Help wanted for some algebraic structures

[skaa he/him]

During the pandemic lockdowns, I started getting obsessed with abstract algebra. Take note that I am not a mathematician nor do I see myself as mathematically-gifted, but you'll be surprised how much a layperson can learn by accident just by watching YouTube videos and daydreaming, and how much more he can learn by going on Wikipedia rabbit holes and pelting random mathematicians with stupid questions.

So anyway, I came up with some algebraic structures that are related to each other, and I decided to write several blog posts about them. Some caveats:

• The whole thing is still a work in progress, mainly because I still have some (probably stupid) questions left unanswered and I don't know how to proceed.
• I have no idea how useful any of this is. I created them for fun, and so far they've been a great motivation to learn more math, so please don't pour cold water over me by complaining about how useless these things are. Corrections will be quite welcome, however.
• If you've seen my cosmere theories, you know how verbose I can be. These blog posts are long and possibly filled with errors, so lots of patience will unfortunately be needed, I think.

With those out of the way, here are links to each blog post:

Part 1: Just a rambling intro. People with limited free time can skip this as I already gave the gist above.

Part 2: This is about what I call a "trirational number". There are several ways to view it:

• A generalization of rational numbers and fractions
• A representation of complex numbers that's different from the rectangular and polar forms
• A number that represents a ratio of three numbers

Part 3: I tried generalizing trirational numbers to "multirational" or "n-rational" numbers.

Part 4: I created a family of unital commutative algebras that I call "polytopic algebras" just for the sake of providing a backbone for multirational numbers (mainly because defining the addition operation for a multirational seems impossible without referencing an underlying algebra).

Part 5: This just describes how one can use polytopic algebras, cyclic groups, and n-simplexes to create multirational numbers.

 

So, why am I sharing this here? I've already talked to a bunch of mathematicians in the Math Stack Exchange as well as in a couple of math-related Discord servers, but so far they've only helped me out with trirational numbers. They all become oddly silent whenever I try to ask about multirationals or polytopic algebras. I'm sharing this here just on the off chance that some people will find these things interesting. I will try to answer your questions given my limited knowledge. And if I'm lucky, someone here might help me fill up the holes in this weird framework.

Edited January 12, 2023 by skaa
typo

[offer]

When you refer to a "trirational number" a>b>c do you demand that a,b,c are integers or do you allow them to be real numbers? The name you gave them implies the first but it seems from what you wrote that you mean the second.

If you are allowing them to be any real numbers (b,c positive) then you might want to rename it to something like tri-fractions 

 

I have read the first 3 parts and I have an idea on a way to look at it that will enable you to extend it to n-dimensions :

Since you are looking at elements of a projective space and define multiplication as element-by-element multiplication it might be usefull to look at it as a Lie group and look at the logarithm map to the Lie algebra - there you have addition as an element-by-elemnt addition. So to define the n-dimension extension of your Lie group you might want to define an n-dimensional Lie algebra.

 

I will try to read the rest when I will have time.

 

 

edit:

In part 4 when you say "

Let's define a "polytopic group" to be a finite cyclic group whose group operation is the multiplication operation of a certain algebra. The span of the group's elements within the algebra's vector space determines the polytopic group's dimensionality, and its convex hull is a polytope with the same dimensionality.

"

In part 4 chapter "Xn and the orthoplxic group" : how do tou define the multiplication in the group?

 

Edited January 8, 2023 by offer

[skaa he/him]

9 hours ago, offer said:

In part 4 when you say "

Let's define a "polytopic group" to be a finite cyclic group whose group operation is the multiplication operation of a certain algebra. The span of the group's elements within the algebra's vector space determines the polytopic group's dimensionality, and its convex hull is a polytope with the same dimensionality.

"

Was there a question attached to this?

9 hours ago, offer said:

 

In part 4 chapter "Xn and the orthoplxic group" : how do tou define the multiplication in the group?

There are multiplication tables in later subsections. Please tell me if my explanation of those multiplication tables was somehow lacking. Alternatively, you could just use matrix multiplication on the matrix representations of Xn (described in another section).

Edit: I forgot your other questions.

Trirational numbers are named after the fact that they represent ratios of three numbers. As I showed in Part 2, any complex number can be represented by "proper trirationals" where all three components are positive reals, but I also described other possible variants. A trirational number where all three components are integers is a "simple trirational".

While rational numbers form a field, simple trirationals don't (adding two simple trirationals does not guarantee the sum will be a simple trirational). But proper trirationals do form a field, so I feel that they are the more natural generalization of rational numbers even if not all three components are integers.

My reason for not calling them "trifractions" is more of personal taste than anything; it just doesn't sound right to me. But I'm open to discussing other alternatives.

Quote

Since you are looking at elements of a projective space and define multiplication as element-by-element multiplication it might be usefull to look at it as a Lie group...

Sorry, but I just want to reiterate that trirationals are complex numbers. I mentioned projective coordinates in my blog post as just another example of homogeneous coordinates, but... are you saying that trirationals can exist in both the complex plane and a projective space at the same time?

Edited January 9, 2023 by skaa
added justification for the "trirational" name

[offer]

13 hours ago, skaa said:

Was there a question attached to this?

Sorry. There was a question but I understood the answer myself and forgot to delete the quate.

 

13 hours ago, skaa said:

Trirational numbers are named after the fact that they represent ratios of three numbers. As I showed in Part 2, any complex number can be represented by "proper trirationals" where all three components are positive reals, but I also described other possible variants. A trirational number where all three components are integers is a "simple trirational".

While rational numbers form a field, simple trirationals don't (adding two simple trirationals does not guarantee the sum will be a simple trirational). But proper trirationals do form a field, so I feel that they are the more natural generalization of rational numbers even if not all three components are integers.My reason for not calling them "trifractions" is more of personal taste than anything; it just doesn't sound right to me. But I'm open to discussing other alternatives.

Okay, I understand what you mean now. I just find the name confusing. I trifractions sounds weird also to me, but I think it is better than a confusing name (of course if I am the only one being confused by it it shoudn`t matter).

 

Quote

Quote

Since you are looking at elements of a projective space and define multiplication as element-by-element multiplication it might be usefull to look at it as a Lie group...

Sorry, but I just want to reiterate that trirationals are complex numbers. I mentioned projective coordinates in my blog post as just another example of homogeneous coordinates, but... are you saying that trirationals can exist in both the complex plane and a projective space at the same time?

What you defined is basically a map from the projective plane (or more precisely a subspace of the projective plane to the comlex numbers). And you are trying to study this map, which is a homomorphism of multiplicative groups.

 

Another thing you might want to do (if you do it in the parts I haven`t read yet foget this comment) is to look at the kernel of this map - when a>b>c=1?

For example:

 

[skaa he/him]

Thanks for your insights, @offer! One of these days I'll delve more into Lie algebras, projective spaces, and kernels. Right now I'm a bit more interested in the stuff I mentioned in Parts 3 to 5. I have a lot of questions about them, but I'll just share a couple here.

 

1. Matrix representations

I really want to know what the deal is with the matrix representations of polytopic algebras. I mean, it would have been so cool if they're just always circulant. There's a whole Wikipedia article about circulant matrices, and they are fascinating. But no, for an even number of dimensions we get this instead (taken from Part 4):

Does this specific kind of matrix even have a name? Wolfram Alpha only describes it as both normal and Toeplitz, which is well and good, but those descriptions also apply to circulant matrices. What do we make of these Toeplitz matrices that are almost circulant but whose entries above the main diagonal are the additive inverses of the ones below? Skew-symmetric Toeplitz matrices are just a small subset of these, but then there's not a lot about those, either.

And yet, the 2x2 matrix representation of complex numbers are of this form. The only reason I discovered the above matrix form in the first place is because the complex numbers are the motivating example for the even cases of Xn, and so I just tried to generalize its matrix form. I just wish I could read more about this Toeplitz variant, particularly its relationship to circulant matrices.

 

2. The Addition Problem

For multirationals, including fractions and trirationals, addition is more complicated than multiplication. And as you move towards higher multirationals, it gets way more complicated way fast. One motivation for constructing the polytopic algebras is that since we can create multirationals in them, we can theoretically convert two such multirationals to the algebra's Cartesian form, add them together (term-wise addition FTW!), then convert them back to a multirational. Theoretically.

I mean, at least I was able to do it for complex numbers, right? The section on trirational addition in Part 2 is probably my favorite section of the whole series. Alas, I'm finding it very challenging to repeat this process for the next polytopic algebra after the complex numbers, which we can call X3 or triplex. I feel that it must be doable with enough mathematical skill. After all, I already know how to make 4-rational numbers in X3 (using the process I explained in Part 5). I also know the exponential formula for X3 so I can actually evaluate 4-rationals to get the Cartesian form. But I have no idea how to convert the X3 Cartesian form to 4-rational form. Do you have any tips on how to do this?

(If you're wondering, I was able to do this for complex numbers because I spent months playing around with trirationals in Wolfram Alpha until I one day had a spark of intuition. I don't think I can do the same thing for algebras that Wolfram Alpha doesn't support.)

Edited January 10, 2023 by skaa

[Going_North_cal]

guys please.

guys my brain.

it hurts.

aaagagahaghahagahgha.

but anyways cool on y'all for figuring out the funny maths.

[skaa he/him]

Haha!  But no, I don't think I could say I "figured out" anything. I rely too much on Wolfram Alpha and on the tiny fraction of Wikipedia math articles that I could understand. My knowledge of abstract algebra is half-baked at best.

My IQ is not that great either. For example, I really should have realized the implications of the kernel way earlier. In Part 3, I said the following:

I'm not even totally certain I used those notations correctly. But either way, I'm an idiot because unwanted simplifications can happen in trirationals as well. Because of the kernel!

So something like 1▶exp(2π/√3)▶exp(-2π/√3) (approximately 1▶37.6223665▶0.0265799) would evaluate to 1, meaning it's not a good representation of the ratio (1 : exp(2π/√3) : exp(-2π/√3)).

I'll go sit in a corner and reflect on my actions.

(Though to be fair, how often do people deal with ratios involving the transcendental number exp(2π/√3) or its integer powers? Maybe I shouldn't beat myself so hard over this.)

Edited January 12, 2023 by skaa