Geometric Bites

📐 What Is Graphjacking? Graphjacking is the creative act of using 2D graphing tools — such as Desmos or GeoGebra — to produce the illusion of 3D or higher-dimensional space. It turns a flat coordinate plane into a window for exploring depth, rotation, and perspective through pure mathematics. ⚙️ Definition Graphjacking is the process of taking a two-dimensional graphing system and manipulating equations to create 3D-like visualizations. It uses projection and trigonometric techniques to simulate a third dimension within the limits of a 2D plane. 🎨 Examples Drawing isometric cubes or dodecahedra on graph paper. Animating a rotating cube using trigonometric functions. Creating optical illusions such as the “Pringle surface.” 📚 Applications Education: Visualizing higher-dimensional concepts intuitively. Art: Designing 2D mathematical works that appear three-dimensional. Mathematics: Exploring projections, transformations, and geometry in creative wa...

🧭 A Primer for Cross Product Calculations Right Hand Rule for Cross Product. Source: https://commons.wikimedia.org/wiki/File:Right-hand_rule_for_cross_product.png 1. Basis Vectors î = (1, 0, 0), ĵ = (0, 1, 0), k̂ = (0, 0, 1) These are unit and mutually perpendicular vectors. 2. Dot Product (for reference) 𝐀 · 𝐁 = |𝐀| |𝐁| cos θ î · ĵ = ĵ · k̂ = k̂ · î = 0 î² = ĵ² = k̂² = 1 3. Definition of the Cross Product 𝐀 × 𝐁 = |𝐀| |𝐁| sin θ n̂ θ is the angle from 𝐀 to 𝐁. n̂ is a unit vector perpendicular to both 𝐀 and 𝐁. The direction of n̂ follows the right-hand rule (anticlockwise = positive, clockwise = negative). 4. Fundamental Basis Cross Products î × ĵ = k̂ ĵ × k̂ = î k̂ × î = ĵ Reversing the order changes the sign: ĵ × î = −k̂ k̂ × ĵ = −î î × k̂ = −ĵ And any vector crossed with itself is zero: î × î = ĵ × ĵ = k̂ × k̂ = 0 5. Expansion in Component Form 𝐀 = a₁ î + a₂ ĵ + a₃ k̂ 𝐁 = b₁ î + ...

3D Rotation Matrix Primer When you rotate a point (x, y, z) in 3D space, you are applying a transformation that changes its coordinates while keeping its distance from the origin the same. This transformation is done using a rotation matrix . 1. Rotation about the x-axis (angle α) Rotation around the x-axis keeps x fixed and rotates the (y, z) plane. (x', y', z') = ( x, y·cos(α) − z·sin(α), y·sin(α) + z·cos(α) ) 2. Rotation about the y-axis (angle β) Rotation around the y-axis keeps y fixed and rotates the (x, z) plane. (x', y', z') = ( x·cos(β) + z·sin(β), y, −x·sin(β) + z·cos(β) ) 3. Rotation about the z-axis (angle γ) Rotation around the z-axis keeps z fixed and rotates the (x, y) plane. (x', y', z') = ( x·cos(γ) − y·sin(γ), x·sin(γ) + y·cos(γ), z ) 4. Combining rotations To rotate a point around all three axes, we combine the rotations. The standard order used in our Desmos cube system is: R = Rz(γ) → R...

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...

Using a geometrical logo that can be reconstructed mathematically offers a variety of benefits, particularly in terms of design precision, versatility, and brand identity. Here's a detailed breakdown of these advantages: 1. Precision and Consistency Exact Reproduction : Mathematical definitions ensure that the logo can be reproduced exactly in any size or medium without distortion or loss of detail. Scalability : Geometrical designs based on mathematical constructs can scale seamlessly, maintaining their proportions and clarity from a tiny icon to a billboard. 2. Aesthetic Appeal Symmetry and Balance : Logos designed using geometric principles often incorporate symmetry and proportion, which are inherently pleasing to the human eye. Timelessness : Geometrical designs tend to feel clean and modern, yet timeless, making them less susceptible to trends. 3. Ease of Modification Flexible Adjustments : Since the logo's structure is based on mathematical principles, chang...