The Pythagram Defined

The Pythagram Defined

Contributors over a 2.5 Year Period:
Tiago Hands (Final Construction), Carlos Luna-Mota (Egyptian Triangles), Andrzej Kukla (The Rhombus with Area of 3)
06 October 2025

Abstract

The Pythagram is a planar geometric structure derived from the 3 : 4 : 5 right triangle and the orthographic projection of a cube. Its construction through defined Cartesian coordinates reveals six interrelated Pythagorean sub-figures that encode proportional symmetry across multiple dimensions. This document formalises the coordinates, connections, and geometric properties of the Pythagram as a reproducible mathematical entity.

Coordinate Data

Group 1 — Central Core
A(16, 18), B(14, 18), C(12, 16), D(12, 14), E(14, 12), F(16, 12), G(18, 14), H(18, 16), I(15, 15), W(15, 17.5), X(12.5, 15), Y(15, 12.5), Z(17.5, 15)

Group 2 — Inner Square
L(15, 20), M(10, 20), N(10, 15), O(10, 10), P(15, 10), Q(20, 10), R(20, 15), S(20, 20)

Group 3 — Outer Shape
A₁(15, 30), A₂(5, 25), A₃(0, 15), A₄(5, 5), A₅(15, 0), A₆(25, 5), A₇(30, 15), A₈(25, 25)

Group 4 — Intermediate Ring
C₁(11, 22), C₂(8, 19), C₃(8, 11), C₄(11, 8), C₅(19, 8), C₆(22, 11), C₇(22, 19), C₈(19, 22)

Group 5 — Fractional Division Points
B₁(13⅓, 16⅔), B₂(13⅓, 13⅓), B₃(16⅔, 13⅓), B₄(16⅔, 16⅔)

Connection Set (Edges)

[A₁, M, R, A₈, A₁]
[A₁, A₂, N, S, A₁]
[A₂, L, O, A₃, A₂]
[A₃, M, P, A₄, A₃]
[A₄, N, Q, A₅, A₄]
[A₅, O, R, A₆, A₅]
[A₆, P, S, A₇, A₆]
[A₇, Q, L, A₈, A₇]
[A₄, A₈]
[A₂, A₆]

Geometric Properties

  • The structure is rotationally symmetric by 45° around the central point (15, 15).
  • Ratios between major and minor edges conform to the 3 : 4 : 5 triangle proportions, preserving Pythagorean scaling.
  • The figure can be inscribed within a 30×30 coordinate grid, enabling reproducible Cartesian construction.
  • When duplicated across six orientations (one per cube face), the complete 3D extension forms the Hexapythagram.

Definition Summary

The Pythagram is a two-dimensional geometric projection composed of five coordinate groups and ten principal connections. It represents a proportional lattice where each line and square derives from the 3 : 4 : 5 Pythagorean relationship, forming the foundational plane for the 3D Hexapythagram. Its geometry unites Euclidean principles of right-angled triangles with cube projection, producing an object that is simultaneously planar and spatial in its mathematical interpretation.

Keywords: Geometry · Pythagorean Theorem · Cube Projection · Coordinate Systems · Mathematical Symmetry · Geometric Constructions

The Pythagram

Figure 1 — The Pythagram Construction from Defined Coordinates and Connections.

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