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On the Curved Patterns Seen in the Graph of PPTs

This paper demonstrates that the curved patterns observed in a graph of Primitive Pythagorean Triples are indeed parabolic curves that arise naturally from the underlying mathematical properties of the triples.

Original authors: James M. Parks

Published 2026-01-22

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: James M. Parks

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 you have a giant, invisible grid stretching out across a field. On this grid, you are plotting points that represent special right-angled triangles. These aren't just any triangles; they are Primitive Pythagorean Triples (PPTs). Think of these as the "purest" building blocks of right triangles, where the three side lengths (like 3, 4, and 5) don't share any common factors that could be divided out.

The paper by James M. Parks takes a look at a massive map of these triangles (up to sides of length 10,000) and asks a simple question: Why do the dots on this map look like they are forming curved lines and circles?

Here is the breakdown of the paper's findings, translated into everyday language:

1. The "Parabolic Train Tracks"

When you plot these triangles on a graph, the dots don't scatter randomly. They line up perfectly on parabolas (the U-shaped curves you might remember from math class, like the path of a thrown ball).

The Rule: Every single dot sits on a specific track. These tracks are defined by a simple formula involving a number called $d$ .
What is $d$ ? Imagine a triangle with sides $a$ $a$ , $b$ $b$ , and $c$ $c$ . If you subtract the two shorter sides ( $c - b$ $c - b$ ), you get a number $d$ $d$ .
- If $d = 1$ , the dots follow one specific U-shaped curve.
- If $d = 2$ , they follow a slightly different U-shaped curve.
- If $d = 8$ , they follow another.
The Surprise: Not every number works as a $d$ . The paper explains that only "special" numbers (like 1, 2, 8, 9, 18, 25...) create valid tracks. If you try to use a number like 3 or 4, the dots won't form a "pure" triangle; they will be messy duplicates of other triangles. The paper identifies a specific sequence of "allowed" numbers (found in the OEIS database as A096033) that act as the keys to these tracks.

2. The "Mirror Image" Effect

The graph is perfectly symmetrical. If you draw a line down the middle (where $x = y$ ), the left side is a mirror image of the right side.

If a triangle has sides $(3, 4, 5)$ , you plot the point $(3, 4)$ .
Because the triangle is the same if you swap the legs, you also plot $(4, 3)$ .
This creates two sets of parabolic tracks: some opening upward (like a bowl) and some opening sideways (like a sideways bowl).

3. The "Downward" Curves

So far, we've talked about curves opening up. But the paper discovers a second set of curves that open downward (like an upside-down bowl).

How they work: These curves are determined by the "parent" of a triangle. If you have a complex triangle, you can often trace it back to a simpler, "root" triangle. The downward curves connect these related triangles.
The Intersection: Here is the magic part: Every single dot on the map sits at the exact intersection of two curves—one opening up and one opening down.
The Right Angle: The paper proves that at every single dot, these two crossing curves meet at a perfect 90-degree angle. It's like a street intersection where one road goes North-South and the other goes East-West, but the roads are curved.

4. The "Starbursts" and "Fake Circles"

When you zoom out and look at the whole map, the dots start to look like they are forming circles or starbursts (like a sun with rays).

The Illusion: The paper clarifies that these are not perfect circles. If you measure them with a ruler, they are slightly off.
The Reality: These "circles" are actually clusters of dots that happen to land on several different parabolic tracks at the same time.
The Starburst: Imagine a central point where four different tracks cross (two opening up, two opening down). Around this center, the dots form a ring. Because the tracks are curved and intersect at right angles, the ring looks like a circle with "spokes" radiating out. The paper shows that these patterns are just the result of these specific parabolic tracks weaving together.

Summary

The paper is essentially a detective story about a pattern.

The Mystery: Why do these triangle numbers form curved lines and circle-like shapes?
The Clue: The numbers are following strict mathematical rules based on the difference between their sides ( $d$ ).
The Solution: The "lines" are actually parabolas. The "circles" are just where these parabolas cross each other at right angles.
The Conclusion: The beautiful, complex patterns seen in the graph are not random art; they are the natural, inevitable result of the math behind right-angled triangles. The "curved patterns" are simply the tracks these numbers are forced to run on.

Technical Summary: On the Curved Patterns in the Graphs of PPTs

Problem Statement
The paper addresses the visual patterns observed in discrete plots of Primitive Pythagorean Triples (PPTs). Specifically, it investigates the "amazing" graph constructed by R. Knott, which plots all PPTs $(a, b, c)$ with legs $a, b < 10,000$ in the first quadrant. While the plot reveals distinct curved structures resembling conic sections, the paper seeks to mathematically characterize these curves, determine their exact equations, and explain the underlying number-theoretic constraints that govern the distribution of PPTs along these paths.

Methodology
The author employs a combination of geometric analysis and number theory. The methodology involves:

Geometric Mapping: Representing each PPT $(a, b, c)$ as an ordered pair $(a, b)$ in the Cartesian plane, where $a < b < c$ .
Parametric Derivation: Analyzing the relationship between the legs and the hypotenuse by defining the difference $d = c - b$ and $d' = c - a$ . The paper derives the equations of curves that contain points sharing a constant difference $d$ .
Parabolic Classification: Identifying that the points lie on specific families of parabolas. The author distinguishes between parabolas opening toward the positive axes (derived from $d = c - b$ ) and those opening toward the negative axes (derived from specific divisibility properties of the coordinates).
Intersection and Orthogonality Analysis: Examining the intersection points of these parabolas to verify that PPTs occur at the intersections of specific curves and proving that these intersecting curves are orthogonal (perpendicular) at the PPT points.
Sequence Verification: Correlating the allowable values of $d$ that generate PPTs with the OEIS sequence A096033.

Key Contributions and Results

Parabolic Equations for PPTs: The paper establishes that PPTs with a fixed difference $d = c - b$ lie on the parabola defined by $y = \frac{x^2 - d^2}{2d}$ . Similarly, PPTs with a fixed difference $d' = c - a$ lie on the reflection $x = \frac{y^2 - d'^2}{2d'}$ .
- For $d=1$ (where $c=b+1$ ), the equation is $y = \frac{x^2 - 1}{2}$ .
- For $d=2$ (where $c=b+2$ ), the equation is $y = \frac{x^2 - 4}{4}$ .
- The paper generalizes this to arbitrary allowable $d$ , showing that the curves are parabolas with their focus at the origin $(0,0)$ .
Allowable Values of $d$ : The paper identifies that not all integer values of $d$ generate PPTs. It proves that for a PPT to exist on a curve with difference $d$ , $d$ must belong to the set of "allowable values" found in OEIS sequence A096033. These values are:
- Squares of odd positive integers ( $1, 9, 25, 49, \dots$ ).
- Twice the squares of positive integers ( $2, 8, 18, 32, 50, \dots$ ).
- The paper demonstrates that if $d$ does not fit these forms, any resulting triple $(a, b, c)$ will share a common factor, thus failing the "primitive" condition.
Secondary Parabolas (Negative Axes): The paper identifies a second set of parabolic curves that open toward the negative $x$ and $y$ axes. These curves are determined by a single PPT point $(a, b)$ and are related to the "base" PPT $(a_1, b_1)$ on the $d=1$ (or $d=2/8$ ) parabola. The equations take the form $y = -\frac{x^2}{2a_1^2/d} + \frac{a_1^2}{2d}$ . These curves also have their foci at the origin.
Orthogonality: A significant result is Proposition 3, which states that at any PPT plot point $(a, b)$ , the parabola opening toward the positive axis and the parabola opening toward the negative axis intersect at right angles. The product of their slopes at the intersection point is $-1$.
Circular and Starburst Patterns: The paper analyzes the "circular" and "starburst" patterns visible in the main graph. It clarifies that these are not perfect circles but are composed of discrete points lying on intersecting sets of parabolic curves.
- Starbursts: These occur near points where multiple parabolas (both positive and negative opening) intersect. The "spokes" of the starburst are segments of these parabolas.
- Circular Clusters: The paper notes that clusters of 8 points often appear to form circles. However, it provides geometric proof (via perpendicular bisectors) that these points do not lie on a single perfect circle. Instead, the "circular" appearance arises because the points lie on the intersection of multiple parabolic families with specific spacing constraints (e.g., $a \equiv s \pmod{2s}$ ).

Significance
The paper claims that the curved patterns observed in R. Knott's graph are not merely visual artifacts but are intrinsic to the mathematics of Pythagorean Triples. The primary significance lies in:

Mathematical Explanation: Providing the exact parabolic equations that govern the distribution of PPTs, moving from observation to rigorous derivation.
Geometric Structure: Revealing that the graph of PPTs is a lattice of intersecting orthogonal parabolas, all sharing a common focus at the origin.
Construction Utility: Demonstrating that these parabolic properties can be used to construct the graph of PPTs systematically.
Clarification of Patterns: Distinguishing between true geometric circles and the "pseudo-circular" patterns formed by the discrete intersection of parabolic families, thereby demystifying the visual complexity of the PPT graph.

The paper concludes that these results are consistent with historical knowledge (referencing Proclus and the Pythagoreans regarding specific triple forms) but extends this to a comprehensive geometric framework for all PPTs.

1. The "Parabolic Train Tracks"

2. The "Mirror Image" Effect

3. The "Downward" Curves

4. The "Starbursts" and "Fake Circles"

Summary

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