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...

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...