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Self-graphing equations

This paper critiques the popular concept of Tupper's self-referential formula for being typographically dependent and trivial, then resolves these issues by formalizing the problem and providing a general solution using computability theory.

Original authors: Samuel Allen Alexander

Published 2026-08-25

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Samuel Allen Alexander

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a world where a mathematical formula, when drawn on a piece of paper, does not just describe a shape like a circle or a wave, but actually draws the very words that make up the formula itself. This is the strange and captivating idea of a "self-graphing equation." The concept gained widespread attention after a specific formula, known as Tupper's self-referential formula, went viral on the internet. That famous formula is capable of drawing any image of a certain size, including the text of the formula itself, but it relies on a specific, pre-chosen number to work. It is a clever trick of coordinates rather than a true self-reference. Mathematicians have long wondered if it is possible to create an equation that inherently contains the instructions to draw itself, without needing a secret code or a specific number to unlock the image. However, the question is tricky because it depends entirely on how we choose to write the letters and symbols. If the font changes, the drawing changes, and the equation might no longer match the picture. Furthermore, if one is allowed to use any function imaginable, the problem becomes trivial and meaningless, as one could simply define a function that draws whatever image they want, including the text of the function itself.

A researcher named Samuel Allen Alexander has now tackled these flaws by turning the problem into a rigorous mathematical question. Instead of focusing on a specific font or a specific set of allowed symbols, he created a general framework that defines what it means for an equation to be "self-graphing" in a way that works for any reasonable system of writing and drawing. He treated the alphabet of symbols, the way they are drawn as shapes, and the way they are interpreted as equations as a formal system. In this system, every string of symbols has a specific meaning as a drawing on a plane. The goal was to find a string of symbols that, when interpreted as a drawing, produces the exact same string of symbols. To solve this, Alexander did not rely on guessing or trial and error. Instead, he used a powerful tool from the field of computability theory, which studies what can and cannot be calculated by machines. He applied a famous result known as the recursion theorem. This theorem, originally used to prove that a computer program can print out its own source code, guarantees that under certain logical conditions, a system can refer to itself.

The paper demonstrates that if a system of equations is "self-constrained"—meaning it has a logical structure that allows it to translate the description of a drawing back into an equation that produces that drawing—then a self-graphing equation is guaranteed to exist. Alexander showed that this condition is met by a very practical system of writing equations. He constructed a specific example using a standard set of letters, numbers, and mathematical symbols, including special tools for handling infinite sums and products. In this system, the symbols are drawn as small, blocky shapes made of tiny pixels, much like letters on a digital screen. The researcher proved that within this system, there is a specific string of characters that, when graphed, draws the exact same string of characters. The proof relies on the fact that the system can express complex logical statements, including the ability to say "there exists" or "for all," which allows the equation to describe its own structure.

The finding is a definitive proof of existence, not a specific recipe for writing such an equation by hand. The paper does not provide the actual string of symbols that solves the problem, because the string would be incredibly long and complex, far beyond what a human could write or read. Instead, the work proves that such a string must exist within any system that meets the logical criteria Alexander established. The research effectively settles the debate about whether self-graphing equations are a meaningless curiosity or a trivial impossibility. It shows that they are neither. They are a genuine mathematical reality that arises naturally in systems capable of expressing their own logic. The work clarifies that the viral formula from the internet was not the only way to achieve this, nor was it a true self-reference in the strictest sense. By formalizing the rules of the game, Alexander has shown that the universe of mathematical equations is rich enough to contain its own image, provided the rules of the game are set up correctly. This result bridges the gap between abstract logic and visual representation, proving that a set of instructions can, in a very real sense, draw itself.

Technical Summary: Self-Graphing Equations

Problem Statement
The paper addresses the concept of a "self-graphing equation": an equation $F(x, y) = 1$ which, when graphed on the Cartesian plane, produces an image of the equation itself. The author identifies two fundamental flaws in the informal pursuit of such equations:

Meaninglessness: The concept is ill-defined without a formal specification of typography (how symbols are rendered) and the mapping from strings to geometric sets.
Triviality: Without restrictions on the class of allowable functions, the problem is trivial. One can simply define a function $F$ that equals 1 on the set of points forming the image of the equation string and 0 elsewhere, making the graph identical to the image by construction.

To resolve these issues, the paper formalizes the problem using concepts from computability theory, specifically aiming to find sufficient conditions under which a self-graphing equation is guaranteed to exist for any valid formalization of equations and typography.

Methodology and Formalization
The paper introduces a rigorous framework called a glyphed notion of equations, defined as a triple $(A, Gr, Gl)$ :

$A$ (Alphabet): A finite set of symbols.
$Gr$ (Graphing Function): A function mapping strings $\sigma \in A^*$ to subsets of $\mathbb{R}^2$ , representing the geometric graph of the equation.
$Gl$ (Glyph Function): A function mapping individual symbols $x \in A$ to subsets of $\mathbb{R}^2$ (the visual representation of the character). This is extended to strings by concatenating the glyphs of individual characters with unit spacing.

A self-graphing equation is defined as a string $\sigma \in A^*$ such that $Gr(\sigma) = Gl(\sigma)$ .

The core methodology relies on computability theory, specifically the Recursion Theorem (also known as the Fixed-Point Theorem). The author defines a property called self-constraint for a glyphed notion of equations. A system is self-constrained if there exists a Gödel numbering of strings and a total computable function $f: \mathbb{N} \to \mathbb{N}$ such that for any computable function $\phi_n$ , if $\phi_n(0)$ outputs the code of a string $\tau$ , then $f(n)$ outputs the code of a string $\sigma$ whose graph $Gr(\sigma)$ is exactly the glyph image $Gl(\tau)$ . Intuitively, $f$ is an algorithm that can take the description of any image (encoded as a string) and construct an equation that graphs that specific image.

Key Results
The central theorem (Theorem 10) establishes that self-constraint is a sufficient condition for the existence of a self-graphing equation.

Proof Sketch: Given a self-constrained system with function $f$ , the author constructs a total computable function $g$ such that $\phi_{g(n)}(0) = f(n)$ . By the Recursion Theorem, there exists an index $n$ such that $\phi_n = \phi_{g(n)}$ . Consequently, $\phi_n(0) = f(n)$ . Let $\tau$ be the string encoded by $\phi_n(0)$ and $\sigma$ be the string encoded by $f(n)$ . By the definition of self-constraint, $Gr(\sigma) = Gl(\tau)$ . Since $\phi_n(0) = f(n)$ , it follows that $\sigma = \tau$ . Therefore, $Gr(\sigma) = Gl(\sigma)$ , proving $\sigma$ is a self-graphing equation.

Concrete Example
To demonstrate that self-constraint is not merely a theoretical abstraction, the paper constructs a specific glyphed notion of equations using a standard alphabet (letters, digits, operators, and symbols for infinite products $\Pi$ ).

The system allows for the encoding of logical statements (including unbounded existential quantifiers $\exists$ ) into equations using infinite products.
The author shows that the condition "a point $(x,y)$ is in the glyph of a string $\tau$ " can be expressed as a logical formula involving the halting behavior of Turing machines and pixel definitions.
Using the closure properties of the system (specifically the ability to express logical conjunction, disjunction, negation, and quantification via infinite products), the paper demonstrates that this logical formula can be converted into an equation $E=0$ using only the allowed symbols.
This construction proves the system is self-constrained, thereby guaranteeing the existence of a self-graphing equation within this specific formalism.

Significance and Claims
The paper claims to provide a general solution to the problem of self-graphing equations by shifting the focus from specific typographic choices to the structural properties of the equation system.

Resolution of Flaws: By formalizing the problem, the paper eliminates the issues of meaninglessness and triviality. It shows that self-graphing equations are not dependent on a specific font or a "trick" function, but are a consequence of the system's ability to encode its own syntax and semantics (self-constraint).
Computability Connection: The work draws a parallel between self-graphing equations and Quines (self-replicating computer programs), utilizing the same Recursion Theorem that proves the existence of Quines.
Generality: The results are presented as sufficient conditions. Any system of equations and typography that satisfies the definition of self-constraint is guaranteed to possess a self-graphing equation. The paper does not claim to produce a specific, human-readable equation for a standard font, but rather proves that such an equation exists within any system capable of encoding the necessary logical machinery (specifically, the ability to express unbounded quantifiers).

Technical Summary: Self-Graphing Equations

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