For decades, mathematicians used a whole arsenal of functions—logarithms, powers, trigonometry—without realizing that they could all be reduced to a single, universal operator. Polish astrophysicist and mathematician Dr. Andrzej Odrzywołek (Jagiellonian University, Kraków) showed how to do this, and his work sparked widespread attention. In an interview for fast-takeoff.com, we discuss mathematics that looks like science fiction, whether analog computers can make a comeback, and how artificial intelligence is changing the way science is done.

***

Your latest paper, “All Elementary Functions from a Single Operator”1 has caused quite a stir online. But if exponential functions, logarithms, and their connections have been known since Euler, why has such a simple, universal operator eluded mathematicians for centuries?

Dr. Andrzej Odrzywołek: Because the scientific procedure may not be clear to the broader community, let’s clarify: this is not a [scientific] paper but a preprint. The results have not yet been formally peer-reviewed and finalized.

It is common knowledge among experts that only a few operations are sufficient to provide basic “elementary” functions, which I define as those known from scientific calculators. For example, Mathematica, used for calculations by theoretical physicists, requires only four operations: addition, multiplication, exponentiation, and the base logarithm. Converting multiplication to addition and vice versa is also, and may have been obvious before calculators appeared. Every schoolchild, student, or engineer had to master the use of tables or a slide rule. The exponential function exp(x), its inverse (the natural logarithm ln(x)), and the four operations (+, −, ×, ÷). That’s enough. Further steps from the recommendation (notice of centuries to exaggeration) are simply not followed and are not implemented. The key formula for the logarithm in EML form requires 7 tools and three applications, which is borderline “natural.” EML itself requires 3 operations on a 36-key calculator, a number that can be checked with a naive algorithm up to about 6 million (2⁷ × 36³). Too many, due to the concatenation at the end, but few enough that someone had to do the math in the first place.

Some people admire the elegance of EML, others ask, “OK, but why?” How do you respond to the accusation that it’s ultimately just mathematical gymnastics, a kind of “art for art’s sake”?

Three questions in one :-)

The concept of elegance is subjective. It’s more accurate to talk about complexity or simplicity. One binary operation and the number 1 are less than a set composed of several functions of a single variable, binary operations, and constants. Formulas compiled into EML can seem ugly, like the pi formula circulating online. I’d call it rather lengthy. However, the EML operator itself is indeed elegant; its form really surprised me. I expected something worse, maybe not necessarily as strange as, say, Minkowski’s ?(x), but rather among special functions rather than school functions.

I needed a single operator for what’s called symbolic regression, i.e., searching for mathematical patterns in data sets. Recently, such techniques have been sidelined, because AI does the same thing much better. The main problem in SR is choosing a base: sine or cosine, tangent or cotangent, addition or subtraction, etc. Using too many operations complicates the search; too few, and we risk missing some valid expressions altogether. For a long time, I’ve had the idea for an automatic verifier that, before running a search, for example, in PySR, checks whether the set of constants (e.g., 2, pi, e), functions (e.g., sin, sinh, cos, cosh), and operations (e.g., +, −, ×) used will actually find any expression. While testing the verifier (VerifyBaseSet), I fed it various things and observed what happened. Then I realized that with just one operator, the problem disappears altogether.

Without a doubt, EML is a kind of intellectual gymnastics. But as Stanisław Ulam wrote:

In your publication, you also write about a prototype EML compiler and the potential for analog computer design. Does this mean that in the future, our processors could operate on a completely different, simplified architecture? Where exactly might EML find the quickest physical application?

Analog computers were long ago replaced by digital ones, but that doesn’t mean they’ve ceased to exist. Their use is limited by technical problems, one of which is the inability to generate arbitrary elementary functions. EML could fill this gap. Generating an exponent is simple: it’s the law of capacitor discharge; subtraction is performed by an operational amplifier. But along the way, we need complex numbers. The path from theory to practice won’t be straightforward. The brain undoubtedly operates in analog mode, and we can’t even replicate the brain of a fruit fly.

Just three years ago, the world of mathematics was captivated by the “einstein” tile: a single prototile that by itself forms an aperiodic set of prototiles. In a sense, you yourself have a similar invention to your name: a single operand that can express any real-world function. Do you think other fields are still hiding their “universal building blocks” from us?

These are related. I myself invented and hold a patent for a three-dimensional polyhedron whose shape allows you to build an igloo. You can find a description in the journal Eureka (”How to Build the Perfect Igloo”2 2014), published by mathematics students at Cambridge, and its model is held by the NKF UJ. I believe that redirecting the computing power of hundreds of thousands of GPUs currently used to train AI to search the solution space of various problems could reveal many surprises.

Apparently, EML is just the beginning, and you’re already exploring its ternary variant, which doesn’t need the constant “1” at all. Why is eliminating this one extra digit so important to you, and what would it take to find a fully self-sufficient operator?

The need to consider the possibility of a specific constant at every level complicates calculations and increases the number of parameters in the EML tree. It also acts as a kind of discrete switch that removes the logarithm from EML. A model without any specific constant, containing only variables x, y, z... would be fully continuous. Such a binary operator would be even more similar to NAND. I haven’t been able to find it. Perhaps it doesn’t exist or is not an elementary function.

At the end of the publication, you openly admit that modern language models have aided you in your work, including translating code into Rust. What’s your day-to-day collaboration with AI like?

Language models have accelerated many things. Firstly, effective translations and multilingual search. Now, on X, I immediately see, for example, posts originally in Japanese that would probably be completely incomprehensible to me without them. From a circuit diagram alone, I wouldn’t have guessed that someone was presenting a proposal for a hardware EML implementation. Secondly, a literature review, including historical texts in Latin or French (Cotes, Liouville). This is very helpful, especially for interdisciplinary research in an era of narrow specialization. In a sense, LLMs fulfill the promise that Google once gave us and then took away: the ability to find any information we need.

Another issue is language proofreading. However, this mainly works with standard texts and requires extreme caution, as the problem of hallucinations still persists. Code translation is the latest AI achievement; a breakthrough occurred in December 2025. Currently, the problem isn’t writing the program itself, but rather clearly defining what it should do and how. It turns out that the best way to explain how a program should work is to have another program do the same thing. An AI agent like GPT Codex or Claude Code can independently run and test programs, allowing for efficient code translation, for example, from Mathematica to Rust. Having two versions allows for easy verification of whether they function identically. The difference is speed: the original VerifyBaseSet requires 40 minutes, but after rewriting it in Rust, it takes just seconds. AI also facilitates working with different operating systems; the repository included with the preprint works on Windows, Linux, and MacOS.

***

Dr. Andrzej Odrzywołek – Polish astrophysicist and mathematician, member of the Department of General Relativity and Astrophysics at the Institute of Theoretical Physics of the Jagiellonian University in Kraków. Author of the paper “All Elementary Functions from a Single Operator,” which caused a stir in the international scientific community. Holder of a patent for a three-dimensional polyhedron for igloo construction (the model is in the Jagiellonian University collection).

Links:

1. UJ employee page

2. LinkedIn

3. X/Twitter