-like settings also lends weight to this (i.e., the crystal dodecahedron dice found in a Roman grave in Patrae that appears to be strongly correlated to “divinatory activities”; the Roman dodecahedron use for “divination games” that was found at the Cathedral of St. Pierre with the Latin inscriptions of the 12 houses of the zodiac; the Roman dodecahedron found in a pit on a hilltop; or the Roman dodecahedra found within caches of money, as though they were used as a good luck charm, etc.).

]]>An analysis of astronomical data suggests not only that the Universe is finite, but also that it has a specific, rather rigid topology. If confirmed,

this is a major discovery about the nature of the Universe. What shape is space? On page 593 of this issue, Luminet et al. suggest

that the topology of the Universe may be a ‘Poincaré dodecahedral space’ — as illustrated on this week’s cover. And this is no idle abstraction: Luminet et al. show that this topology, unlike many others, is supported by data from NASA’s Wilkinson Microwave Anisotropy Probe

while the article to which *that* article refers is entitled "Dodecahedral space topology as an explanation for weak wide-angle

temperature correlations in the cosmic microwave background", is by Jean-Pierre Luminet, Jeffrey R. Weeks, Alain Riazuelo,

Roland Lehoucq & Jean-Philippe Uzan and commences

]]>The current ‘standard model’ of cosmology posits an infinite flat universe forever expanding under the pressure of dark energy.

First-year data from theWilkinson Microwave Anisotropy Probe (WMAP) confirm this model to spectacular precision on all but

the largest scales. Temperature correlations across the microwave sky match expectations on angular scales narrower than 608

but, contrary to predictions, vanish on scales wider than 608. Several explanations have been proposed. One natural

approach questions the underlying geometry of space—namely, its curvature and topology. In an infinite flat space, waves from

the Big Bang would fill the universe on all length scales. The observed lack of temperature correlations on scales beyond 608

means that the broadest waves are missing, perhaps because space itself is not big enough to support them. Here we present a

simple geometrical model of a finite space—the Poincaré dodecahedral space—which accounts for WMAP’s observations with

no fine-tuning required. The predicted density is Q0 1, and the model also predicts temperature correlations in matching

circles on the sky.

The article to which David Marjanović was referring is indeed in the issue of *Nature* whose link you supplied.

http://georgehart.com/virtual-polyhedra/vp.html

Which has the nice property that the illustrations can be rotated in 3-space.

]]>Idean cave of Zeus dodecahedron:

https://archiv.ub.uni-heidelberg.de/propylaeumdok/1865/1/Chanotis_Dodecahedron_2006.pdf

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