A Mathematical Theory for the Existence of Angels, Part 2. A Cosmology of the Platonic Solids & Lichfield Cathedral.
“Both before and after the Beginning, there was another Beginning, and before that, another Beginning … all the way down. This is the story of your soul, and there will be no end, because there is no end.”
Introduction
“I sink beneath the surface of the Sun falling to the very centre and, like a new stone, I find myself in all things.”
In this article, we further develop our mathematical theory for the existence of angels, as begun in our earlier article. The link to this article is here below, and it may be helpful, though not essential, to study this first.
Axioms
Based on earlier articles, we have the following axioms:
- Lichfield Cathedral is designed on the principles of Kabbalah as laid down in the oldest extant Jewish mystical text, the Sefer Yetzirah, written by the patriarch Abraham.
- Angels are naturally occurring entities that form part of the structure of the Universe.
- Angels are infinitely small, likely smaller than or equivalent to sub-atomic particles, perhaps similar to the size of the strings of superstring theory, and like them, are present throughout the Universe.
- The angels described in the structure of heaven by Ezekiel chapter 1 in the Torah/Old Testament are higher dimensional Platonic solids (tetrahedron=seraphim, octahedron=cherubim, cube=throne, sphere=orphanim). In addition, the vision of the “likeness with a human appearance” (Ezekiel 1:26–27) is his own “higher / divine self”, equivalent to the principle of the Tzelem as described in Genesis 1:26–27.
- Their function is to carry information between and across all of nature (including human consciousness) and the Planck Scale around 1.616 x 10^-35 m (the basic Planck units are length, mass, temperature, time and charge, derived from five universal physical constants: the speed of light, the gravitational constant, the reduced Planck constant, the Boltzmann constant, and the elementary charge).
- Based on the similar concepts of loop quantum gravity, which suggests that spacetime is made up of discrete, indivisible units or “atoms” of space, often called “spin networks” or “quantum loops”, we explore the possibility that the Platonic Solids operate across all dimensions simultaneously, holding and transferring information.
- There are correspondences between the principles we explore here, considering Platonic Solids acting as angels and the medium they work within and the various five universal physical constants found in the Planck Scale.
- The complex biology of life demonstrates an attempt by evolution to replicate the Infinite dimensions of God within humans (and all life). We are, therefore, infinitely dimensional beings tethered together by one Single Soul.
Platonic Solids
Plato’s cosmology views the universe as an expression of mathematical perfection and order. Plato’s description of the Platonic solids is found in his work “Timaeus”. According to Plato, the Platonic Solids are the ‘root’ or fundamental building blocks of the physical universe. They exist as infinitely small, infinitely present harmonic shapes that form the structure of the Universe, transmit the Plan of the Universe and form the perfect harmonies that make up our souls. They are, like our analysis of the architecture of Lichfield Cathedral, a “theory of everything”.
The regular solids, later known as the Platonic solids, are regular polyhedra, or three-dimensional shapes, whose faces are identical regular polygons (the tetrahedron, hexahedron or cube, octahedron, dodecahedron, and icosahedron) were, to Plato, the embodiment of this perfection. In this article, we also review the hyperdimensional sphere.
He assigned each of these solids to a classical element: fire, earth, air, the heavens, and water, respectively. The Platonic solids can be seen as an early philosophical attempt to find a unifying theory of everything, combining mathematics, natural philosophy, and metaphysics.
- Tetrahedron (Fire): “The tetrahedron is the simplest form of spatial enclosure, and it represents the manifest light of creation.” — Buckminster Fuller
- Cube (Earth): “In the cube we find an image of creation, stability, and integrity. Its six faces offer us the directions of sacred space.” — Unknown
- Octahedron (Air): “The octahedron, with its eight faces, speaks to us of the organizing principle of the mind, the mediator between heaven and earth.” — Unknown
- Dodecahedron (Aether/Universe): “Twelve, the number of faces on a dodecahedron, is the symbol of cosmic order, the perfect number that represents the universe.” — Unknown
- Icosahedron (Water): “The twenty faces of the icosahedron echo the fluidity and transformative power of water, and the many paths we might take in life.” — Unknown
However, it is essential to note that the Platonic solids were more than representations of the classical elements to Plato. They symbolized the perfection and harmony that he believed underlies the cosmos. Plato considered the universe to be an ordered whole, with the perfection of the Platonic solids reflecting this cosmic order.
Plato’s approach to cosmology and physics was deeply philosophical. His descriptions are not identical to the modern scientific understanding of physical matter and the elements. However, they allow us to see how ancient philosophers tried to make sense of the world, seeing connections between geometry, nature, and philosophy. In this way, these remain highly relevant today.
The below Platonic Solids below exist in all dimensions and the sphere:
Lichfield Cathedral
In earlier articles, we used the design of Lichfield Cathedral to help us:
Lay out a mathematical argument synthesising the geometry of Lichfield Cathedral with the theory of music to find the symbolism of the Ark of the Covenant in the Old Testament and also its contents. We then looked at how this model could be applied to the “structure” of all the major religions of the world.
One conclusion from this article was a superimposition of the Platonic Solids on the floor plan of Lichfield Cathedral as such.
We argued that human constructs of morality were a natural consequence of the structure of consciousness and so essentially universal and the same across all cultures and religions throughout time, but with a particular focus for the times they were established.
We considered that the mystical experience of the presence of God that we find in prayer and meditation might actually be the “local consciousness” of the Milky Way Galaxy and found clues in the design of Lichfield Cathedral that suggested its architects shared this view.
Let’s delve deeper into the mathematics of the Platonic Solids to explore these ideas further.
The Philosophical or Divine Attributes of the Platonic Solids
Tetrahedron
- “For our God is a consuming fire.” — Hebrews 12:29
- “The fire shall ever be burning upon the altar; it shall never go out.” — Leviticus 6:13
- “For the LORD your God is a consuming fire, a jealous God.” — Deuteronomy 4:24
Hypothesis: The tetrahedra, as angels, have the function of transmitting information from Unity to Nature and then back again. It is like the song of the birds.
The tetrahedron, the simplest of the solids, is composed of four equilateral triangles. It has the largest surface area for its volume. This gives it the property of being able to convey the most information from the Unity from which it was formed. This is why we assigned it to the Seraphim in our “Mathematical Theory of for the Existence of Angels”.
Plato assigned the tetrahedron to represent the element of fire. This association is thought to symbolize the piercing nature of fire, as the tetrahedron, with its pointed vertices and edges, is considered the “sharpest” of the Platonic solids.
Plato’s approach to explaining the natural world is, in essence, geometric. He proposes that the physical world was made up of these tiny, invisible shapes. In this way, fire consists of many tiny tetrahedra, a piece of earth is composed of many tiny cubes, and so on. Plato’s universe is an orderly one, where the nature of the physical world is eflected the mathematical perfection of the geometric forms.
He believed that the underlying order and harmony of the cosmos were symbolised in these geometric shapes. Fire’s qualities of heat and light, as well as its tendency to rise or ascend, were reflected in the form of the tetrahedron. This is also why we place it in the Quire of Lichfield Cathedral. So, in a sense, the tetrahedron is fire’s essence, and its physical properties were the expression of this essence in the perceptible world.
Octahedron
- “The wind blows wherever it pleases. You hear its sound, but you cannot tell where it comes from or where it is going.” — John 3:8
- “He unleashes his lightning beneath the whole heaven and sends it to the ends of the earth.” — Job 37:3
- “He caused the east wind to blow in the heavens and by his power he directed the south wind.” — Psalm 78:26
Hypothesis: The octahedra provide the frequency channels for Unity to broadcast to the mind. It creates a musical unison, a secret chord, a mediator between Heaven and Earth.
The octahedron is a regular polyhedron that consists of eight equilateral triangles. Given its smooth and slightly rounded shape, Plato considered it less sharp and penetrating than the tetrahedron. It represents air, which Plato deems to be less “sharp” or “penetrating” than fire.
The nature of air is primarily its capacity for optimising space. In our Ark of the Covenant model of Lichfield Cathedral we ascribed it to the Transepts, which were also found to represent unison in music. Its central point reaches out through the infinite space created by its dual shape, the cube, so that Unity has a channel through which the flow of information can be directed.
Cube
- “The LORD is the everlasting God, the Creator of the ends of the earth.” — Isaiah 40:28
- “The world is firmly established; it cannot be moved.” — Psalm 93:1
- “He set the earth on its foundations; it can never be moved.” — Psalm 104:5
Hypothesis: “The cube has the function of creating infinite space across which the tetrahedra communicate. In this way the mind can evolve through this space as music progresses through octaves.”
The cube, or hexahedron, has six square faces. For Plato, it represented the element of earth. The cube, being the most stable and solid among the Platonic solids, reflected the properties of the earth, which is solid, stable, and enduring. This stability might also symbolize the foundational role earth plays in the physical world, providing a base for the rest of the elements.
Icosahedron
- “He draws up the drops of water, which distil as rain to the streams.” — Job 36:27
- “The sea is his, for he made it, and his hands formed the dry land.” — Psalm 95:5
- “But let justice roll on like a river, righteousness like a never-failing stream!” — Amos 5:24
Hypothesis: “The icosahedra is like a Divine Chamber that we seek as we follow the many paths we might take in life, which echo the fluidity and transformative power of water.”
The icosahedron, with its twenty equilateral triangles, was associated with the element of water by Plato. Given its many faces and rounded shape, it represents the fluidity and adaptability of water, which takes the shape of any container. The icosahedron’s multitude of triangular faces, flowing smoothly into each other, mirrors water’s capacity to flow and change form.
Dodecahedron
- “The heavens declare the glory of God; the skies proclaim the work of his hands.” — Psalm 19:1
- “Lift up your eyes and look to the heavens: Who created all these? He who brings out the starry host one by one and calls forth each of them by name.” — Isaiah 40:26
Hypothesis: “The Dodecahedron is the heavenly foundation of the chamber of the icosahedron. It is the Universe and Cosmic Cathedral.”
The dodecahedron is a twelve-faced solid where each face is a regular pentagon. Plato assigned this solid to represent aether (also spelt “ether”), or the cosmos. The dodecahedron doesn’t correspond to a tangible classical element like the previous four, but rather the substance making up the heavens and the constellations. Its complex shape symbolizes the intricate and harmonious order of the universe.
N-Sphere
- “Hear, O Israel: The LORD our God, the LORD is one.” — Deuteronomy 6:4
- “I am the Alpha and the Omega,” says the Lord God, “who is, and who was, and who is to come, the Almighty.” — Revelation 1:8
- “Have we not all one Father? Has not one God created us?” — Malachi 2:10
Hypothesis: “The n-sphere is at first the lungs and voice that the Platonic Solids use to give life and movement to their functions and second the eyes through which they see everything.”
Surface Area to Volume Ratios in 3-Dimensions
The surface area (called skin or boundary in higher dimensions) to volume (called content in higher dimensions) ratios for Platonic solids decreases as the number of faces increases. And conversely, the volume-to-surface area ratio increases as the number of faces increases. Here they are listed in order from highest to lowest ratio, which also means from smallest to a largest volume given a constant edge length:
- Tetrahedron (4 faces)
- Cube (Hexahedron, 6 faces)
- Octahedron (8 faces)
- Dodecahedron (12 faces)
- Icosahedron (20 faces)
- Sphere (infinite faces)
Properties of the Tetrahedron (n-Simplex) in Higher Dimensions
“Skin” Area to Volume Ratio
Since we hypothesise the tetrahedron’s function as an angel is to convey information, we will first look at its surface area to volume ratio as this is that part of its geometry that would convey such information.
Surface area to volume ratio is a very important concept in mathematics, and it has different implications depending on the context, especially when we’re considering different dimensions. In 3 dimensions, the skin-to-volume ratio of the tetrahedron decreases as the size of the shape increases. This is true of all 3-dimensional shapes. Conversely, the skin-to-volume ratio of the tetrahedron increases as it becomes ever smaller.
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a tetrahedron to arbitrary dimensions. The simplex is so named because it represents the simplest possible polytope in any given dimension. For example,
- a 0-dimensional simplex is a point and has no surface area or volume
- a 1-dimensional simplex is a line segment and has no surface area or volume
- a 2-dimensional simplex is a triangle and has a surface area but no volume
- a 3-dimensional simplex is a tetrahedron and is the first simplex with both surface area and volume
- a 4-dimensional simplex is a 5-cell from where we start calling surface area “skin” or “boundary” and volume “content”
In three-dimensional space, the surface area to volume ratio of a tetrahedron is the highest among the Platonic solids. This is due to the fact that a tetrahedron, with its few faces and sharp angles, has less volume relative to its surface area compared to other shapes.
When we move to higher dimensions, we change how we think about “volume” and “surface area”. The 3D equivalents in higher dimensions are often referred to as “content” and “skin / boundary”. The precise definitions get a bit more complicated due to the increased complexity of higher-dimensional spaces, but we can still apply the intuition of surface area to volume ratios.
In four-dimensional space, for example, the simplest analogue of the tetrahedron is the “5-cell” or “pentachoron”. The 5-cell is a four-dimensional polytope (the four-dimensional analogue of a polyhedron) consisting of five tetrahedra.
As you go to higher dimensions, the boundary-to-content ratio of a simplex will tend to increase. However, the mathematical descriptions and comparisons in dimensions higher than three get progressively more abstract and are less directly related to our intuitive understanding of these ratios. Another interesting property of the n-simplex is that in all dimensions, the number of its vertices (0-faces), edges (1-faces), faces (2-faces) and cells (3-faces) show a symmetrical pattern. In this way, it has a kind of geometric symmetry.
Mathematics of the Skin-to-Content Ratio n-Simplex in All Dimensions
In any dimension, the skin-to-content ratio of an n-simplex can be computed by taking the ratio of the (n-1)-dimensional content (boundary) and the n-dimensional content (volume). These formulas are not straightforward and require a solid understanding of higher-dimensional geometry. Let’s consider a unit regular n-simplex, with all edge lengths equal to 1. The formula for the content of an n-simplex is given by:
V_n = sqrt(n+1) / [n! * sqrt(2^n)]
The skin of an n-simplex is composed of n+1 (n-1)-simplices. The content of each (n-1)-simplex (let’s call it V_(n-1)) is given by the same formula, substituting (n-1) for n.
V_(n-1) = sqrt(n) / [(n-1)! * sqrt(2^(n-1))]
To find the skin measure, multiply the content of an (n-1)-simplex by the number of (n-1)-simplices:
B_n = (n+1) * V_(n-1)
You can just about see in the above two graphs that the skin area decreases just a little more slowly as dimensions increase when compared to its volume in the same dimensions.
Now, to get the skin-to-content ratio, divide B_n by V_n:
B_n / V_n = (n+1) * V_(n-1) / V_n
Plugging in the formulas for V_n and V_(n-1), we get a complex formula involving factorials and square roots. These equations and the conceptual understanding of higher-dimensional geometry are advanced mathematical concepts and usually require a background in fields such as topology or abstract algebra to comprehend fully. This is beyond the capacity of the author.
Properties of the Octahedron (Cross-Polytope) in Higher Dimensions
“Skin” Area to Volume Ratio
Just like the tetrahedron inn 3-dimensions, the skin-to-volume ratio of the octahedron decreases as the size of the shape increases. The higher-dimensional analogue of an octahedron is part of a family of shapes called cross-polytopes.
In four dimensions, the cross-polytope equivalent to an octahedron is a 16-cell (also known as a “hexadecachoron”). Like the octahedron in three dimensions, the 16-cell is composed of regular tetrahedra. Moving up in dimensions, the shapes become increasingly complex, but they retain a common feature: they are composed of lower-dimensional “slices” that resemble octahedra.
It is typically the case that the skin-to-content ratio increases as we move up in dimensions for regular cross-polytopes, just as the surface area to volume ratio of the tetrahedron is greater than that of the cube in three dimensions.
Mathematics of the Skin-to-Content Ratio Cross-Polytope (Octahedron) in All Dimensions
As with the tetrahedron, the boundary to content (i.e., skin to volume) ratio of an n-dimensional cross-polytope can be computed by taking the ratio of the (n-1)-dimensional content (boundary) and the n-dimensional content (volume).
In general, for an n-dimensional unit cross-polytope (with all edge lengths equal to 1), the volume V_n is given by:
V_n = 2^n * n! / (2n)!
The boundary of an n-dimensional cross-polytope consists of 2n (n-1)-dimensional cross-polytopes. The volume of each (n-1)-dimensional cross-polytope (let’s denote it as V_(n-1)) is given by:
V_(n-1) = 2^(n-1) * (n-1)! / (2n-2)!
The total boundary measure B_n is then given by multiplying the volume of an (n-1)-dimensional cross-polytope by the number of such cross-polytopes:
B_n = 2n * V_(n-1)
The boundary-to-content ratio is then given by:
B_n / V_n = 2n * V_(n-1) / V_n
Substituting the formulas for V_n and V_(n-1) and simplifying, we can find the skin-to-content ratio for an n-dimensional cross-polytope.
Properties of the Cube (n-Cube / Hexahedron) in Higher Dimensions
Skin-to-Volume Ratio
The cube, or hexahedron, is a three-dimensional object, and its surface area to volume ratio is a well-defined concept in this dimension. In three dimensions, the cube’s surface area to volume ratio decreases as the cube gets larger, as is true for all three-dimensional objects.
The cube’s equivalent in higher dimensions is known as a hypercube, hexahedron or n-cube. A 4-cube, or tesseract, is the four-dimensional analogue of the cube. In five dimensions, it would be a 5-cube, and so on. As with other regular shapes, the skin-to-content ratio of an n-cube tends to increase as we move to higher dimensions.
Mathematics of the Skin-to-Content Ratio of the Cube in All Dimensions
The boundary to content (i.e., skin to volume) ratio of an n-cube can be computed by taking the ratio of the (n-1)-dimensional content (boundary) and the n-dimensional content (volume). As before, these formulas are nontrivial and require a good understanding of higher-dimensional geometry.
For an n-dimensional unit cube (with all edge lengths equal to 1), the content V_n is straightforward: it’s just 1^n, or simply 1, because the “volume” of a unit cube in any dimension is always 1.
The boundary of an n-cube consists of 2n (n-1)-dimensional cubes (i.e., each face of the cube is another cube, one dimension lower). The content of each (n-1)-dimensional cube (let’s denote it as V_(n-1)) is also 1. And so the volume of any n-cube of any dimension is unity. This is why it is “grounded”, “stable”, and immoveable.
The total boundary measure B_n is then given by multiplying the content of an (n-1)-dimensional cube by the number of such cubes:
B_n = 2n * V_(n-1) = 2n
Since the content of the n-cube is 1, the skin-to-content ratio is then just the boundary measure:
B_n / V_n = B_n / 1 = B_n = 2n
So the skin-to-content ratio for an n-dimensional unit cube is just 2n. This interesting finding correlates with our earlier model of the octave when we synthesised music theory with the geometry of Lichfield Cathedral and found the dimensions of the Ark of the Covenant (as per our “Mathematical Theory of Angels” that intuits Ezekiel’s vision of the heavens to the Platonic and other solids).
This means that as the dimensions increase, the skin-to-content ratio also increases linearly. This reflects the fact that in higher dimensions, cubes become “more boundary and less volume”, so to speak.
Properties of the Sphere (n-Sphere) in Higher Dimensions
The sphere, like other geometric objects, has a specific surface area to volume ratio in three dimensions. For a sphere with radius r, the surface area is 4πr² and the volume is 4/3πr³, so the surface area to volume ratio is 3/r. This means that as the sphere gets larger, the surface area to volume ratio decreases.
When we move to higher dimensions, the notion of a “sphere” generalizes to a hypersphere. The “volume” (really, content) and “surface area” (really, skin or boundary measure) of a hypersphere in n-dimensions can be calculated. However, the formulas are more complex and involve the gamma function, a function that generalizes the factorial.
As you go up in dimension, an interesting thing happens: the volume of the hypersphere increases up until the 5th dimension and then begins to decrease, even eventually tending towards zero as the number of dimensions goes to infinity. The surface area of the hypersphere, on the other hand, reaches a maximum at the 7th dimension and then also decreases, tending towards zero as the number of dimensions increases.
This results in the surface area to volume ratio of the hypersphere tending towards zero as the number of dimensions goes to infinity. It’s a counterintuitive result.
Mathematics of the Skin-to-Content Ratio of the Sphere in All Dimensions
The skin-to-content ratio of a hypersphere can be computed by taking the ratio of the (n-1)-dimensional content (skin) and the n-dimensional content (volume). However, these formulas are quite complicated and involve advanced mathematical functions.
Specifically, the formulas for the volume of an n-dimensional hypersphere of radius r involve the gamma function, which is a function that extends the factorial function to real and complex numbers. For a unit hypersphere (r = 1), the formula simplifies a bit.
The content (or volume) V_n of a unit n-dimensional hypersphere is given by:
V_n = pi^(n/2) / Gamma(n/2 + 1)
The skin of an n-dimensional hypersphere is an (n-1)-dimensional hypersphere. The content (or boundary measure) of an (n-1)-dimensional hypersphere (let’s denote it as V_(n-1)) is given by:
V_(n-1) = pi^((n-1)/2) / Gamma((n-1)/2 + 1)
The skin-to-content ratio is then given by:
B_n / V_n = V_(n-1) / V_nTetr
Substituting the formulas for V_n and V_(n-1), we get a complicated ratio involving pi and the gamma function.
Practical Application
“Angels only have one leg and they cannot sit down.”
Physics
The author suggests that since the principles involved in this discussion involve the infinitely small, that correlations may be found in the Planck Scale mentioned at the beginning of the article. By way of a “starter for ten” the author has brainstormed possible correspondences in the crude image below.
Films and TV
If you have access to Netflix and other streaming services and the time to watch these series and film (The OA and Another Earth), then you will gain much understanding from their narratives. They have the same principal actor, Brit Marlin.
The OA
An interview with Brit Marlin.
Another Earth
“There’s another you out there …”
Dr Nick Stafford
Eye of Heaven — Lichfield Cathedral, a Theory of Everything
Sometime too hot, the eye of heaven shines. William Shakespeare, Sonnet 18
Footnotes
Artificial intelligence was used to research and write this article. Wikipedia was also used for text and figures.
The ideas expressed in this article are pure speculation, and the author claims no truth or originality in them.
