Phinite elements, and fun with other algebraic integers

[follymath]

About the author

I'm a software engineer who's been exploring various spaces and geometries which arise naturally as consequences of modular arithmetic.

Quick Summary

So much material exists about complex numbers, and treat i as if it must be handed down from on high. It's nothing more than an element that squares to -1, by definition. Not a number, but an algebraic concept, with properties that can be realized in a variety of ways.

The same could be said for the number φ. People have been gushing over its incredible algebraic properties for ages. But these properties are not exclusive to the number 1.618..., the golden ratio. There are phi-like residues living in any ring with an element squaring to 5 and with 2 invertible (that is, an element representing one-half).

All you need to find real, concrete examples of these elements, and many others like them, is to explore times tables in the integers mod n, for various values of n. All of this richness gets obscured, and all these extra square roots disappear, once you take n to ∞. Once you demand all of ℤ. Yet this is what we do on purpose. Routinely. And we don't even know what we're missing.

Target medium

A video, most likely. You can completely capture this concept with simple times tables. Though if you really dig into the geometry, there's an even more interesting stack of hyperboloids -- like a can of Pringles in a quadratically stretched cylinder. Or something along these lines.

What I could really use in a collaborator is someone who understands how to organize and simplify a jumbled mess of interrelated ideas. Someone who understands how to simplify a presentation, and tease out the key aspects. I've been tooling around with this stuff in isolation for so long that, frankly, every time I sit down to write it all out, it has a bit too much of this energy:

Thus, I've never really written about this publicly. I have a bunch of other somewhat related explorations I'd like to write up. For example, I did a brief writeup detailing a novel topological space which represents any positional notation for any base on another issue. But it's hard not to sound like a crank, or worse, a numerologist, when you're literally talking about the geometry of numerals.

More details

Here is an observablehq notebook with a random assortment of notes from back when I last messed around with these elements. Below is a longer explanation/very rough first draft of a possible script I'd put together to give a sense of these elements, the space they inhabit, and why it might be worth exploring, as a kind of gateway drug into algebraic number theory...

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Have you ever seen an imaginary object -- a square root of -1? Could you pick one out of a lineup? "Of course not", you might say. It's "imaginary", after all. Okay, what about an object like φ, the beloved golden ratio? It's "real" enough, sure, but can you really get ahold of something as abstract as a root of a quadratic equation? Is it really so irrational?

As it happens, there exist analogs for both of these quadratic integers, along with a whole zoo of other algebraic integers, hiding in plain sight, in finite rings. The integers mod n, for particular values of n.

The properties of, say, imaginary units in the integers mod n are well known. We know which n will have elements which square to -1, and why. We may even know exactly where to look to find these elements, assuming we know the prime factors of n. And these really are objects we can literally point to. Like, as entries of a finite multiplication table. Any entry with a -1 along the main diagonal. We teach third graders how write out a times table, and fourth graders enough about fractions for them to understand how to take a modulus. These are the only prerequisites necessary to see an imaginary unit first-hand.

And while you're looking down the diagonal, you can touch and feel some other fascinating objects. For example, if you see two 1 elements straddling the diagonal, you've found a φ element and its associate. Alternatively, you could find a 5 along the diagonal, which immediately gives you a square root of 5. You can work backward from here, navigating around the times table corresponding to algebraic operations, to find the corresponding φ element. There can be multiple elements which square to 5 -- each one corresponding to its own φ.

There are also many connections between the various rings of integers mod m. Much of this becomes clear simply by observing how the times table changes as you vary the modulus m. Here, have a times table and a slider for m.

For example, any for any ring of n elements containing a φ element, it must contain a square root of 5. If you look at the ring n+2, you can find an element associated with √-3. From here you can conjure ω, Eisenstein's first cube root of unity:

$$ω = \frac{-1+\sqrt{-3}}{2}$$

All of this from a table you could teach a young child how to read.

You can play around with all the same algebraic properties revered about the golden ratio, but without having to understand algebra, let alone how to work with abstract objects, like solutions to algebraic equations. You could even learn some basic algebra just from observing and tinkering with the mechanics of elements in these tables.

But why stop there -- the silver ratio is also fascinating, and is just a unit away from any element which squares to 2. A jumping off point to the PV numbers.

Or, taking a slightly different path, you could explore the metallic means. You can't miss the obvious symmetry about the diagonal (thanks, commutativity), but there's an even stronger kind of symmetry -- that gives us some interesting geometric to explore, and exploit.

Let's examine the first differences of along this antidiagonal.

We first found "golden" elements by looking for the pattern 1 1 straddling the main diagonal. We then found "silver" elements by looking for a 2 on the main diagonal, but if you look closer, you'll notice the pattern 1 2 1 appears on the antidiagonal. We can find "bronze" elements by looking for 1 3 3 1.

So what's the next pattern? 1 4 5 4 1. No, this is not a typo -- no binomial coefficients here, just parabolas. Specifically, alternating pronic and square parabolas, for half-integer and integer indices, respectively. Which makes sense -- of course you'd find parabolas within times tables.

But the well-defined structure and position of these parabolas puts some tight constraints on the "shape" of our mod tables, the "distance" between them, and the kinds of elements which can exist and where. To every antidiagonal we can find the associated metallic mean. From here we can infer the presence and precise position of the associated square root element.

There's actually separate geometric relationship tying all the multiple square root elements together, but that's a story for another day.

Contact details

Right here on github is a fine place to start. Or @follymath on discord.

Additional context

If someone's interested in doing something with this, just chime in and I can elaborate further.

Again, here are some random notes from back when when I was tooling around with this. A bit of a mess, but if anyone was interested in picking this up as a general concept to explore I could provide better info, visuals, and general tooling to aid exploration and discovery.

People seem to love them some crazy phidentities, so this oughtta be like manna from click-bait heaven. But this is more than just recreational math fodder -- there's actually a real bridge to algebraic number theory that's accessible to just about anyone, regardless of education level.