The Secret Truth of the Universe

A beautiful geometric visualization of positive-valued binomial expansions, the algebraic expressions produced by raising sums of variables (a, b) to natural-valued powers n. The algebraic structures and procedures of binomial expansion are described by the binomial theorem–which, in turn, is proved by the above figures. 

Given (a+b)^n, there will be (n+1)-many terms, c(a^(n-m))(b^m), where c is a constant, and a and b are variables. 

By tradition, these terms are arranged in descending order by powers of the leading term, a. Accordingly, the binomial expansion of (a+b)^n begins with the term having the highest power of a, which is a^n for all n. 

The remaining n-many terms are ordered such that the exponent of each successive a term (n-m) decreases by one. Correspondingly, the exponent of the b term (m) increases by one, such that (n-m)+m=n. 

The binomial coefficients c for each successive term c(a^(n-m))(b^m) are described by Pascal’s triangle. Given (a+b)^n, the nth row of Pascal’s triangle contains (n+1)-many numbers, which are the coefficients c, listed in the order described above.

Note that the exponent n is the dimension of the figures pictured above. This is no coincidence; the term (a+b)^n can be depicted geometrically by a figure whose measure (length, area, volume, hypervolume, etc.) is the quantity produced by multiplying (a+b), n-many times. 

Additionally, observe that the coefficients c give the quantity of n-dimensional figures required to fill in the missing bits of area, as illustrated by the figures. For example, in order to fill the volume (a+b)^3, we use two cubes of volume a^3 and b^3, and the remainder is filled by 3 “plate-like” figures and 3 “tube-like” figures (each of whose dimensions are (a^2)b and a(b^2), respectively).

The frieze groups are used to classify patterns which are repetitive in one direction, according to their symmetries. Of course, such designs occur frequently in architecture and decorative art. Mathematical study of such patterns reveals that essentially, only seven types of symmetry can occur.

The first frieze group, coined HOP by John Conway, is generated by a single translation and therefore contains only translational symmetries:

The second group, STEP, contains translation and glide reflection symmetries. This group is singly generated as well, by a glide reflection (as a translation can be simulated by two glide reflections).

SIDLE contains translation and vertical reflection symmetries.

Next, a SPINNING HOP is generated by a translation and a 180° rotation.

SPINNING SIDLE contains translation, glide reflection and rotation (by a half-turn) symmetries.

JUMP contains translation and horizontal reflection symmetries.

Finally, SPINNING JUMP contains all symmetries (translation, horizontal & vertical reflection, and rotation). This group requires three generators, for instance a translation, the reflection in the horizontal axis and a reflection across a vertical axis.

The structure underlying quaternion multiplication.

Sir William Rowan Hamilton famously carved this revelation into a bridge:

These numbers may feel familiar to those acclimated to complex numbers. The quaternions extend them–hence their similarity (i.e., both feature negative squares, etc.). Whereas complex numbers are visualized as vectors in the complex plane, quaternions can be imagined as vectors in a 4D space. 

Beginning from any “interior” block, a red arrow out (into another block) multiplies the original by i. Going in reverse (opposite the arrow), we divide by i. A green arrow out is multiplication by j. Likewise, blue corresponds to k.

Starting from the 1 block in the center, we move right-forward (towards our viewpoint) along the red arrow, obtaining i. Then we turn clockwise along the green arrow, into the k block. Lastly, we take the blue arrow left-forward, to the -1 block on the outside. Thus ijk=-1.

The other identities can be deduced visually.

This animation took me three weeks! My senior thesis advisor does some math with braids (I’m doing number theory with him) so I asked him to send me one to draw as an anti-anxiety project. I’m really happy with how it turned out! Here are some of the intermediate products…

And bonus! My cat Lily (who I adopted from the aforementioned thesis advisor) being VERY “helpful.”

Basically you can take any shape you want to start with, and then scale it and rotate it down according to a set of functions that map points in a bigger domain to other points in a more restricted domain, which you then plug back in and repeat until your fixed points make an interesting pattern, which Falconer points out is usually a fractal. 

The reason I asked about IFS is because you can stretch, skew, and shrink shapes using affine transforms:

link to the code

(The first large part of the code (before “/////…///”) is simply something to render with motion blur.)

I don’t know much about IFS but this is implemented using recursivity. Each triangle has 3 children, and you start with one large triangle. To show one triangle you show its 3 children, excepted when you reached the maximum depth : in that case you just show one white triangle. The function “show” takes the positions of the 3 vertices of the triangle as input, and calls “show” on its 3 children (excepted when you reach max depth). The inputs for the new calls are computed with some interpolation between vertices with a random value (called “noise”) that loops through time.

If you have any question regarding the code, let me know.

Here’s an alternative version with other random values

Basically you can take any shape you want to start with, and then scale it and rotate it down according to a set of functions that map points in a bigger domain to other points in a more restricted domain, which you then plug back in and repeat until your fixed points make an interesting pattern, which Falconer points out is usually a fractal. 

The reason I asked about IFS is because you can stretch, skew, and shrink shapes using affine transforms:

@the-real-numbers I don’t know about this picture, but today I learned that

you don’t need to start with a triangle when constructing it via subdivision rules (any closed and bounded area in the plane suffices to start with; successive iterations will converge on the equilateral Sierpinski triangle), and

There is a way to iteratively generate a set of points that describes the sober Sierpinski triangle (or at least converges to it).  I don’t know if it’s technically fair to call it an IFS since it involves an element of randomness.

Let me know if you figure out how this picture got generated!  I’d love to know.