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The Euclidean algorithm, lotuses and singularities

This introductory paper explores the correspondence between the anthyphairetic process (a subtraction-based variant of the Euclidean algorithm) and the resolution of singularities for plane curves, demonstrating how "lotuses"—special two-dimensional simplicial complexes—serve as computational architectures to visualize and reconstruct the sequences of integers and pairs generated by these processes.

Original authors: Patrick Popescu-Pampu

Published 2026-06-05

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Patrick Popescu-Pampu

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

The Big Picture: A Mathematical Rosetta Stone

Imagine you have two different languages describing the same physical object. One language is ancient arithmetic (subtracting numbers over and over), and the other is modern geometry (smoothing out crumpled paper shapes).

This paper argues that these two languages are actually speaking the exact same story. The author, Patrick Popescu-Pampu, introduces a visual tool called a "Lotus" (a flower-shaped geometric diagram) that acts as a Rosetta Stone. It allows you to translate a simple number game directly into a complex shape-smoothing process, and vice versa.

1. The Number Game: "Anthyphairesis"

The paper starts with a very old way of finding the greatest common divisor (GCD) of two numbers, which the Greeks called anthyphairesis.

The Analogy: Imagine you have two sticks of different lengths, say 11 inches and 4 inches.
The Process: You take the shorter stick (4) and cut it out of the longer one (11). You are left with a 7-inch piece. Now you have a 7-inch stick and a 4-inch stick. You cut the 4-inch stick out of the 7-inch one. You are left with a 3-inch stick.
The Loop: You keep doing this: subtract the smaller from the bigger, over and over, until the two pieces are the same size.
The Result: This process is the "slow version" of the famous Euclidean algorithm. It generates a sequence of numbers (like 11, 7, 3, 1) that tells you exactly how the sticks relate to each other.

2. The Shape Game: Smoothing a "Kinked" Curve

On the geometry side, the paper looks at a specific type of curve defined by the equation $y^a - x^b = 0$ (where $a$ and $b$ are the numbers from our stick game).

The Problem: If $a$ and $b$ are bigger than 1, this curve has a "kink" or a sharp point (a singularity) at the center, like a piece of paper crumpled into a tight corner.
The Solution (Blow-ups): To fix this, mathematicians use a process called "blowing up." Imagine zooming in on the sharp corner. When you zoom in, the sharp point expands into a small line or circle. You then look at the new shape. If it's still kinked, you zoom in again.
The Connection: The paper shows that the exact same sequence of steps you took with the sticks (subtracting 4 from 11, then 3 from 4, etc.) is the exact same sequence of zoom-ins needed to smooth out the curve.
- If you subtract the smaller number from the bigger one in the number game, you are performing a specific type of "zoom-in" (called a blow-up) on the curve.
- When the numbers finally become equal (1 and 1), the curve is perfectly smooth.

3. The "Lotus": The Visual Translator

This is the paper's main invention. How do you see the connection between the numbers and the zoom-ins without doing all the math? You draw a Lotus.

What is a Lotus? It's a flower made of triangles (petals) glued together.
How it works:
- From Numbers to Lotus: If you take the sequence of numbers from your stick game (the quotients from the Euclidean algorithm), you can build a lotus. Each number tells you how many petals to add in a specific direction.
- From Lotus to Numbers: Conversely, if you look at a finished lotus, you can "read" the numbers back out of it.
The "Anthyphairetic Rectangle": Before turning the numbers into a lotus, the author suggests drawing them as a stack of rectangles. Imagine stacking blocks where the height and width change based on your subtraction steps. If you "squash" or contract this stack of blocks, it magically transforms into the flower-shaped lotus.

4. Why This Matters (According to the Paper)

The paper doesn't claim this will cure diseases or build better bridges. Instead, it claims that the Lotus is a "computational architecture."

The Metaphor: Think of the Lotus as a physical calculator or a map.
- If you want to know the history of how a curve was smoothed (the sequence of zoom-ins), you don't need to do the complex geometry. You just look at the Lotus.
- If you want to know the sequence of numbers generated by the ancient subtraction game, you don't need to do the arithmetic. You just trace the petals of the Lotus.
The "Dual Graph": The paper also explains that the Lotus contains a hidden "skeleton" (a dual graph) that tells you exactly how the different parts of the smoothed curve intersect and how "twisted" they are (mathematicians call this "self-intersection numbers"). The Lotus makes these invisible numbers visible and easy to count.

Summary

The paper is a guidebook showing that:

Subtracting numbers (Anthyphairesis) and smoothing curves (Blow-ups) are two sides of the same coin.
The Lotus is the flower-shaped diagram that holds the map between them.
You can build the Lotus from the numbers, or read the numbers from the Lotus, making complex geometric problems solvable by simple visual counting.

The author dedicates this to Karine Chemla, a historian of science who loves colorful drawings, because this paper turns abstract math into a colorful, flower-like picture.

Technical Summary: The Euclidean Algorithm, Lotuses and Singularities

Problem Statement
The paper addresses the structural relationship between two distinct mathematical domains: the classical number-theoretic process of anthyphairesis (the iterative subtraction of the smaller of two quantities from the larger, leading to the greatest common divisor) and the geometric process of minimal embedded resolution for singularities of plane binomial curves defined by $y^a - x^b = 0$ . While it is established that the sequence of blowups required to resolve the singularity of such a curve mimics the sequence of subtractions in the anthyphairetic process, the paper seeks to provide a unified, graphical framework that explicitly localizes the numerical data of both processes within a single combinatorial object.

Methodology
The author employs a combination of algebraic geometry and combinatorial topology to construct and analyze "lotuses," a specific type of two-dimensional simplicial complex. The methodology proceeds through the following steps:

Geometric Analysis of Blowups: The paper first reviews the standard representation of the blowup of the origin in an affine plane. It demonstrates how the strict transform of a binomial curve $C_{a,b}$ under a blowup corresponds to a reduction in the exponents $(a, b)$ via subtraction (e.g., transforming $C_{a,b}$ to $C_{a, b-a}$ if $a < b$ ). This establishes the direct parallel between the Euclidean algorithm (via subtraction) and the sequence of blowups leading to the minimal embedded resolution.
Construction of the Lotus: The paper introduces the lotus $\Lambda(b/a)$ $Λ (b / a)$ associated with a positive rational number.
- Standard Construction: It recalls the existing method of constructing the lotus from the continued fraction expansion of $b/a$ (derived from the Euclidean algorithm), where the quotients determine the triangulation of a base triangle.
- Alternative Construction (Anthyphairetic Rectangle): The paper proposes a novel construction starting directly from the sequence of pairs generated by the anthyphairetic process. This involves defining a "subtraction rectangle" $S(a,b)$ for each step of the process and stacking them to form an "anthyphairetic rectangle" $R(a,b)$ .
Contraction and Isomorphism: The author defines a contraction operation on the anthyphairetic rectangle $R(a,b)$ to produce an "anthyphairetic triangle" $T(a,b)$ . By contracting specific vertical edges (where vertices share numerical decorations) and the top edge of the unit square, $T(a,b)$ is shown to be isomorphic to the lotus $\Lambda(b/a)$ .
Reverse Reconstruction: The paper details a converse algorithm (Proposition 6.1) that reconstructs the sequence of coprime integer pairs from a given lotus. This involves decorating the lotus vertices starting from the top (the unit pair) and descending petal-by-petal, summing the values of adjacent vertices to determine the values of the third vertex in each triangular petal.

Key Contributions

The Anthyphairetic Rectangle: The introduction of the anthyphairetic rectangle $R(a,b)$ as a graphical representation of the subtraction-based Euclidean algorithm. This object serves as an intermediate structure that visually encodes the entire history of the anthyphairetic process.
Isomorphism of Structures: The proof (Proposition 5.3) that the contracted anthyphairetic triangle $T(a,b)$ is isomorphic to the lotus $\Lambda(b/a)$ constructed via continued fractions. This establishes a direct bridge between the subtraction-based and division-based views of the algorithm within the same topological framework.
Computational Architecture: The demonstration that lotuses function as "computational architectures." The paper shows that one can compute the sequence of integer pairs (the anthyphairetic history) and the intersection numbers of the exceptional divisors simply by iterating local operations (summing vertex decorations) within the petals of the lotus.
Visualization of Resolution Data: The paper provides a method to read the dual graph of the exceptional divisor and the self-intersection numbers of its components directly from the structure of the lotus (Proposition 4.5). Specifically, the order of petal attachment corresponds to the order of blowups, and the number of petals adjacent to a vertex determines the negative of the self-intersection number of the corresponding exceptional divisor.

Results

The paper successfully constructs the lotus $\Lambda(b/a)$ for the pair $(4, 11)$ using both the continued fraction method and the anthyphairetic rectangle method, demonstrating their equivalence.
It establishes that the minimal embedded resolution of the curve $C_{a,b}$ is achieved by a sequence of blowups that exactly mirrors the anthyphairetic process on $(a, b)$ .
It proves that the dual graph of the exceptional divisor of the minimal embedded resolution is isomorphic to the boundary edges of the lotus (excluding the base vertices), and that self-intersection numbers can be derived from the local connectivity of the lotus.
The reverse algorithm (Proposition 6.1) is validated, showing that the sequence of coprime pairs can be recovered from the lotus structure without prior knowledge of the original integers.

Significance and Claims
The paper positions itself as an introductory article intended to make these concepts accessible to readers without prior expertise in algebraic geometry or singularity theory. Its primary significance lies in:

Unification: It unifies the arithmetic of the Euclidean algorithm (specifically the anthyphairetic view favored by David Fowler) with the geometric resolution of singularities through the visual language of lotuses.
Computational Utility: It illustrates how lotuses serve as "computational architectures" where complex invariants of singularities (such as intersection numbers and resolution sequences) can be computed through simple, iterative local operations on a simplicial complex.
Pedagogical Value: By using the specific case of $(4, 11)$ as a paradigmatic example, the author aims to clarify the general construction of lotuses and their relationship to blowup processes, offering a concrete alternative to the more abstract continued fraction constructions previously established in the literature.

The author explicitly notes that these results are not new discoveries of the underlying mathematics (which dates back to Max Noether and Enriques) but rather a new graphical formalism (the lotus and the anthyphairetic rectangle) that clarifies the relationship between the arithmetic and geometric processes.