Geometric Bites

The Pythagorean Hexagon — Proof and Definition © Tiago Hands — 18 October 2025 (UTC +8) Definition and Construction Six points A, B, C, D, E and F form a hexagon whose sides are all equal in length. Point G lies at the centre of a circle with radius AB = x > 0. Points F, B and D lie on the circumference of this circle, forming a right-angled triangle at G where angle FGB = 90°. Quadrilateral ABGF forms a square. Quadrilaterals FGDE and BCDG form rhombuses that share the same side length x. The internal angles satisfy α + β = 270°. Each figure has a centre: H for square ABGF, I for rhombus BCDG, and J for rhombus FGDE. When these three figures are joined edge-to-edge around the circle, their areas obey a Pythagorean relation. Areas of the Component Figures Let AB = x. Square ABGF: A 1 = x 2 Rhombus FGDE: A 2 = 2x 2 cos(β/2)sin(β/2) = x 2 sin(β) Rhombus BCDG: A 3 = 2x 2 cos(α/2)sin(α/2) = x 2 sin(α) Algebraic and Trigonometric Proof We aim to show A...

🜛 The Unicursal Pythagram 08 October 2025 The unicursal pythagram is a geometric figure drawn on a 6×6 grid . It is made from sixteen points connected by thirty-two straight lines , and the whole pattern can be drawn without lifting the pen . That is what makes it unicursal — it is one continuous path that begins and ends at the same point. The figure is related to the 3:4:5 right triangle , just like the standard pythagram. But while the standard version is about static shapes, the unicursal version adds movement. It shows how all the parts of the pattern connect together in a single, flowing line. Points used (A–P) A(2,2), B(3,4), C(4,2), D(2,3), E(4,4), F(3,2), G(2,4), H(4,3), I(0,3), J(1,5), K(5,5), L(6,3), M(3,0), N(1,1), O(3,6), P(5,1) Drawing order (32 lines) A→B, B→C, C→D, D→E, E→F, F→G, G→H, H→A, A→I, I→J, J→B, B→K, K→L, L→C, C→M, M→N, N→D, D→J, J→O, O→E, E→L, L→P, P→F, F→N, N→I, I→G, G→O, O→K, K→H, H→P, P→M, M→A In simple terms The unicursal pythagram shows...

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...