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Geometry

Parametric surfaces, Voronoi tessellations and quantum-mechanical orientations — geometry as the language of form and space.

10+ simulations Three.js · WebGL · GLSL Voronoi · L-System · Quaternion

Category Simulations

Simulations in development — stay tuned

Parametric geometry — describing shapes through parameter functions (u, v). Torus, bread surface, Boy surface — any smooth surface can be built with two parametric equations and GPU tessellation.

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★★☆ Moderate
Torus Knot
Reshape a (p,q) torus knot with iridescent fresnel shading — change p, q and tube thickness.
Three.js GLSL Geometry
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★★★ Advanced
Menger Sponge — 3D Fractal
The canonical 3D fractal: subdivide each cube into 27 children and keep 20, recurse. Rendered as a GLSL raymarched SDF with iteration level 0–5 and Hausdorff dimension log20/log3.
Three.js GLSL SDF Fractal
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★★★ Advanced
Delaunay & Voronoi
Compute the Delaunay triangulation (Bowyer–Watson) of a draggable point set live, with the Voronoi dual overlaid. Hover a triangle to see its empty circumcircle — the defining Delaunay property.
Canvas 2D Delaunay Voronoi Bowyer-Watson
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★★★ Advanced
Bézier, B-Spline & NURBS
The parametric curves behind every vector graphic and font. Drag control points, watch the de Casteljau scaffold sweep, adjust spline degree and NURBS weights, and build an exact circle.
Canvas 2D Bezier NURBS de Casteljau
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★★★ Advanced
Geodesic Dome — Icosahedral Subdivision
Build Buckminster Fuller geodesic spheres by subdividing an icosahedron and projecting onto the sphere. Vary the frequency, switch to a dome, and verify Euler's formula V−E+F=2.
Canvas 2D Polyhedron Icosahedron Euler
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★★★ Advanced
Sphere & Circle Packing
How densely can equal circles and spheres fill space? Compare square, hexagonal and random packings and FCC/HCP/BCC/SC lattices with live packing fractions and kissing numbers.
Canvas 2D Packing Density Lattices FCC/HCP
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★★★ Advanced
Gyroid & Minimal Surfaces
Triply-periodic minimal surfaces in 3D — Gyroid, Schwarz P, Diamond and Neovius — rendered live with marching cubes. Adjust isovalue, resolution and tiling, then orbit the structure.
Three.js TPMS Marching Cubes Minimal Surface
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★☆☆ Beginner
Spirograph
Hypocycloid and epicycloid curves traced by a point on a rolling circle. Adjust radii and offset to generate Lissajous-like star and petal patterns.
Canvas 2D Cycloid Parametric
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★☆☆ Beginner
Sierpiński Triangle
Fractal self-similar triangle generated by the Chaos Game: pick a random vertex and jump halfway — the attractor reveals the Sierpiński pattern.
Canvas 2D Fractal Chaos Game
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★★☆ Moderate
Fractal Explorer
GPU-accelerated Mandelbrot and Julia set explorer. Zoom infinitely into the boundary of the Mandelbrot set using GLSL escape-time colouring.
GLSL Mandelbrot Complex Plane
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★☆☆ Beginner
Number Spirals
Ulam spiral and Sacks spiral reveal hidden prime patterns. Visualise how integers arranged on a spiral expose number-theoretic structure.
Canvas 2D Primes Ulam Spiral
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★☆☆ Beginner
Kaleidoscope
Symmetry groups in action: n-fold mirror reflections generate kaleidoscopic patterns. Change fold count and seed geometry in real time.
Canvas 2D Symmetry Reflection
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★★☆ Moderate
L-Systems
Lindenmayer string-rewriting grammars rendered as turtle graphics. Grow ferns, trees, fractals and space-filling curves from simple rules.
Canvas 2D L-System Turtle Graphics
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★★☆ Moderate
Mechanisms & Linkages
Four-bar linkages, Peaucellier cell, Watt’s linkage and Chebyshev’s mechanism. Rigid-body kinematics with constraint solving.
Canvas 2D Kinematics Linkages
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★☆☆ Beginner
Pythagoras Tree
Recursive fractal tree built from squares following the Pythagorean theorem. Adjust branch angles to morph between symmetric and asymmetric growth forms.
Canvas 2D Recursion Fractal
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★☆☆ Beginner
Lissajous Curves
Parametric curves from two sinusoids at different frequencies and phases. Trace the beautiful loops and figure-eights that appear on oscilloscopes and pendulum tables.
Canvas 2D Parametric Math Art
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★☆☆ Beginner
Fibonacci Spiral
Sunflower phyllotaxis and golden ratio spirals. See how 137.5° (the golden angle) produces perfect packing with Fibonacci numbers.
Canvas 2D Golden Ratio Nature
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New ★★☆ Moderate
Tessellations
Regular tilings (triangular, square, hexagonal) and Penrose kite-dart aperiodic tilings. Explore symmetry groups with pan and zoom.
Canvas 2D Penrose Tiling
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New ★★★ Advanced
Quaternions & Rotation
Side-by-side Euler angles vs quaternion rotation. See gimbal lock in action, smooth SLERP interpolation, and compare 3D orientation methods.
Canvas 2D Quaternion SLERP
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New ★★★ Advanced
Parametric Surfaces
Torus, Klein bottle, Möbius strip, sphere and hyperboloid. Interactive wireframe rendering on Canvas 2D with adjustable mesh density.
Canvas 2D Parametric Topology
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New ★★★ Advanced
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New ★★★ Advanced
Topology Surfaces
Topological surfaces: genus, Euler characteristic, Klein bottle and Mobius strip interactively explored.
topology genus Euler characteristic
Origami Folding
Miura-ori, Yoshimura and Kresling crease patterns with animated step-by-step rigid folding on Canvas 2D.
Canvas 2D Origami Rigid Folding

Key Concepts

Geometry as the language of form and space

Parametric Curves
Points defined by a parameter t: x=f(t), y=g(t). Spirograph traces hypo/epicycloids via two radius parameters. More expressive than y=f(x) — can describe closed curves and self-intersecting forms.
Fractal Dimension
Fractals have non-integer Hausdorff dimension. Sierpiński triangle: d = log3/log2 ≈ 1.585. The Mandelbrot boundary has dimension 2. Self-similarity at all scales is the defining property.
L-System Grammar
String rewriting: axiom + production rules applied repeatedly. Turtle interprets the result as draw commands (F=forward, +=turn, [=push, ]=pop). Captures the recursive branching structure of plants.
Symmetry Groups
A set of transformations (rotations, reflections) closed under composition. Kaleidoscopes use dihedral group D_n. There are exactly 17 distinct wallpaper groups for 2D periodic tilings of the plane.

Learning Resources

Articles and tutorials about the algorithms in this category

Article Parametric Surfaces in Three.js Equations of torus and hyperboloid. BufferGeometry via parameter (u, v). Normal-mapping. → Article Voronoi Diagram: from 2D to Sphere Fortune's algorithm. Event queue structure. Voronoi-Delaunay on a spherical surface. → Article Quaternions Without Fear Complex numbers in R⁴. Gimbal lock in practice. SLERP for smooth animation. →
All articles →

About Geometry Simulations

Symmetry, curves, patterns, and space — explored visually

Geometry simulations explore the visual and mathematical properties of shapes, symmetry, and spatial structures. Chladni pattern simulations vibrate a virtual plate at resonant frequencies, revealing the nodal lines where sand accumulates — the same physics Chladni demonstrated to Napoleon. Spirograph generators trace hypotrochoid and epitrochoid curves from coupled rotating circles, producing thousands of distinct Lissajous-like figures.

Kaleidoscope, Sierpiński triangle, and L-system simulations demonstrate how reflection symmetry, recursive subdivision, and rewriting rules produce endlessly complex patterns from elementary operations. These visualisations connect geometry to art, architecture, crystallography, and computer graphics. Adjusting the ratio of circle radii, fractal depth, or reflection count gives immediate feedback on how geometry parameters govern visual outcomes.

Geometric simulations connect pure mathematics to the real world. Voronoi diagrams appear in city planning (nearest hospital routing), materials science (grain boundary modelling), and computer graphics (texture synthesis). The Delaunay triangulation is the backbone of finite-element mesh generation. Lissajous curves are used in oscilloscope calibration and musical harmony analysis. Exploring these interactively makes abstract geometry immediately applicable.

Key Concepts

Topics and algorithms you'll explore in this category

Voronoi DiagramsNearest-neighbour space partitioning
Delaunay TriangulationDual graph of Voronoi; maximises minimum angles
Spirograph / EpitrochoidParametric curves from rolling circles
Pythagoras TreeRecursive right-triangle fractal
Sierpiński TriangleIFS fractal via chaos game
Lissajous CurvesParametric curves from perpendicular oscillations

Frequently Asked Questions

Common questions about this simulation category

What is a Voronoi diagram?
A Voronoi diagram partitions a plane into regions, each containing all points closer to one seed point than to any other. The construction is the dual of the Delaunay triangulation. Voronoi diagrams model natural phenomena like cell division, territorial boundaries, and crystal grain structures.
How are Spirograph curves generated?
Epitrochoids and hypotrochoids are traced by a point on a circle rolling around another circle. By varying the ratio of radii and the pen distance, you can generate any Spirograph pattern. The parametric equations are x(t) = (R-r)cos(t) + d·cos((R-r)t/r).
What makes Sierpiński Triangle a fractal?
The Sierpiński Triangle has Hausdorff dimension log(3)/log(2) ≈ 1.585 — more than a line but less than a plane. It can be generated by the chaos game (random mid-point iteration), by Pascal's triangle mod 2, or by iterating an IFS (Iterated Function System) with three contraction maps.
What are parametric surfaces and how are they rendered?
Parametric surfaces are defined by functions x(u,v), y(u,v), z(u,v) over a 2D parameter domain. The simulation samples a grid of (u,v) values, evaluates the positions, and renders the mesh using Three.js with lighting and normal mapping. Torus, Klein bottle, Möbius strip, and minimal surfaces such as the catenoid are all parametric examples you can explore.

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