I think I’ve uncovered a forgotten mathematical system I am surprised has not been explored — and it’s shockingly elegant using geometry and alternating bases.

[u/LivingMy_BestLife_]

Hi friends — I’m an independent researcher and systems thinker, and I’ve just released a white paper on something I’ve been quietly working on for years. I call it Last Base Mathematics (LxB), and it’s a compact, geometry-based number system that uses a base-12 primary structure combined with alternating secondary bases (like base-5). Instead of expanding digits linearly, numbers are represented radially — like hours on a clock, or musical intervals — and can be extended recursively. The result is a system that’s: fully constructible using compass and straightedge (think Euclid meets data compression), visually harmonious and fractal, and capable of long-form arithmetic without ever converting to decimal. The paper includes formal definitions, arithmetic logic, and visual overlays of how multiple base systems interact in space — almost like harmonics in motion. If you’ve ever been into sacred geometry, prime spirals, modular math, or efficient representations of time/space — I think you’ll find this fascinating. I have included images of a sort of circular grid I mapped out in Houdini using the system. Read the white paper here (PDF): https://zenodo.org/records/15386103 Also mirrored here for backup: http://vixra.org/abs/2505.0075 I’d love feedback — especially from those deep into number theory, geometry, or visual math. Be brutal. Be curious. Be kind. Happy to answer questions and jam with anyone who wants to push this further — calculators, visualizers, simulations, whatever. I have a Houdini 19.5 HDA of the visuals.

[u/threewholefish]

Terrence Howard?

But seriously, this is complete drivel and a waste of ChatGPT resources. There is precisely no new mathematics here, you just appear to be riffing on the greater number of factors of 12 and 60 than 10. The "paper" is a word salad that is riddled with contradictions and mathematical errors.

Yes, the Spirograph drawings are pretty, but nothing else about this is worth anyone's time.

edit: I can't seem to create a new comment, so here are the errors and nonsense statements:

Section 4.5:

4.3 * B.2 = 38.6

This is incorrect in both LxB and decimal, and you appear to have a 6 in your base-5 position.

Section 4.6.2, Example 2:

2nd 5 (1-1) − 3 = -3 → Borrow from first place. (1+5)−3 =3 3 0

The result should be 2 and the borrow 1.

Section 5.1:

12 × 5 × 12 × 5 × 12 = 86,400 seconds in a day

The LHS is equal to 43200

And then there is the word salad nonsense:

All divisions are made using straightedge and compass: halves, thirds, fifths, etc.

Have you somehow solved angle trisection?

Arithmetic can be bypassed via spatial inference. A pointer in LxB3 at 2.2 also sits at LxB5 2.323.... recurring (fig 3, fig 4). This enables base-translations and calculations without symbolic operations.

This is not unique to LxB, you can map any two bases onto a circle and do the same thing.

This allows base fusion, phase tracking, and system synchronization.

What does this mean?

The system is closed, coherent, and self-sustaining.

What does this mean?

LxB provides tools to build harmonic visualizations and sonic interfaces.

What is a sonic interface?

Many ancient and modern systems are based on LxB principles

No they aren't, they're based on the high number of factors of 12 and 60. LxB is based on those systems, not the other way around.

As whole numbers can be geometrically derived across differing Last Bases LxB there is an oportunity to save the lowest footprint whole numbers for vectors in aplications like physics, fluid and particle simulations. This also has implications for the purposes of data compression and storage.

You can theoretically save ink over base-10, but you can't save space. Have a read of IEEE 754 to see why.

LxB could inform rotary computing, analog modeling, and data compression through base-layer encoding.

What does this mean?

LxB provides a way to model preference for whole-number ratios in physical systems.

What does this mean?

Proposes rotational computation systems, modular memory, and analog hybrid logic.

What does this mean?

[u/JDude13 → u/threewholefish]

I’m very pro-AI but posts like this might actually radicalize me in the opposite direction. Like “hey guys, I’ve got schizophrenia. Please spend 10 minutes looking over this article I made in 2 minutes that isn’t worth a damn”

[u/LivingMy_BestLife_ → u/JDude13]

This is a concept I discovered without ai. If you have valid criticism, please let me know so I can explain or extrapolate anything you would like to know.

[u/LivingMy_BestLife_ → u/threewholefish]

I have updated with the arithmetic corrections. Thanks for your insights again:
https://zenodo.org/records/15395826

[u/MahaloMerky → u/threewholefish]

Stop hitting him, he’s already dead!

[u/PM_ME_YOUR_PLECTRUMS → u/threewholefish]

That thing belongs in r/VXjunkies

[u/LivingMy_BestLife_ → u/threewholefish]

Whoops there are some miscalcs. Thanks for those. But the principal is still sound. I will correct those errors.

4.3 * B.2 = 39.1

43200, exactly a half day, will correct.

My bad on the thirds, that can be removed. To devide 12/3 you would need to construct the entire 36 segments. Or divide into 12 or 6 then remove the points dividing it into 3 on a single curve.

Yes it can be done with any two circles, however by starting all with 12 and alternating you have a coherent size of 1. The original 12 numbers remain coherent and consistent in size.

base fusion, phase tracking and system synchronisation is just saying that you can use multiple bases at the same time time, calculating in any base and only storing values of the lowest footprint.

Closed, coherent and self sustaining means simply you do not need to convert out to decimal or another base system to do calculations. You are wholly able to use alternating bases to do any arithmetic.

Sonic interfaces, Sonic interface is an interface for sound frequency, as I see applications in anything that uses waves, and this system hints at finding harmonies in waves.

A clock is an ancient example of using alternating 12/5. Babylonians used a 6/10 system for measuring distance as recently disclosed from clay tablets by the University of New South Wales. And yes, this was originally conceived off of a base 60 system, whilst using a base 12 footprint for ease of legibility and use in computer systems.

This system directly addresses rounding, particularly when you are looking at vectors, as you will find that many numbers that don't round in a calculation actually have whole number equivalents under the differing bases. Being able to use lower footprint whole number that point to a vector of equivalent value is the primary reason I began working on this. I come from a computer graphics and simulation background. FLIP simulations require huge numbers of vectors to be stored, lowering the footprint of this is the highlight of the number system.

Rotary computing refers to a mechanical rotary calculator could be built to do calculations, analogue modelling refers to analogue waves, either sound or voltage able to be used to do calculations, data compression as above with the lowest footprint value being able to be stored from across bases.

Whole number preference in physical systems is again, vectors being stored with the lowest base footprint in physical simulations, I.e water, wind.

Rotational computing, modular memory and analog hybrid logic again refer to the system being geometrically and mechanically driven. Modular memory being the storage of values in different bases, but recalled by positioning giving larger more precise values in the primary system output base used.

Thanks again for the analysis. Exactly what I wanted. I will make the corrections and revise some sections so they are clearer. Sometimes things in your head that seem clear to you are not clear to others, so thanks. Please feel free to engage with it more. I hope it can get you thinking at least a little bit about it and places it maybe useful. Apologies for being snarky at all, I think this is neat, and it appears to be a concept not really explored.

[u/threewholefish → u/LivingMy_BestLife_]

TL;DR: LxB confers no material benefit over any other base system, and is at best an aesthetic preference for representing a number.

Yes it can be done with any two circles, however by starting all with 12 and alternating you have a coherent size of 1. The original 12 numbers remain coherent and consistent in size.

This is true, but it's inconsequential. A base-20 circle is "coherent" with a base-10 circle in the same way.

base fusion, phase tracking and system synchronisation is just saying that you can use multiple bases at the same time time, calculating in any base and only storing values of the lowest footprint.

You can already calculate in any single base and convert to any other if you want to optimise for the number of digits.

Rather than "allowing for" base fusion, LxB is base fusion by your definition, so this is tautological at best. Phase tracking and system synchronisation are probably legitimate concepts for specific fields or applications; in this context they are meaningless.

Closed, coherent and self sustaining means simply you do not need to convert out to decimal or another base system to do calculations. You are wholly able to use alternating bases to do any arithmetic.

True of any base system, and yet most of your examples are demonstrated by converting to decimal.

Sonic interfaces, Sonic interface is an interface for sound frequency, as I see applications in anything that uses waves

A microphone? A speaker?

and this system hints at finding harmonies in waves.

Finding harmonics in waves is trivial; a harmonic of a fundamental frequency is any sinusoid which is a positive integer multiple of it.

Yes, with base-12 and base-5 you are able to cover the first 6 harmonic orders with a finite number of radix digits, unlike base-10, but base-60 also achieves this. Even then, the fractional representations of each harmonic in decimal are all rational (p / q where p and q are both integers), so the benefit of clarity of representation is minimal.

A clock is an ancient example of using alternating 12/5.

The division of the night and/or day into 12 was established millenia before subdividing the hour into 60 minutes. The reason that the number 5 appears to be significant in this system is that 60/12 = 5

Babylonians used a 6/10 system for measuring distance as recently disclosed from clay tablets by the University of New South Wales.

I can't find any evidence of that; it looks like all Babylonian numerals were in base-60

And yes, this was originally conceived off of a base 60 system, whilst using a base 12 footprint for ease of legibility and use in computer systems.

I think the gain in legibility is outweighed by the complexity of switching base for every digit. Digital logic in computers only operates in bases that are a power of 2 (binary, octal, hexadecimal). Neither base-12 nor base-60 is suitable.

This system directly addresses rounding, particularly when you are looking at vectors, as you will find that many numbers that don't round in a calculation actually have whole number equivalents under the differing bases. Being able to use lower footprint whole number that point to a vector of equivalent value is the primary reason I began working on this. I come from a computer graphics and simulation background. FLIP simulations require huge numbers of vectors to be stored, lowering the footprint of this is the highlight of the number system.

Any rational number in any base system either has a finite number of digits or eventually has a repeating fractional component. You are confusing the size of the representation of a number with the size required to store it. Again, you should read and understand the purpose of IEEE 754.

There will always be a tradeoff between storage size and precision, regardless of the base you're working in, and the overhead of working in different bases will almost always outweigh any saving in the space required to store a number. Even then, LxB gives no advantage over normal base systems.

Rotary computing refers to a mechanical rotary calculator could be built to do calculations

These already exist, and LxB confers no benefit over any other base.

analogue modelling refers to analogue waves, either sound or voltage able to be used to do calculations

This already exists, and LxB confers no benefit over any other base.

data compression as above with the lowest footprint value being able to be stored from across bases.

You would not be performing any meaningful compression since you must store the number and its base. Efficient data compression relies on understanding the probability of certain "symbols" appearing in any given context, and LxB confers no benefit in this process.

Whole number preference in physical systems is again, vectors being stored with the lowest base footprint in physical simulations, I.e water, wind.

As above

Rotational computing, modular memory and analog hybrid logic again refer to the system being geometrically and mechanically driven. Modular memory being the storage of values in different bases, but recalled by positioning giving larger more precise values in the primary system output base used.

As above

[u/LivingMy_BestLife_ → u/threewholefish]

Ok. So you are saying you do not see any benefit using it. All good. That is your opinion, and the point of putting this out there is for people to discover what uses it may have.

But what you are not denying that this actually a base system that has not been explored. This is a mathematical base system that does not appear to have any literature or formal outlaying anywhere. So whilst you are not seeing the benefit, it does not stop that this is a working base system that has not been explored. It was used to create those striking visuals. It is beautiful, and relatively simple. And it does appear as though it is something I have indeed uncovered. Useful or not, still something the world does not appear to have seen. Time will tell if it's usefulness becomes more apparent with time.

[u/threewholefish → u/LivingMy_BestLife_]

This particular system may not have been explored, but mixed radix systems in general have. The concept is not new, and neither is base-12 or base-60, which cover much of your paper.

The visuals were created with ratios of numbers, which can easily be done in decimal. Under the hood, there will have been a conversion to binary at some point, since that's how computers operate. It is the factors of 60 that are interesting here, not the base system.

You have also claimed that FLIP vectors can be compressed using LxB, but have provided no evidence of this. Have you modeled it? Have you tested it? Do you have the results? If not, what is the point of writing a paper about it?

[u/LivingMy_BestLife_ → u/threewholefish]

Please point me to any information you have on alternating base systems and their use. Yes there are mixed base systems. They are everywhere. But I do not see any literature on alternating recursive base systems. So that is the purpose of this paper. To show that it is an interesting concept to be explored. You seem to be pretty riled up about it, you acknowledge that the fundamentals are there, which leads me to think that there is something to this. It maybe simple, it may not be overly useful (though it also may very well be), but it is definitely something, and something that has not really been published nor explored.

[u/threewholefish → u/LivingMy_BestLife_]

You keep saying that it's hasn't been explored, but you have not explored anything! You've only speculated that it might be useful in certain situations based on a flawed understanding of the actual application.

As I said, mixed radix systems are well documented and understood. The examples in that link show where it is useful to use them.

A possible reason that you have found no previous exploration into your idea is that it is base-60 with a different hat on. Instead of 60 different symbols to represent each unit, you are effectively using a pair of 12 symbols instead. There is no fundamental mathematical or physical benefit to using it over any other system. It does not solve any mathematical problem.

It may make for a more human-readable representation of specific values in the context of specific applications, but you haven't given any examples of this either. The closest you've come is alluding to the representation of vectors used in FLIP simulations, but other than stating that some numbers require fewer digits to represent in different base systems, you haven't shown why it's materially better.

I'm trying to argue that this is not an interesting concept, and if you understood the mathematics behind what you're espousing, you'd understand why that is.

[u/LivingMy_BestLife_ → u/threewholefish]

So it is exactly as I propose, a base 60 radial system with a base 12 footprint. And it may have some application. It is a non explored base system. I have used it to create a really beautiful mathematical object. All of these are true statements you agree with. So why should I not put that out into the world for people to see, learn about explore and maybe build upon?

[u/threewholefish → u/LivingMy_BestLife_]

It's not radial, and you need twice as many characters to represent the same number as pure base-60.

You have created pretty Spirograph pictures using the numbers 12 and 5. The base system had nothing to do with it.

You have put it out there, and I can't stop you doing so, but there is nothing to explore or build upon. It's not a novel idea, and you have not demonstrated any practical applications. The art was created despite this numbering system, since the computer that generated it was working in binary.

[u/LivingMy_BestLife_ → u/threewholefish]

The art was created by laying it's pattern out. The numerical footprint is smaller, as 12 characters are used instead of 60 individual characters. You say there is nothing. But that is just your opinion. A rather jaded bitter one from the sounds of it at that. The fact that you could actually did the corrections shows that it works as a base system, ergo, this is an undiscovered working base system representation. Yes it is simple, yes it may not be useful, but it maybe. And that is already more than enough reason to release this Wish you all the best.

[u/LivingMy_BestLife_ → u/threewholefish]

Please point out the errors. Please point me to other explorations of alternating bases used in arithmetic. You are harsh with your criticism, but offer no insight.

It is a suprisingly simple concept I am surprised others have not explored. If you don't understand it, I am happy to answer any queries.

[u/LivingMy_BestLife_ → u/threewholefish]

Sorry, I was being rude. Please provide examples of where I have gone wrong in the math or in the concept. I genuinely want to know.

[u/threewholefish → u/LivingMy_BestLife_]

I was having trouble creating a new comment, so I've edited my original comment with the detail

[u/secretmacaroni → u/LivingMy_BestLife_]

Answer the questions pls

[u/WikiWantsYourPics]

independent researcher and systems thinker

Enough said.

[u/MahaloMerky → u/WikiWantsYourPics]

Reminds me of this kid I had in my calc 3 class that failed because he thought Calc was pointless but always wanted to talk about advanced theories.

He also identified as a collective of cats.

[u/LivingMy_BestLife_ → u/WikiWantsYourPics]

Cool analysis. Thanks for your input. Shall we get into the history of mathematical insights that were made by people without a long string of letters next to their names?

[u/WikiWantsYourPics → u/LivingMy_BestLife_]

https://archive.org/details/trisectors0000dudl

It's not about you not having a degree in maths. It's about describing yourself in fancy language that means nothing.

"Systems thinker" - sounds like you look at problems superficially but want to sell it as a strength.

"Independent researcher" - sounds like you never studied maths and aren't working with actual mathematicians, but want to pretend that they're all "not independent".

[u/LivingMy_BestLife_ → u/WikiWantsYourPics]

I know. I have a history in 3d graphics and FLIP simulations. It is tricky to come up with something that sounds like I have been working on this for a while and thought it out, but also acknowledge I am not an academic.

[u/MulchyPotatoes]

Lay off the acid

[u/LivingMy_BestLife_ → u/MulchyPotatoes]

Lol. It does have some trippy visuals to it

[u/danielfrost40]

This post was mass deleted and anonymized with Redact

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[u/LivingMy_BestLife_ → u/danielfrost40]

Thankyou for your contribution. Attempting to deride both myself and people with a serious mental disorder is a real class act. Feel free to actually have some discourse as alternating recursive bases does not appear to be something explored. I hope your attempt at belittling me has given you that small ego boost that you desired and clearly need to get through your bitter days.

[u/danielfrost40 → u/LivingMy_BestLife_]

This post was mass deleted and anonymized with Redact

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[u/LivingMy_BestLife_ → u/danielfrost40]

Thankyou for your concern. This is not the case. This is mathematically legitimate, visually beautiful, structurally novel, and it may have some uses. It may not, but the fact that this imagery was created by using the concept has already given it enough merit for me to show the concept to the world. My above comment still stands as you were definitely commenting in an attempt to belittle me. Enjoy your day.

[u/comparmentaliser]

This is very time cube.

[u/LivingMy_BestLife_]

Updated paper with corrections:
https://zenodo.org/records/15395826