Combinatorics, Set Theory & The Technologies of Information: The Mathematics of the Kabbalah Tree of Life in the Key of Lichfield Cathedral.
“There’s this beautiful dictionary that really teaches you about these very old questions, of how you count the subsets of a set? Something as basic as that. But then, if you really just take that idea first, you end up finding a [higher dimension in geometry].” — Professor Federico Ardila, San Fransisco State University
First, watch this video to familiarise yourself with the mathematical field of polyhedral combinatorics.
Great. So Federico describes the study of possibilities, using combinatorics, looking to find the number of subsets in a set of “n” elements, and finds that the answer is the number of vertices in an n-cube.
You should really then try and read the below article, to set the context, linked here:
n- Cube: Counting the Number of Sequences
{0} = 1, {1} = 2, {3} = 4, {4} = 8, {5} = 16 and so on …
Check out the table of vertices in the link below for more answers to this series:
So, in combinatorics, the number of subsets of a set of n elements is given by the formula 2^n, which also equals the number of vertices of an n-cube. This is because, for each element in the set, you have two options: either include it in the subset or don’t. Therefore, for n elements, you have 2^n possible combinations, which is the number of subsets of the set.
n-Simplex: Counting all distinct ordered lists (or sequences) of length n chosen from a set of size n+1
When it comes to an n-simplex, each simplex has n+1 vertices. This means that the number of vertices of an n-simplex is given by n+1.
Therefore, if you are trying to find a combinatorial concept that is equivalent to the number of vertices of an n-simplex, you would be looking for the set of:
All distinct ordered lists (or sequences) of length n chosen from a set of size n+1
For example, if you have a set of n+1 elements {1, 2, …, n+1}:
- A 0-simplex (a point) has 1 vertex, which corresponds to all sequences of length 0 (i.e., the empty sequence).
- A 1-simplex (a line) has 2 vertices, which corresponds to all sequences of length 1, chosen from a set of size 2.
- A 2-simplex (a triangle) has 3 vertices, which corresponds to all sequences of length 2, chosen from a set of size 3.
- A 3-simplex (a tetrahedron) has 4 vertices, which corresponds to all sequences of length 3, chosen from a set of size 4.
- And so on.
A “sequence of length n” just means a list that contains n items. When I say “distinct sequences,” I mean that we’re considering the order of the items in the list to be important. So, for example, the list {1, 2} is considered different from the list {2, 1} because even though they contain the same items, the items are in a different order.
For example, let’s say you have a set of three objects, which we’ll call A, B, and C. We want to make sequences that are two items long from this set. Here are the different possibilities:
- {A, B}
- {A, C}
- {B, A}
- {B, C}
- {C, A}
- {C, B}
As you can see, there are 6 distinct sequences of length 2 that we can make from a set of 3 items.
A table of the number of vertices in any given n-simplex can be found by clicking the link below:
In a real-world context, consider a simple example, such as choosing a combination for a lock. If the lock uses two numbers from 0 to 2 (i.e., 0, 1, or 2), and numbers cannot be repeated, then the sequences can be considered as possible combinations for the lock. Here, the combinations would be {0, 1}, {0, 2}, {1, 0}, {1, 2}, {2, 0}, and {2, 1}.
Alternatively, think about a race with three runners (A, B, and C). Suppose you want to predict the first two finishers. In that case, the sequences represent the different possibilities: {A, B} means A finishes first and B finishes second, {A, C} means A finishes first and C finishes second, and so on.
This concept also applies in many areas, such as computer science (sequences of commands or operations), music (sequences of notes), genetics (sequences of genes), and many other fields.
When it comes to representing data or certain mathematical problems, simplices and their higher dimensional analogues often come into play in a number of contexts:
Computational Geometry and Topology:
Questions about triangulations, tessellations, convex hulls, Voronoi diagrams, and simplicial complexes often involve n-simplices. These can arise in many fields, such as computer graphics, machine learning, and data analysis.
Machine Learning and Data Science:
In machine learning, particularly in problems where we are trying to find a simple yet effective representation of complex, high-dimensional data. One approach is to try to approximate the data as a cloud of points in high-dimensional space and then use simplicial complexes to create a network of interconnected simplices that captures the overall shape of the data.
Game Theory:
In certain types of games, such as those involving mixed strategies, the set of all possible strategy profiles can be represented as a simplex. Each vertex of the simplex represents a pure strategy, and points inside the simplex represent mixed strategies.
Optimization Problems:
The simplex method is a commonly used algorithm for solving linear programming problems, which are optimization problems with linear objective functions and linear constraints. In this context, the vertices of the simplex represent feasible solutions to the problem.
Therefore, whenever you encounter a problem that requires dealing with subsets, combinations, complex geometric relationships, high-dimensional data approximation, strategy profiles in games, or linear optimization, you might find it helpful to think in terms of simplices and their higher-dimensional analogues.
Octahedron: The Principle of Doubling (Signed Permutations)
In geometry, an n-octahedron is an n-dimensional analogue of an octahedron. In three dimensions, an octahedron has 6 vertices. Higher dimensional analogues of an octahedron, often called cross-polytopes, have 2n vertices, where n is the dimension of the space.
Suppose you are looking for a combinatorial concept that is equivalent to the number of vertices of an n-octahedron. In that case, it’s worth noting that this number (2n) corresponds to the number of signed permutations of a set of n elements.
Let’s clarify what a signed permutation is. A permutation of a set is an arrangement of its elements into a sequence or linear order. A signed permutation allows for each element to be in a positive or negative state, hence the factor of 2.
For example, for the set {1, 2}, there are 2n = 2*2 = 4 permutations: {1, 2}, {-1, 2}, {1, -2}, {-1, -2}. These 4 permutations correspond to the vertices of a 2-octahedron or a square in this case.
This is more of a metaphorical comparison than a direct mathematical correspondence, and signed permutations aren’t as commonly referenced in combinatorics as subsets are. Still, they do provide a way to associate the vertices of an n-octahedron with a combinatorial concept.
Here are some real-world examples to illustrate the idea of signed permutations:
Coding Theory:
Suppose you’re creating a simple error detection scheme for a digital communications system. You decide to use a simple sign-based scheme: you send a positive bit to represent a ‘0’ and a negative bit to represent a ‘1’. In this scenario, the signed permutations of your message bits correspond to the different messages you can send. If your message length is n, then there are 2n possible messages corresponding to the vertices of an n-octahedron.
Genetics:
Consider the genes of an organism. Certain genes can be either activated (positive) or deactivated (negative). If you’re studying n genes, then there are 2n possible states for those genes, corresponding to the vertices of an n-octahedron.
Investment Strategies:
Consider a simple investment strategy that involves n different assets. For each asset, you can either go long (buy, represented as positive) or short (sell, represented as negative). Then, there are 2n different strategies corresponding to the vertices of an n-octahedron.
Physics:
In physics, particularly in quantum mechanics, particles can have properties that exist in two states, often described as spin up (+) or spin down (-). If you’re studying a system with n particles, then there are 2n possible states for those particles, corresponding to the vertices of an n-octahedron.
Machine Learning:
In machine learning, especially in classification problems, we often deal with feature vectors. If we consider simple binary features (either present (+) or not present (-)) for n features, there are 2n possible feature vectors corresponding to the vertices of an n-octahedron.
These are metaphoric comparisons rather than direct mathematical correspondences. The actual real-world situations can be far more complex, but these examples should give you a sense of how the concept might be applied.
Parable Synthesis (The Discovery of Language, Music & Trading)
Once upon a time, in the world of Geometry, lived three wise beings — n-cube, n-simplex, and n-octahedron. They were renowned for their wisdom and their unique ways of interpreting the universe.
One day, the three of them were having a conversation about the world beyond Geometry. They were curious about how their characteristics could be seen in this foreign realm.
“I have heard of a place,” started n-cube, “where beings communicate using something called ‘language’. They have words — the elements, and their combinations form sentences, much like our subsets. For example, if they have n words, they can create 2^n different sentences. Just as I, in the world of Geometry, have 2^n vertices for a set of n elements.”
N-simplex, intrigued, chimed in, “That’s fascinating, n-cube. But let me tell you about the world of music. Musicians pick notes, much like choosing vertices on my body. If a melody is made of n notes, chosen from a set of n+1, they create beautiful sequences, each unique, just like the vertices of my form.”
N-octahedron, not to be outdone, added, “In the world of finance, investors trade assets. They either buy, represented by a positive state, or sell, represented by a negative state. If an investor trades in n assets, there can be 2n strategies. These strategies represent the vertices of my structure, each vertex a unique combination of buys and sells.”
As they reflected on their revelations, they marvelled at the ways their geometric principles manifested in such diverse fields in the world beyond.
“You see, friends,” said n-cube, “we are not confined to our geometric world. Our principles and properties permeate throughout the universe.”
N-simplex added, “Indeed, our vertices and faces, while concrete in our world, can represent abstract concepts in other realms.”
N-octahedron concluded, “Our existence in Geometry is but a reflection of the countless possibilities that exist elsewhere. As we understand more about ourselves, we reveal more about the world at large.”
And so, they continued their conversation, each with a newfound appreciation for their unique geometric structures and how they connected to the wider universe. They understood that, despite their differences, they were all united in their shared ability to illustrate and understand complex principles, both within Geometry and beyond.
Application To The Kabbalah Tree of Life: It Grows From Within As A Reflection of Itself
In Kabbalah, Gematria is a traditional Jewish system of assigning a numerical value to a word or phrase. The ten sephiroth in the Tree of Life, along with their corresponding Gematria values, are as follows:
- Keter (Crown) — 620: Calculated from the Hebrew spelling כתר
- Chokmah (Wisdom) — 73: Calculated from the Hebrew spelling חכמה
- Binah (Understanding) — 67: Calculated from the Hebrew spelling בינה
- Chesed (Mercy) — 72: Calculated from the Hebrew spelling חסד
- Gevurah (Strength/Judgment) — 216: Calculated from the Hebrew spelling גבורה
- Tipharet (Beauty) — 1081: Calculated from the Hebrew spelling תפארת
- Netzach (Eternity) — 148: Calculated from the Hebrew spelling נצח
- Hod (Glory) — 15: Calculated from the Hebrew spelling הוד
- Yesod (Foundation) — 80: Calculated from the Hebrew spelling יסוד
- Malkuth (Kingdom) — 496: Calculated from the Hebrew spelling מלכות
Hidden Coincidences
Let’s explore some numerical overlaps between gematria values of the sephirot and the properties of the n-cubes, n-simplexes, and n-octahedrons:
- Binah (Understanding) has a gematria value of 67. A 6-cube or 6-dimensional hypercube has 64 vertices, which is very close numerically. However, if we separate the numbers 6 and 7 (which is plus 1 from 6), we get the number of vertices in an octahedron with its central point, from which it extends in all 6 dimensions.
- Chesed (Mercy) has a gematria value of 72. A 3-cube (a regular cube) has 12 edges, and a 6-cube has 72 edges.
- Malkuth (Kingdom) has a gematria value of 496. In the realm of mathematics, 496 is a perfect number; that is, it is equal to the sum of its proper divisors. However, this doesn’t directly connect to the properties of n-cubes, n-simplexes, or n-octahedrons. But if we separate the numbers to: 4, 9 and 6, which are equivalent in our model to a 1-cube, 3-octahedron & 8-simplex, respectively.
- Yesod (Foundation) has a gematria value of 80. This doesn’t directly connect to the vertices, edges, or faces of n-cubes, n-simplexes, or n-octahedrons in low dimensions. But a 4-dimensional simplex (or 4-simplex, also known as a 5-cell) has 5 vertices, 10 edges, 10 faces, and 5 cells (5+10+10+5 = 30). If you consider all the elements (vertices, edges, faces, and cells) of two disjoint 4-simplexes, you get 60 elements. Add to these the 20 elements of a third 4-simplex without the cells (5 vertices, 10 edges, and 5 faces), you get 80, matching the gematria value of Yesod.
Of course, these are numerical coincidences rather than mathematical equivalences, and the author suggests that these findings are pure speculation, but that’s where the fun lies.
Both the principles of gematria and the properties of these geometric shapes have their own distinct mathematical and symbolic meanings. Nonetheless, it’s always fascinating to explore the interplay of numbers across different systems!
Artwork Meditation
Lichfield Cathedral
We have explored many aspects of Lichfield Cathedral in respect of its connection to the Kabbalah, more specifically, the Sefer Yetzirah. That said, take a look at these & feel free to explore more!
Dr Nick Stafford
Eye of Heaven — Lichfield Cathedral, a Theory of Everything
