Tessellation - Wikiparency

# Tessellation  ## Microsims — three.js ### Tessellation (three.js) <div class="microsim-player"> <iframe src="https://wikitube-3d-microsims.netlify.app/Tessellation_microsim.html" width="100%" height="620" frameborder="0" loading="lazy" sandbox="allow-scripts allow-same-origin" title="Tessellation — three.js microsim"></iframe> </div> *Tiling the plane with repeating patterns that share edges with no gaps or overlaps.* **Open it full-screen:** [Tessellation_microsim.html](https://wikitube-3d-microsims.netlify.app/Tessellation_microsim.html) · library `threejs` · route `microsim/threejs/` ### Related microsims Live sims on neighbouring articles: - [[Euler_characteristic]] *Sim hosted off-article; the article owns the reference, not the runtime (WIKI_RULES §10.4). Placed by `g08_place_microsims.py`.*  > **Two builds exist for this concept.** The player above is the build that is actually served by the CDN. A second, richer build (`Tessellation.html`) is written up further down but is still staged — its player is disabled until the deploy lands. §5: these promote into one shelf, they are not forks.  ## Microsim (three.js) Canonical file: `Three-JS-MicroSims/Tessellation.html` (self-contained, three.js r180 import map, built to THREE JS SWEEP STANDARD / Three js Microsim Master Class). **Live:** [wikitube-3d-microsims.netlify.app/Tessellation.html](https://wikitube-3d-microsims.netlify.app/Tessellation.html) **Description (100 words):** Two steppers set p, the sides of each regular tile, and q, how many meet at every corner. The HUD computes (p−2)(q−2) live and the scene rebuilds to the verdict: below 4 the tiles close into a sphere — one of the five Platonic solids, drawn as a spherical tiling with great-circle edges; exactly 4 and the pattern lies flat — the square, triangular, or hexagonal tabletop; above 4 it overflows into the Poincaré disk, whole polygons crowding an infinite rim. One inequality, three geometries — the whole classification in two small integers. Spin and Speed included, keyboard orbit, reduced motion always honored. <div class="microsim-player">  <p class="wt-pending"><strong>Microsim staged, not yet on the CDN.</strong> <code>Tessellation.html</code> is built and deploy-ready in <code>Microsims for Dissemination/</code>, but the Netlify project still serves the geometry+spintronics set only. The player is disabled until the deploy lands; the explanatory text below is unchanged.</p>   </div> | Control | Range | Effect | |---|---|---| | p stepper | 3–8 sides | polygon of each tile; verdict and world rebuild live | | q stepper | 3–8 at a vertex | how many tiles share each corner | | HUD map strip | — | (p−2)(q−2) vs 4 with the active world named: sphere (5) · plane (exactly 3) · hyperbolic (∞) | | Spin + Speed | 0.05–1.0 | fixed-timestep turntable rotation | | drag / arrow keys | — | orbit (keyboard path included) | **Poster:** `Three-JS-MicroSims/Tessellation.png` — the {5,3} dodecahedron as a spherical tiling. **Verified:** world label asserted headless for {5,3}→sphere, {4,4}→plane, {3,6}→plane, {7,3}→hyperbolic; all 36 {p,q} pairs (3–8 × 3–8) rebuild with verdicts matching the formula; zero pageerrors and zero console messages; the hyperbolic tilings are built by genuine (2,p,q) mirror reflections on the Minkowski hyperboloid, capped at ~500 triangles of whole faces, then projected to the Poincaré disk. ## Links (Wikipedia order)  `3-7_kisrhombille` · `4-5_kisrhombille` · `A_Mathematician's_Apology` · `A_Treatise_on_Painting` · `Adolf_Zeising` · `Alan_Turing` · `Alba_Rojo_Cama` · `Albrecht_Dürer` · `Alexei_Vasilievich_Shubnikov` · [[Algorithm]] · `Algorithmic_art` · `Alhambra` · `Alternated_octagonal_tiling` · `Alternated_order-4_hexagonal_tiling` · `American_Mathematical_Society` · `Ammann–Beenker_tiling` · `Anamorphosis` · `Anatoly_Fomenko` · `Ancient_Rome` · `Andradite` · `Andrea_Palladio` · `Anduriel_Widmark` · `Andy_Lomas` · `Anisohedral_tiling` · `Annals_of_Mathematics` · `Antakya` · `Anthony_Hill_(artist)` · `Antiprism` · `Antoine_Pevsner` · `Apeirogonal_antiprism` · `Apeirogonal_hosohedron` · `Apeirogonal_prism` · `Aperiodic_set_of_prototiles` · `Aperiodic_tiling` · `Archeological_Museum_of_Seville` · `Architectonic_and_catoptric_tessellation` · `Architecture_of_Mesopotamia` · `Aristid_Lindenmayer` · `Ars_Mathematica_(organization)` · 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`Order-6_square_tiling` · `Order-7_triangular_tiling` · `Order-8_triangular_tiling` · `Origami` · `Owen_Jones_(architect)` · `Oxford_University_Press` · `Packing_problems` · `Pantheon,_Rome` · `Paolo_Uccello` · `Paraboloid` · `Parallelogram` · `Parastichy` · `Parmigianino` · `Parthenon` · `Pattern` · [[Pattern_formation]] · `Pattern_recognition_(psychology)` · `Patterns_in_nature` · `Pavers_(flooring)` · [[Penrose_tiling]] · `Pentagonal_tiling` · `Pentahexagonal_tiling` · `Perfect_rectangle` · `Permutation` · `Perspective_(graphical)` · `Peter_Forakis` · `Philip_Ball` · `Phyllotaxis` · `Physical_Review_Letters` · [[Physics]] · `Pi_(art_project)` · `Pi_in_the_Sky` · `Piero_della_Francesca` · `Pinwheel_tiling` · `Plane_(mathematics)` · `Plastic_ratio` · `Plateau's_laws` · `Plato` · `Plesiohedron` · `Poincaré_disk_model` · `Polygon` · [[Polyhedron]] · `Polyiamond` · `Polykleitos` · `Polyomino` · `Polyominoes:_Puzzles,_Patterns,_Problems,_and_Packings` · `Princeton_University_Press` · 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`Truncated_infinite-order_triangular_tiling` · `Truncated_octagonal_tiling` · `Truncated_octahedron` · `Truncated_order-3_apeirogonal_tiling` · `Truncated_order-4_apeirogonal_tiling` · `Truncated_order-4_heptagonal_tiling` · `Truncated_order-4_hexagonal_tiling` · `Truncated_order-4_octagonal_tiling` · `Truncated_order-4_pentagonal_tiling` · `Truncated_order-5_hexagonal_tiling` · `Truncated_order-5_pentagonal_tiling` · `Truncated_order-5_square_tiling` · `Truncated_order-6_hexagonal_tiling` · `Truncated_order-6_octagonal_tiling` · `Truncated_order-6_pentagonal_tiling` · `Truncated_order-6_square_tiling` · `Truncated_order-7_heptagonal_tiling` · `Truncated_order-7_square_tiling` · `Truncated_order-7_triangular_tiling` · `Truncated_order-8_hexagonal_tiling` · `Truncated_order-8_octagonal_tiling` · `Truncated_order-8_triangular_tiling` · `Truncated_pentahexagonal_tiling` · `Truncated_square_tiling` · `Truncated_tetrahexagonal_tiling` · `Truncated_tetrapentagonal_tiling` · `Truncated_triheptagonal_tiling` · `Truncated_trihexagonal_tiling` · `Turing_machine` · `Uniform_coloring` · `Uniform_honeycomb` · `Uniform_honeycombs_in_hyperbolic_space` · `Uniform_polyhedron` · `Uniform_tiling` · `Uniform_tilings_in_hyperbolic_plane` · `University_of_Wollongong` · `Uruk` · `Vastu_shastra` · `Vertex_(geometry)` · `Vertex_configuration` · `Viewpoints:_Mathematical_Perspective_and_Fractal_Geometry_in_Art` · `Vitruvian_Man` · `Vitruvius` · `Voderberg_tiling` · `Voronoi_diagram` · `Wallpaper_group` · `Wang_tile` · `Waterfall_(M._C._Escher)` · [[Wave]] · [[Wayback_Machine]] · `Weaire–Phelan_structure` · `Widmanstätten_pattern` · `William_Blake` · `Wilson_Bentley` · `Woodcut` · `Wythoff_construction` · `Zellij` > **Room:** [[Geometry]] · **Status:** ✅ validated · three.js flagship · ← back to GEOMETRY PORTAL ## Overview A tessellation covers a surface with tiles — no gaps, no overlaps — and the strictest version of the game is the *regular* tiling: one kind of regular polygon, arranged identically at every corner. Write it in Schläfli shorthand as {p,q}: p-sided tiles, q of them meeting at each vertex. Whether such a pattern can exist at all comes down to corner bookkeeping. A regular p-gon carries an interior angle of 180°·(p−2)/p, and q of those angles have to share each vertex. Compare their total to the 360° a flat corner offers and the whole classification compresses into one product: (p−2)(q−2) measured against 4. Below, equal, above — three verdicts, and not one candidate {p,q} escapes the trichotomy. The verdict is curvature in disguise. When (p−2)(q−2) < 4 the corners fall short of 360°, the surplus angle forces the pattern to curl, and it closes into a sphere: exactly five solutions, and they are precisely the Platonic solids — {3,3}, {4,3}, {3,4}, {5,3}, {3,5} — each a spherical tiling wearing a polyhedron's name. When the product equals 4 the corners fit perfectly and the pattern lies flat forever: exactly three regular tilings of the Euclidean plane, the {4,4} grid of squares, the {3,6} field of triangles, the {6,3} honeycomb. (Loosen the rules to allow several polygon types with identical vertices and the plane grants just eight more — the semiregular, or Archimedean, tilings.) And when the product exceeds 4 the corners overflow: no sphere, no flat plane can hold the pattern, but the hyperbolic plane holds them all — infinitely many regular tilings, the ones M. C. Escher compressed into his *Circle Limit* woodcuts. Humans tiled long before they classified. The zellige craftsmen of the Alhambra explored an astonishing share of the seventeen Wallpaper group symmetries in cut tile, centuries before group theory named them, and Escher walked out of Granada with notebooks that became a career of interlocking birds and fish. From here two doors open wide: Hyperbolic geometry, where the microsim's overflowing tilings live natively, and [[Penrose_tiling]] — the lawless cousin, tiles that cover the plane completely yet can never repeat. ## The trichotomy at a glance | (p−2)(q−2) | World | Regular tilings | Examples | |---|---|---|---| | < 4 | Sphere — positive curvature | exactly 5 | {3,3} tetrahedron, {4,3} cube, {3,4} octahedron, {5,3} dodecahedron, {3,5} icosahedron | | = 4 | Flat plane — zero curvature | exactly 3 | {4,4} squares · {3,6} triangles · {6,3} hexagons (plus 8 semiregular when mixing tiles) | | > 4 | Hyperbolic plane — negative curvature | infinitely many | {7,3} heptagons, {4,5}, {8,8}, … Escher's *Circle Limit* country | ## See also - Read on Wikipedia: [Tessellation](https://en.wikipedia.org/wiki/Tessellation) - On Wikitube: https://en.wikitube.io/wiki/Tessellation - Room: GEOMETRY PORTAL · siblings [[Penrose_tiling]], Hyperbolic geometry, Wallpaper group, Regular polygon - Three.js library page: THREE JS GEOMETRY MICROSIMS --- *Built by the GEOMETRY_PORTAL sweep (S-geosweep-016) from row 16 of the worklist, 2026-07-23.*  *Built to the [[WT!Three_js_Microsim_Master_Class|three.js Master Class]].*  ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Tessellation) : [Wikitube](https://en.wikitube.io/wiki/Tessellation) ## Previous hub tags Tree parents: [[Cellular_automaton]] · [[Self-organization]]. Legacy hubs: `GENERATIVE`. --- *Sources: 1 legacy note. Minted wave 1, 2026-07-30 (v1.6 order).*