hexponents!

This page displays the patterns of repeated multiplication in certain hexagonal number systems, as shown in my video.

You can generate new patterns by changing the settings. Use the die icon to randomize or the hearts icon to see my favorites. If you find something cool, copy the code and paste it as a comment on the video, and I might add it to the list of favorites!

There are three primary settings behind each pattern:

crow squared

These calculations are done within 2-dimensional systems of quadratic integers. The horizontal dimension is made up of real numbers, and the diagonal dimension is spanned by c, which is called 'crow' in my video. These are called quadratic integers because c is the root of a quadratic equation. Changing this equation changes the behavior of the system.

I set this equation to always have the form:

c² = c + n

One could also change the coefficient of the c term, but the resulting system may then be rectangular instead of hexagonal, so I limited to just that case.

modulus

All calculations are done using modular arithmetic. The purpose of this is to create a system with only finite different values, meaning that repeated multiplication is forced to enter a cycle. Changing the modulus will change the length and shape of these cycles.

orbits

The core idea of this page is to display orbits. An orbit is a set of elements that can reach each other through a certain action. For example: if we begin with the value 1 and repeatedly multiply by c, we get all the powers of c. The system only has finite values because we use modular arithmetic, so these powers of c must eventually repeat, entering a cycle. We say this set of powers is 'the orbit of 1 under multiplication by c'.

Given some multiplier x, all values can be partitioned into orbits under multiplication by x. However, instead of allowing you to choose that multiplier x, I simply allow you to choose of the amount of orbits. This is because there may be multiple multipliers that result in the same amount of orbits, and if they do, those orbits will be identical (because I have restricted the modulus to be prime). Therefore it is simpler to just choose the desired amount of orbits.

about me

I am TheGrayCuber, a creator of mathematical websites and videos!

The 'hexponents!' video that I made on this topic was experimental, a challenge to write entirely in rhyme. While this was a lot of fun, it lacked depth and clarity of the underlying mathematics.

Lately I have been grappling with this decision of how to create videos. Ultimately they are meant as entertainment, and so it performed its purpose better than a serious video 'Orbits of Eisenstein Integers' would have. However, I understand that some viewers would prefer that educational style, so for now I will use these websites as a place to provide more clarity.

If you're interested in learning, I recommend that you try calculating some of these orbits by hand. Use c² = c - 1 with a small modulus such as 5, and try out different multipliers. This works best when the modulus is prime, so you may find it helpful to watch my video on the Eisenstein Primes, and then utilize my visualizer to identify the primes. That's what I did while preparing for this video.

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