The Crossroads of Heaven & Earth is Time, the Mathematics of Lichfield Cathedral.
Our Sages aroused us with this rule: “The kingdom of earth is the same as the kingdom of heaven.” Rabbi Joseph ben Abraham Gikatilla (1248 — after 1305), Gates of Light
Introduction
“Space, the final frontier.” Leonard Nimoy
In this, the first of a series of articles, we expand on our earlier discovery of the higher dimensional geometry in the design of Lichfield Cathedral. We have explained in this sense that we believe Lichfield Cathedral is a “theory of everything” constructed in stone, one that has a life of its own when viewed this way. We describe a mathematical proof showing the single measurements from which the Cathedral is built are units of ‘infinite space’, which in the modern world we mistake as time.
Here we expand one mathematical proof for this idea. We begin by explaining that the Cathedral is a 225-dimensional hypersphere nested in a hypercube as it would look in 3 dimensions. We will show a Pythagorean proof for this. We examine what the architects may have been trying to tell us by this design and what spiritual messages might be encoded in this. This will help us understand the ‘ideas’ of Heaven, Earth and Time and how they might work together. We will look at more recent mathematical, scientific and technical discoveries in this area that give us practical applications to the ideas in this design.
Lichfield Cathedral
Lichfield Cathedral is a unique and beautiful three-spired mediaeval cathedral in the heart of Great Britain. The current Cathedral is around 800 years old, but other earlier sacred buildings have been on this site. It was consecrated to St Chad of Mercia in 664 AD, who introduced Christianity to much of this part of England.
What are a sphere and a hypersphere?
“There is geometry in the humming of the strings, there is music in the spacing of the spheres.” Pythagoras
A sphere is a geometric shape defined by points exactly the same distance from the same central point. It is an extension of a circle from 2-dimensions, which in turn is an extension of a point in space. A circle symbolises evolution as a process of transformation from death to birth, ending, and beginning. The circle has no beginning and no end; in this sense, it represents eternity. In many spiritual beliefs, a circle represents the Divine life force or Spirit that keeps our reality in motion.
A hypersphere is a sphere that exists in any dimension above 3-dimensions. It is difficult to visualise objects in higher dimensions above the experience of our own 3-dimensions. Despite this, it is common in mathematics to study higher dimensions, and the use of coordinates in numbers makes this easier. If we construct higher dimensional proofs by using a combination of mathematics and geometric visualisations, they often give counterintuitive results. This is especially true when dealing with the hypersphere, as you will see.
Consider first how a sphere in 3D might look to someone living in a 2D world. As you can see in the image, a circle of ever-increasing radius appears in the 2D plane, which reaches a maximum size and then vanishes to a point.
A 4D sphere has the three axial geometries of a 3D sphere (x, y, z) and an additional axis at 90 degrees to all three of these (w). Therefore the surface of a 4D sphere is itself 3-dimensional. If a 4D sphere were to pass through 3 dimensions, we would see a point in space become a small sphere, expand into a larger sphere, shrink back to a point and then vanish again.
What do higher-dimensional hyperspheres look like?
“O King, through the country there are royal roads and roads for the common citizens, but in geometry there is one road for all.” Menaechmus to Alexander the Great when he asked him to teach him geometry concisely.
It becomes difficult to visualise the effects of even higher dimensional hyperspheres. Artists have attempted to represent them, and this renders a rather wirey-looking image, as below.
In other graphical representations, hyperspheres look spikey. This is because as the number of dimensions increases, the edges of the bounding box containing the ball begin to warp inwards, and the corners project more outwards, as is perceived in 3 dimensions.
The Cathedral Crossing is the Crossroads of Heaven, Earth and Time
“From the position of Earth, Heaven is both at the centre and also all around.”
The design of Lichfield Cathedral is centred on the Crossing, the square structure underneath the main central spire. The surrounding structures of the Cathedral project out from this crossing. To the west, there is the Nave. To the east, the Choir and Lady Chapel, and north and south, the Transepts.
A view of the plan of the Cathedral shows that the Crossing is a square divided into four smaller squares. If you visit the Cathedral, you will see that when you stand in the Crossing, the main pillars are divided into four sections, with the arches above making the fifth section. By comparison with the widths of the projecting segments in the Nave, Choir and Aisles, you can fit four circles onto the floor space of the Crossing. Projecting this into three dimensions gives us eight spheres stacked 2 x 2 x 2. Of interest here is that in an earlier article, we discovered the importance of the numbers two, four and eight in the distances of the cathedral from Jerusalem (2480 Roman miles) and Mecca (2480 ancient Arabic miles).
We will now look at how the geometry of these circles as they are packed into this area of the Cathedral and place a smaller packed circle in the centre of them. Let's have a look at the geometry of these five circles.
Symbolically the four circles represent the Earth. The number four had always been symbolic of the cross and the square. Almost from prehistoric times, the number four has represented solid things that can be touched and felt. Four is a symbol of wholeness and universality, a symbol which draws all to itself. Where lines of latitude and longitude intercept, they divide the Earth into four portions.
The smaller circle represents the size of a metaphorical central ‘chamber’ within Earth pushing against its limitations. At the same time, the central circle stabilises the outer four by being in contact with all four. It is an Earth within the Earth.
The unit measurements can be seen on the diagram below. The radius of the larger circles is one. The central circle completes the pattern, and we now need to calculate the radius of the smaller circle. We can use the Pythagorean formula (a² + b² = c²) to calculate the distance between the centres of the two circles, as shown. Here we get the square root of 2, which is approximately 1.414. It is then easy to see that the radius of the smaller circle is the square root of 2 minus one, which gives us 0.414.
Now we need to think about extending this model into three dimensions. When we do this, we get eight stacked spheres, as in the image below. This is the model that the Cathedral architects would have had in mind as the unit structure of the Crossing, from which the whole Cathedral plan extends from. This simple model of eight stacked spheres is richly symbolic and universally spiritual in its meaning.
Universally eight is the number of cosmic balance. It is the number of cardinal points on a compass and pointers on the weather-vane of the Tower of the Winds in Athens. There are eight spokes in the Buddhist Wheel of the Law, eight petals on the Lotus, eight trigrams in the I Ching and eight pillars in the Temple of Heaven. There are eight angels who support the Throne of Heaven, and eight is the number of the Mirror of Amaterasu (a sacred bronze mirror that is part of the Imperial Regalia of Japan).
Spheres are representative of the Earth, as compared to the cube, which is representative of Heaven. This is a basic principle in church design from before Norman times and is seen in the nearby St Chad’s church.
Compare these eight stacked spheres to the below four-dimensional tesseract, a hyperdimensional cube, as it might look in three dimensions. With visual imagination, you can see how these eight spheres can be centred on the outer cube’s vertices. The central smaller stacked sphere would then fit into and touch the vertices of the central cube. As this model is a 4D hypercube, the smaller central cube within which the smaller sphere occupies is actually in one higher dimension than the surrounding eight spheres. Applying spiritual language to this geometric symbolism, we get something like, “From the position of the Earth, Heaven is at the centre and is all around.” This may sound quite mystical, but it is also very literal, though it depends on how we define “Heaven” and “Earth”. In order to understand this, we must consider ourselves as “A mirror within the mirror”.
The escalator down to higher dimensions
Rabbi Akiva said: Who is able to contemplate the seven palaces, and behold the heaven of heavens, and see the chamber of chambers, and say: “I saw the chamber of YH?” Ma’aseh Merkavah, Synopsis, 554
Okay, let’s see how we get from this core position all the way down to 225 dimensions. The progression of this mathematical and geometrical story is now understood by stepping up the dimensions of the tesseract. In three dimensions, considering the two-dimensional geometry lifted into the stacked spheres, we now have a central sphere with a radius of the square root of 3 minus 1. And so, in 4 dimensions, the radius of the 4D hypersphere is the square root of 4 minus 1. In 5 dimensions the hypersphere radius is the square root of 5 minus 1. This gives the general equation for the radius of a hypersphere as below:
This corresponds to the multidimensional expansion of the Pythagorean theorem discovered by Riemann in 1854:
So let’s use this formula to unpack a few other Cathedral design features.
Infinite space is the unit measure building block of Lichfield Cathedral
“Time does not exist, it is only in every other moment that we experience all things.”
Continuing our journey down to higher dimensions in our model of stacked spheres, we find, counterintuitively, that in 4 dimensions, the central sphere is now the same radius as the surrounding eight spheres. This is true even though all spheres are still in contact with each other. This makes no sense in 3 dimensions but is possible in 4 dimensions because the shape of the 4D hypersphere is determined by its bounding 4D cube (tesseract), whose edges and surfaces start to bow inwards to make more space within its corners.
One way to understand this is that in the 4D tesseract, the central cube is an integrated 3D cube (just as a 3D cube is an integrated square). As a 3D cube defines space, then mathematically, the inner 4D cube must have infinite volume even though it sits inside the larger 3D cube.
The God Consciousness
“We must accept that we are not separate beings but rather the same single consciousness perceiving different places across all of time and space.”
Therefore, our model tells us that the unit measure of the Cathedral is 4-dimensional space. In common thought, the fourth dimension is considered to be time. Whilst this can be helpful, it may only be part of what the Cathedral architects were trying to tell us. Another perspective on the 4th dimension is that it is infinite in space. Even higher dimensions are simply this infinite space continually wrapping itself up in tighter knots. One view here is that time itself doesn't exist but is a consequence of these higher dimensional knots moving through infinite space. Everything is about space, nothing else. The catch here is that for this mathematical idea to work, we must accept that we are not separate beings but rather the same single consciousness moving around at different places in time and space.
One way to transform this geometry into language would be to say that combining time and all of space is the equivalent to where Heaven and Earth overlap, or are equivalent, or that time and space are the consequence or interface of Heaven and Earth. Plato said this more poetically, “Time is the moving image of eternity.”
What is it like to walk around in higher dimensions?
“Our Sages aroused us with this rule: “The kingdom of earth is the same as the kingdom of heaven.” Rabbi Joseph Gikatilla, Gates of Light
The main problem with visualising higher dimensions is that we try to do so from the position of being trapped in a lower dimension. Plato described this well in his allegory of the cave. It is hard to see the whole world from inside a prison or from the bottom of a well. The key is to recognise that what we see in our ‘existence’ is just a shadow of our ‘reality’. ‘Reality’ is the thing that makes things move in the prison of our lower-dimensional ‘existence’.
You cannot see the air but feel a breeze on your face. A fish cannot see water but knows when it is not in it. You cannot see gravity, but you can see the autumn leaves start to fall. Gravity and acceleration feel the same because, as Newton showed us, they are equivalent. Later, Einstein showed us that gravity and acceleration are the same from the perspective of a higher dimension, the warping of space. As discussed in an earlier article, we looked at how physicists have approached theories of everything by using the tool of higher dimensions. Kaluza-Klein theory, Superstring theory and Quantum mechanics all use the tool of higher dimensional mathematics to explain the world.
You actually don’t need to visualise the worlds of higher dimensions. You can use mathematics and geometry like a dashboard to show what is going on. In the same way, an aircraft pilot can fly a plane at night over land and sea, using a combination of his instruments and the wisdom of his experience.
The Ninth and Tenth Dimensions
Let's look at this in regard to the ninth and tenth dimensions. Our model tells us that the central sphere has the following radii in these higher dimensions. It tells us that the 9D hypersphere fills the 3D cube and the odd reality that in 10D, the central hypersphere becomes larger than the original bounding box.
One possible way physics views these higher dimensions based on the principle that the 4th dimension is equivalent to time is that in the ninth dimension, we can compare all the possible universe histories, starting with all the different possible laws of physics and initial conditions. The spiritual language of our ninth-dimensional geometry tells us that, at this point everything in Heaven and on Earth is equivalent.
In religious symbolism, the number nine has often been associated with completeness. Christ died at hour nine of the day, to make the way of salvation open to everyone. The Jewish Day of Atonement (Yom Kippur) is considered to be the holiest of the year. It begins at sunset on day 9 of the seventh Hebrew month (Leviticus 23:32). The number 9 represents the fruits of God’s Holy Spirit, which are Faithfulness, Gentleness, Goodness, Joy, Kindness, Long suffering, Love, Peace and Self-control (Galatians 5:22–23). The Baha’i writings say that the number nine is “the number of perfection” and “the highest digit, hence symbolizes comprehensiveness, culminations.”
Higher dimensions all the way down
“Behold, you are fair, my love! Behold, you are fair! You have dove’s eyes behind your veil. Your hair is like a flock of goats, Going down from Mount Gilead.” Song of Solomon 4:1 (NKJV)
The Milky Way Galaxy
It is a wonderful coincidence that the limit of the Lady Chapel mathematically marks a 225-dimensional hypersphere. The galactic year of the Milky Way is, by coincidence, 225 million years. As we discovered in our last article, at midnight on St Chad’s day in 664 AD, the longitudinal axis of the earlier Lady Chapel aligned with the Milky Way Galactic disc as it intersected the Earth’s horizon. Moreover, later that day, Saturn set on the Earth’s horizon bearing 130 degrees, and this lined up with Jerusalem on Earth. At this time, Saturn was close to Sagittarius A*, a point in the sky now known to be the supermassive black hole at the centre of our Milky Way. The vertical angle of the Central Spire is also the combined angles of the Earth’s tilt and the tilt of the Solar System against the galactic disc. The architects must have known all of these facts and built these features into the Cathedral. A Black Hole is also a type of hypersphere, being spherical in “shape” as seen from the outside in 3 dimensions but also having increasing dimensions of time upon its surface (the event horizon). Could this all be a coincidence?
Is the Universe a hypersphere?
One theory of physics suggests that the Universe might be a hypersphere. The Hypergeometrical Universe Theory (HU) proposes the Universe as the 3D hypersurface of a lightspeed expanding hypersphere. HU not only proposes the 3D Universe to be the hypersurface of a lightspeed expanding hypersphere, but it also proposes much more. The resulting picture could guide us in using the information about the space elasticity tensor to create matter and create an easier path to thermonuclear fusion, antimatter production, EM propulsion, Dark Matter mining, and space travel.
The final picture of the Universe may be infinitely simpler than this, everything being made of space. The physicist Louis de Broglie made this connection when he postulated the de Broglie equation (below) that made the link between the wavelength and mass (actually momentum) of an electron. He did this at a time when physics realised that light photons had both particle and wavelength properties. He took Einstein’s equation E=mc² and proposed a formula that gave actual momentum to a given wavelength.
Do higher dimensional hyperspheres create subatomic particles?
As already explained, a hypersphere is a higher-dimensional analogue of a sphere, with the defining property being that the distance from the centre to any point on the surface is always constant. These geometric shapes are now playing a significant role in our understanding of the physics of matter, particularly in the fields of quantum mechanics and particle physics.
One of the most important applications of hyperspheres in the physics of matter is in the study of quantum mechanical systems, which are characterized by the wave-particle duality of matter. In quantum mechanics, the behaviour of particles is described using wave functions, which can be thought of as mathematical representations of the probability of finding a particle at a particular location. These wave functions are often represented using complex numbers, which can be visualized as points in a two-dimensional plane. When these wave functions are plotted in this way, they often form roughly spherical shapes, leading to the use of hyperspheres as a mathematical tool for describing quantum systems.
Hyperspheres have also played a significant role in the study of particle physics, which is the branch of physics that deals with the fundamental building blocks of matter. In particle physics, the behaviour of particles is often described using theories such as quantum field theory, which involves the use of higher-dimensional spaces to describe the interactions between particles. These higher-dimensional spaces can be thought of as abstract mathematical constructs, but they have been found to have a number of important physical consequences. For example, the existence of extra dimensions beyond the three dimensions of space that we are familiar with has been proposed as a possible explanation for the observed properties of certain subatomic particles.
Overall, the use of hyperspheres in the physics of matter has played a crucial role in our understanding of the fundamental nature of the universe. From studying quantum systems to exploring higher-dimensional spaces in particle physics, these geometric shapes have provided a powerful mathematical tool for describing the behaviour of matter at the most fundamental level.
The volume and surface area of the hypersphere
Although the radius of the hypersphere has expanded, its volume and surface areas have become counterintuitively infinitesimally small. It becomes ever more refined in higher dimensions. Could this ever-refined higher dimensional space be the thing that we call ‘Heaven’? What does this mean? Could it be that whilst the canopy of Heaven covers all things, is at the centre of all things and is available to all, it is a difficult and rare thing to find and extremely valuable? Could it also indicate that the influence of our central black hole, at a distance of 25,800 light years, from the Earth is both subtle to our senses but powerful enough to guide our whole solar system in a vast circle?
The Other Platonic Solids
How many regular polytopes are there in successive higher dimensions? Counterintuitively we get the following sequence: 0, 1, infinite, 5, 6, 3, 3, 3, 3, 3 … and so on forever. Remember that a regular polytope is simply a geometric object with flat sides with the highest degree of symmetry of all the polytopes. Because of this symmetry, we have chosen to use them in our geometric calculations. The number of polytopes in any given dimension are shown in a table form below:
These three regular polytopes are:
- Simplices, including the equilateral triangle and the regular tetrahedron.
- Hypercubes or measure polytopes, including the square and the cube.
- Orthoplexes or cross polytopes, including the square and regular octahedron.
In further articles in this series on higher dimensions in the design of Lichfield Cathedral, we will explore what role these additional polytopes have in our model of the Universe as being made up only of space in increasingly higher dimensions.
Dr Nick Stafford
Eye of Heaven. Lichfield Cathedral a Theory of Everything
Unicordia Forest Publishing UK
The ideas expressed in this article are pure speculation, and the author does not claim any truth or originality.
Footnotes
The Penguin Dictionary of Symbols. Jean Chevalier and Alain Gheerbrant 1996.
Beholders of Divine Secrets. Mysticism and Myth in the Hekhalot and Merkavah Literature. Vita Daphna Arbel. State University of New York Press. 2003.
Gates of Light (Sha’are Orah) by Rabbi Joseph Gikatilla. Translated by Avi Weinstein. Harper Collins. 1994.
https://www.biblestudy.org/bibleref/meaning-of-numbers-in-bible/9.html
https://bahaiteachings.org/spiritual-meaning-significance-number-nine/
