Symmetries in The Pattern of Faces in n-Simplex of Any n-Dimension

N-simplex progression from a 0-simplex (point), through a 1-simplex (line), 2-simplex (triangle), 3-simplex (tetrahedron), 4-simplex (pentachoron), all the way up to a 10-simplex.

In an n-dimensional simplex (also called an n-simplex), the number of k-dimensional faces (which we denoted as F_x in the previous article) is given by the binomial coefficient, specifically f_k = “n choose k” where n is the dimension of the simplex and k is the dimension of the face. The binomial coefficient is defined as:

nCk = n! / [k!(n-k)!]

where n! is n factorial, or the product of all positive integers up to n, and k! and (n-k)! are defined similarly.

This formula represents the number of distinct sets of k elements that can be chosen from a set of n elements, which matches the geometric interpretation of selecting k vertices from the n+1 vertices of an n-simplex to form a k-dimensional face.

Interestingly, the binomial coefficients exhibit a symmetry, such that nCk = nC(n-k). This is due to the symmetry of the formula for the binomial coefficient: swapping k and (n-k) in the formula doesn’t change the result.

This means that an n-simplex has the same number of k-dimensional faces as it has (n-k)-dimensional faces.

So, in a 3-simplex (tetrahedron), for example, there are 4 vertices (0-dimensional faces), 6 edges (1-dimensional faces), 4 faces (2-dimensional faces), and 1 3-simplex (the tetrahedron itself). The sequence is symmetrical: 4, 6, 4, 1. See the table below n-dimensions up to 10.

To calculate the total number of faces (including the simplex itself) in an n-dimensional simplex, you can add up the binomial coefficients for k = 0 to n. This is equivalent to calculating the sum of the elements in the (n+1)th row of Pascal’s Triangle, which is 2^n.

So to sum up there is a symmetrical pattern in the number of faces of different dimensions in an n-simplex, due to the symmetry of the binomial coefficients.

However, keep in mind that this symmetry is with respect to the middle of the sequence (for n is odd) or between the two middle elements of the sequence (for n is even), not the ends of the sequence (see below).

Artwork for meditation

Table showing f_k for simplex of n-dimensions

The Binomial Coefficient in Simple Terms

The binomial coefficient is a mathematical term that describes the number of ways you can choose a certain number of items from a larger set, without worrying about the order in which you choose them.

So, let’s say you have a group of friends and you’re choosing a team for a game, or you’re choosing a committee from a larger group. The binomial coefficient gives you the number of different teams or committees you could form.

The formula for a binomial coefficient is written as “n choose k”, and is often represented like this: nCk or as (n k) or even as a fraction like this:

n!/k!(n-k)!

So if you have a group of 5 friends (n=5) and you’re choosing a team of 2 (k=2), the binomial coefficient tells you there are 10 possible teams you could form.

Work out:

5!/2!(5–2)! = 120 / 2 (6) = 120 / 12 = 10

Note that you could reframe this question as, “If you had a group of 5 dimensions (say for example, three (considered as space in totality, i.e. units of “infinite space”) of space & one of time & one of soul), and you are choosing a dual of those dimensions, you will have 10 possible dual dimensions.” You may find the below math workout on Lichfield Cathedral a useful link here:

The Crossroads of Heaven & Earth is Time, the Mathematics of Lichfield Cathedral.

In simpler terms, the binomial coefficient answers the question: “If I have n items, in how many ways can I choose k items?” And keep in mind, the order in which you choose these items doesn’t matter. So choosing Alice then Bob is considered the same as choosing Bob then Alice.

Even Dimensions

The symmetry in the number of k-dimensional faces in an n-dimensional simplex does hold for both even values of n as well. That is, for any n-dimensional simplex, the number of k-dimensional faces is equal to the number of (n-k)-dimensional faces.

To see why this formula gives the number of k-dimensional faces, think of each face as being determined by a unique subset of the vertices of the simplex. An n-dimensional simplex has (n+1) vertices, and a k-dimensional face is determined by a unique set of (k+1) vertices. So the number of k-dimensional faces is equal to the number of ways to choose (k+1) items from a set of (n+1) items, which is given by the binomial coefficient “(n+1) choose (k+1)”, or (n+1)C(k+1).

The total number of faces (not including the outermost n-dimensional face itself, which is always equal to 1) of an n-dimensional simplex is then given by the sum of the binomial coefficients “n choose k” for k = 0 to (n-1). This quantity does show a symmetrical pattern when written out as a sequence.

For example, for a 4-dimensional simplex, or 4-simplex, this sequence would be:

• 0-dimensional faces (vertices): 5C0 = 5
• 1-dimensional faces (edges): 5C1 = 10
• 2-dimensional faces (triangular faces): 5C2 = 10
• 3-dimensional faces (tetrahedra): 5C3 = 5

Notice the symmetry: 5, 10, 10, 5.

So the symmetry in the number of faces of different dimensions in an n-dimensional simplex does hold for both even and odd values of n, with the exception of the outermost n-dimensional face.

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Dr Nick Stafford — Lichfield Cathedral, a Theory of Everything

“Sometime too hot, the eye of heaven shines. William Shakespeare Sonnet 18