A Mathematical Theory for the Existence of Angels and its Practical Application in the Modern World
In this theosophical discussion, the author draws parallels between ‘theories of everything’ in physics, ideas of consciousness, the function of, and mechanisms that angels may use to communicate. Finally, we contemplate some of the modern-day applications found in these ideas.
“And I looked, and, behold, a whirlwind came out of the north, a great cloud, and a fire infolding itself, and a brightness was about it, and out of the midst thereof as the colour of amber, out of the midst of the fire.” Ezekiel 1:4 (KJV)
A theosophical hypothesis of consciousness
“True teaching is not an accumulation of knowledge; it is an awakening of consciousness which goes through successive stages.” Ancient Egyptian proverb
In an earlier article, we touched upon a theory that the ‘hard problem’ of consciousness may come from a mechanism of resonance between networks of brain neurones that form multidimensional networks and the ‘strings’ described in superstring theory. In this sense, consciousness would be a type of ‘energy’ (just like heat) arising from sizes at the quantum level being ‘tuned into’ by the brain. If true, this could also explain why, as some faiths believe, all consciousness is at some point unified. Indeed, the brain may have evolved to reach this level of self-awareness, and in so doing, the universe has become self-aware.
At extremely small sizes, time and space are thought, by the physics of loop quantum gravity (LQG), to be granular, that is to say, they cannot be divided anymore. At quantum levels, reality is ‘wrapped up’ in higher dimensions, as described in superstring theory, and this is either in ten or twenty-six dimensions at unbelievably small scales. The mathematics of LQG doesn’t work in higher dimensions, but rather suggests, that time (or space-time) is a network. Superstring theory and LQG, thought initially to be very different, may be linked, as two sides of the same coin, and attempts have been made to combine them using field theories, something that describes the universe according to values at each point in space-time.
Einstein’s theory of relativity explains the universe's geometry at very large scales, and physics is working step by step to resolve the differences between these frameworks to come up with a ‘theory of everything’. Whatever the answer is to combine these ideas, it does seem that in all their extremes, these theories suggest that space and time do come to an end, or collapse in on themselves, in very small (superstring and LQG) and very large (in black holes) scales. Stephen Hawkin theorised in 1974 that relativistic quantum effects at the event horizon of a black hole cause it to release ‘radiation’ or ‘information’ that would lead it to eventually exhaust itself over many trillions of years. A theosophical meditation on these ideas would suggest that there might therefore be a “circulation of energy/information/light” between these very small and very large natural phenomena. Cycles seem to appear throughout nature, and it could be that this is another on a more grand scale.
Some esoteric ideas posit that everything has a consciousness: animal, vegetable and mineral. Indeed, all moving life forms would fit this idea from the perspective of a human. It may also be true that plants and inanimate objects have some sentience, as do vast structures such as planets, Solar Systems, Galaxies, groups of Galaxies and perhaps the whole universe. The nature of this consciousness may be shaped by the complexity and structure of the object. Cultures have recognised this possibility all the way back to the ancient Egyptians and their Neters, and the earlier civilisations from which they evolved. Common cultural (some now say religious) beliefs extend this idea to the communication and connection between these different consciousness.
If all things in the universe came from the same place, say the Big Bang, it makes sense to assume they are all connected in space and time. At very small scales, this may be the higher dimensional strings of superstring theory, the granular LQG, and the universe's black holes. In this way, they effectively join everything together like the root system of the universe in their strings and singularities. While these singularities are all separate and discrete from our perspective, they are all actually in the same place. At the time of writing, a new fundamental force appears to be discovered at CERN, and astronomical studies suggest earlier indications of dark matter acting as a skeleton throughout the cosmos. Our speculations will be tested as science progresses, and we can rework them accordingly.
Phillip Pullman is interviewed here about ‘dust’ or ‘Rusakov particles’ from his ‘His Dark Materials’ series of stories. There are parallels in what he says here with what we are about to discuss further.
The mathematics and geometry of angels
“If you search for the laws of harmony, you will find knowledge.” Ancient Egyptian proverb
So, where do angels come into all of this? The author speculates that what we recognise as angels are naturally occurring entities, no doubt with their own consciousness, just like everything else. Angels are believed, in religious frameworks, to have existed from the time of creation, to have assisted God in the creation, design and management of the universe, and since man has been around, to assist God in communicating with humankind. Ezekiel’s vision of the heavens, as described in the chapter of the same name in the Old Testament, will require some study to understand where the author is going next with this discussion.
The author speculates that what Ezekiel saw in his vision was the natural behaviour of higher-dimensional shapes projected into three-dimensional space and showing their movement projected in the 4th dimension of time. Ezekiel describes the human perspective of what angels may look like and their geometric relationship with God. In summary, Ezekiel saw various classes of angels, and the theory here is that they represent particular higher-dimensional shapes: seraphim (tetrahedron), cherubim (octahedron), ophanim (n-spheres), and the throne (hypercube/tesseract). Geometric shapes are also used to describe the Chakras in eastern faiths, and there may also be a link here with these ideas.
Only three regular polytopes (tetrahedron, octahedron, hypercubes) and n-spheres can exist in the 5th dimension and higher dimensions. Let’s have a look at each of these geometries.
“And above the firmament over their heads was the likeness of a throne, in appearance like a sapphire stone; on the likeness of the throne was a likeness with the appearance of a man high above it. Also from the appearance of His waist and upward I saw, as it were, the color of amber with the appearance of fire all around within it; and from the appearance of His waist and downward I saw, as it were, the appearance of fire with brightness all around. Like the appearance of a rainbow in a cloud on a rainy day, so was the appearance of the brightness all around it. This was the appearance of the likeness of the glory of the Lord.” Ezekiel 1:26–28 (NKJV)
The hypercube is unique as a geometric object. As it extends into ever higher dimensions, its volume increases exponentially, such that even a tiny cube will eventually have a volume equivalent to the size of the universe. The author is not skilled in complex mathematics but also understands that the surface area of a hypercube also extends to infinity in the same way. This is similar to the idea behind the TARDIS in the popular BBC TV science fiction series Dr Who, and the Tesseract in the Marvel Studio films. Watching these films is a valuable way to understand the properties of these shapes in how they are used to describe the stories in which they find themselves.
This property would make it the geometric metaphor for the ‘house of God’, as it may act theoretically as the scaffolding of the universe. In this structure, God can be both everywhere and in each point of space-time at all times. This geometric shape may be the ‘seed idea’ behind the ‘kingdom of heaven’ found throughout different cultures and religions.
Above it stood seraphim; each one had six wings: with two he covered his face, with two he covered his feet, and with two he flew. And one cried to another and said: “Holy, holy, holy is the Lord of hosts; The whole earth is full of His glory!” Isiaih 6:2–3 (NKJV)
Some believe Seraphim to be the highest order of angels. They are characterised artistically and described in Ezekiel’s vision as having three pairs of wings. The author suggests that tetrahedrons, known as the simplex series in higher-dimensional geometry, are what are seen projected in 3 dimensions in this vision. As their name suggests, they are made of ‘fire’, which parallels the ancient element ascribed to the tetrahedron in the series of 5 Platonic solids in 3 dimensions.
The volume of ever-higher dimensional tetrahedron vanishes quickly in on itself, as if into its own black hole, as a reciprocal of the factorial (1/n!) of its n-dimension. The author is not skilled in complex mathematics but suggests that the surface area to volume ratio of ever higher-dimensional simplices increases to ever higher values as the volume simultaneously vanishes. This surface area gives it the property of containing information and having the ability to carry this information across dimensions, in the same way as Hawkin radiation has the property of drawing information from a black hole.
Tetrahedrons have other interesting properties. When stacked face to face, they form a Boerdijk–Coxeter helix, a shape consisting of 3 intertwined helices, like DNA and the symmetry we described earlier in the Baldachin at the centre of the St Peter’s Basilica in the Vatican. Unlike any other stacking of Platonic solids, the Boerdijk–Coxeter helix is not rotationally repetitive in 3-dimensional space. Even in an infinite string of stacked tetrahedra, no two tetrahedra will have the same orientation because the helical pitch per cell is not a rational fraction of the circle. In 4-dimensional space, this helix repeats in rings of exactly 30 tetrahedral cells that tessellate the 3-sphere surface of the 600-cell tetrahedron, one of the six regular convex polychora (we discuss the 4th dimension in greater detail elsewhere, but a visit to the 600-cell Wikipedia page has animations of this shape that give an idea of what Ezekiel may have seen).
“Each one had four faces: the first face was the face of a cherub, the second face the face of a man, the third the face of a lion, and the fourth the face of an eagle.” Ezekiel 10:14
Some angelologists believe Cherubim to be the next most senior angel after seraphim. They are characterised by four-faced geometry, with four faces, four pairs of wings and legs that do not ‘steer’ as they move around. The author suggests that what Ezekiel sees is the octahedron series as projected into three-dimensional space and time.
The volume of an ever-higher dimensional octahedron quickly becomes zero, as if into its own black hole. The author is not skilled in complex mathematics but suggests that the surface area to volume ratio of ever-higher dimensional octahedron increases to ever higher values as the volume vanishes. This surface area gives it the property, or ability, of containing information carrying this information across dimensions, like the simplices.
“Now as I looked at the living creatures, behold, a wheel was on the earth beside each living creature with its four faces. The appearance of the wheels and their workings was like the color of beryl, and all four had the same likeness. The appearance of their workings was, as it were, a wheel in the middle of a wheel. When they moved, they went toward any one of four directions; they did not turn aside when they went. As for their rims, they were so high they were awesome; and their rims were full of eyes, all around the four of them.” Ezekiel 1:15–18
In Ezekiel's vision, the ‘lowest’ order of angels is the ophanim, described as ‘wheels within wheels’. The author suggests these are higher dimensional spheres (or n-balls). As the ophanim are the lowest order of angels, they would ‘presumably’ be tasked with communicating with humankind, perhaps in some combination of geometries with the other higher dimensional shapes. The geometry of spheres in higher dimensional space could indicate this mechanism. As the graph below shows, the volume of an n-sphere is maximum in the 5th dimension, and its surface area is maximum in the 7th dimension. In addition, the ratio of surface area to volume becomes briefly negative in only the 6th dimension.
In the Kabbalah, the Tree of Life, described by the patriarch Abraham, is a five-dimensional construct. Its five dimensions are 3 of space, 1 of time and 1 of spirituality. Therefore, these ‘fill’ the universe, which is expressed in the mathematics of the volume of the five-dimensional n-sphere. The Tree of Life's lower 6 Sephirot (divine emanations) can be metaphorically described as the ‘emotions of the heart’ or consciousness. There is a hidden virtual Sephirot called ‘Daat’ (Knowledge). The author suggests a parallel with the geometry of the surface area of the 7n-sphere. The surface of a sphere would be that part which would contain any information provided by a ‘higher voice’, like the waves on the surface of a globe of water. In this parallel, the 5th dimension provides the ‘breath’ and the 7th dimension the ‘voice’. The visibility of angels to humankind would therefore be in the 6th dimension, acting as a ladder with a reversed polarity allowing the flow of information from the ‘heavens’ to the ‘earth’.
The 9th Sephirah of Yesod and the hidden virtual ‘Sephirah’ of Daat may conceal the idea that the authentic experience of the divine is achieved through the love between two people as they combine, in one, their mutual male and female aspects, and in so doing share in the divine experience. This is the ‘Bridge Over Troubled Water’. More on this in later articles.
The author describes elsewhere the 6-dimensional geometry that he sees in the design of the Gothic Lichfield Cathedral in the UK, something he hypothesises is universal in the sacred architecture of most faiths.
The ideas expressed in this article might seem unbelievable, and the author accepts that completely. Still, it is no less odd than believing in the existence of angels without seeking any proof beyond that.
However, there are also broad practical applications in using these higher dimensional shapes.
Modern-day uses of simplices (tetrahedron) include their mathematics in industrial statistics arising in problem formulation and algorithmic solutions, such as calculating the correct mixtures of yeast, flour, water and sugar in the large scale manufacture of bread.
In operations research, linear programming problems can be solved using simplices. Simplices are used in the geometric design of computer graphics, allowing for the animation of very complex shapes in computer games.
In chemistry, the hydrides of most elements in the p-block of the periodic table resemble simplices, which have applications in chemical engineering.
The author discusses the application of this mathematics and geometry in the underlying principle of information management, data science, machine learning and artificial intelligence in future articles.
Simplices have many interesting mathematical and geometric features. One particularly interesting feature is a symmetry in the number of their edges, faces and cells regardless of what dimension they are described in. For example, an 8-simplex has thirty-six 1-faces and thirty-six 6-faces, and this property is also seen when comparing its 2-faces and 5-faces, and 3-faces and 4-faces. This mimics the symbolism of the Jewish menorah, which shows reflections of its lateral lamps around its central light.
Meditating on geometric shapes can lead to interesting ‘discoveries’. The geometry of the other forms described also has wide practical usage, and the author would encourage the reader to explore these for themselves.
Dr Nick Stafford
Excerpt from “Eye of Heaven. Lichfield Cathedral a Theory of Everything”, published by Unicordia Forest Publishing UK.
The ideas expressed in this article are pure speculation, and the author does not claim any truth or originality in it.
Hierarchies of angels differ in different religions, within the same faith and between various authorities on this topic. There would appear to be no universally agreed system in this area. The author approximately aligns with the De Coelesti Hierarchia that Thomas Aquinas follows in the Summa Theologica for this article.