Polytope - Wikiparency

# Polytope ## Microsim (three.js) <div class="microsim-player">  <p class="wt-pending"><strong>Microsim staged, not yet on the CDN.</strong> <code>Polytope.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> *The polytope is the geometry hiding beneath both 3D rendering and optimization: the feasible region of a [[Combinatorial_optimization|combinatorial-optimization]] problem is a convex polytope whose corners the simplex method walks, while a [[Graphics_processing_unit|GPU]] draws each of its faces as mesh triangles. In three dimensions a polytope is simply a [[Polyhedron|polyhedron]].* > A **polytope** is a shape with perfectly flat sides — like a cube's six squares — generalized to any number of dimensions. In two dimensions it is a polygon; in three, a polyhedron; the idea continues into four dimensions and beyond. This microsim shows the five most symmetric three-dimensional examples, the **Platonic solids**, and lets you pull their faces apart to see that each solid is nothing but identical flat polygons joined edge to edge. Slide the faces back together and the closed convex object reappears. ## About this microsim Pick a solid with the **Tetrahedron**, **Cube**, **Octahedron**, **Dodecahedron**, or **Icosahedron** button, then drag **Explode Faces** from 0 to 1.4 to push every polygonal facet outward from the center. At 0 the object is the closed, familiar solid; as the value grows the faces drift apart, so you can count them, read their shape — triangle, square, or pentagon — and see how neighboring faces were once hinged along shared edges. Exploding as you switch solids makes the pattern land before any formula does: the tetrahedron flowers into 4 triangles, the cube into 6 squares, the dodecahedron into 12 pentagons. That is the whole abstract idea made physical — a polytope is just a set of flat faces bounding a region. ## Related microsims - [[Polyhedron]] — a polyhedron is precisely a 3-polytope; this sim's five solids are its most symmetric cases. - [[Combinatorial_optimization]] — linear and integer programs optimize over a polytope; the simplex method walks its vertices. - [[Graphics_processing_unit]] — GPUs render polytopes as triangle meshes of vertices, edges, and faces. - Bounded set — a convex polytope is exactly a bounded intersection of finitely many half-spaces. - [[Set_theory]] — the half-space form constructs a polytope as an intersection of half-space sets. - [[Monte_Carlo_method]] — related ALGORITHM microsim - Process optimization — related ALGORITHM microsim - Square of opposition — related ALGORITHM microsim ## Links (Wikipedia order)  `10-cube` · `10-demicube` · `10-orthoplex` · `10-simplex` · `120-cell` · `16-cell` · `1_22_polytope` · `1_32_polytope` · `1_42_polytope` · `24-cell` · `2_21_polytope` · `2_31_polytope` · `2_41_polytope` · `3_21_polytope` · `4-polytope` · `4_21_polytope` · `5-cell` · `5-cube` · `5-demicube` · `5-orthoplex` · `5-simplex` · `6-cube` · `6-demicube` · `6-orthoplex` · `6-simplex` · `600-cell` · `7-cube` · `7-demicube` · `7-orthoplex` · `7-simplex` · `8-cube` · `8-demicube` · `8-orthoplex` · `8-simplex` · `9-cube` · `9-demicube` · `9-orthoplex` · `9-simplex` · `Abstract_polytope` · `Affine_space` · `Alicia_Boole_Stott` · `Apeirogon` · `Apeirotope` · `Arthur_Cayley` · `August_Ferdinand_Möbius` · `Bernhard_Riemann` · `Boundary_(topology)` · `Bounded_set` · `Bounding_volume` · `Branko_Grünbaum` · `Cambridge_University_Press` · `Canonical_form` · `Cayley–Dickson_construction` · `Chaim_Goodman-Strauss` · `Codimension` · `Complex_polytope` · `Computer_graphics` · `Configuration_(polytope)` · `Contractible_space` · `Convex_Polytopes` · `Convex_body` · `Convex_hull` · `Convex_polytope` · `Cosmology` · `Coxeter_group` · `Cross-polytope` · `Cube` · `Cubic_honeycomb` · `David_Richeson` · `Degrees_of_freedom` · `Demihypercube` · `Digon` · `Dimension` · `Dimension_(vector_space)` · `Dimension_of_an_algebraic_variety` · `Dodecahedron` · `Dover_Publications` · `Dual_polyhedron` · `E6_(mathematics)` · `E7_(mathematics)` · `E8_(mathematics)` · `Edge_(geometry)` · `Eight-dimensional_space` · `Equilateral_triangle` · `Eric_W._Weisstein` · `Euclidean_space` · `Euler's_Gem` · [[Euler_characteristic]] · `F4_(mathematics)` · `Face_(geometry)` · `Five-dimensional_space` · `Flag_(geometry)` · `Flat_(geometry)` · `Four-dimensional_space` · `Fractal_dimension` · `Free_module` · `G2_(mathematics)` · `Geoffrey_Colin_Shephard` · [[Geometry]] · [[George_Boole]] · [[German_language]] · `Günter_M._Ziegler` · `Half-space_(geometry)` · `Hausdorff_dimension` · `Henri_Poincaré` · `Hermann_Grassmann` · `Hexagon` · `Hilbert_space` · `Honeycomb_(geometry)` · `Hypercomplex_number` · `Hypercube` · `Hypercubic_honeycomb` · `Hyperplane` · `Hyperpyramid` · `Hyperrectangle` · `Hyperspace` · `Hypersurface` · `Icosahedral_honeycomb` · `Icosahedron` · `Imaginary_number` · `Inductive_dimension` · `Integer_matrix` · `Integral_polytope` · `Internal_and_external_angles` · `Internet` · `John_Horton_Conway` · `Kepler–Poinsot_polyhedron` · `Krull_dimension` · `Lebesgue_covering_dimension` · `Linear_programming` · `List_of_regular_polytopes` · `Ludwig_Schläfli` · `Manifold` · `MathWorld` · `Michael_Guy` · `Minkowski–Bouligand_dimension` · `Monogon` · `N-sphere` · `Octahedron` · `One-dimensional_space` · `Order-5_pentagonal_tiling` · `Partially_ordered_set` · `Pentagon` · `Pentagonal_polytope` · `Peter_McMullen` · `Platonic_solid` · `Polygon` · [[Polyhedron]] · `Polytrope` · `Projective_space` · [[Quantum_mechanics]] · `Regular_4-polytope` · `Regular_Polytopes_(book)` · `Regular_dodecahedron` · `Regular_icosahedron` · `Regular_octahedron` · `Regular_polygon` · `Regular_polytope` · `Regular_tetrahedron` · `Schläfli_symbol` · `Search_engine_(computing)` · `Semiregular_polytope` · `Seven-dimensional_space` · `Shape` · `Simple_Lie_group` · `Simplex` · `Simplicial_complex` · `Six-dimensional_space` · `Slack_variable` · `Spacetime` · `Square` · `Square_tiling` · [[Tessellation]] · `Tesseract` · `Tetrahedron` · `The_Symmetries_of_Things` · `Theoretical_physics` · `Three-dimensional_space` · `Topology` · `Toroid` · `Toroidal_polyhedron` · `Twistor_theory` · `Two-dimensional_space` · `Uniform_10-polytope` · `Uniform_1_k2_polytope` · `Uniform_2_k1_polytope` · `Uniform_5-polytope` · `Uniform_6-polytope` · `Uniform_7-polytope` · `Uniform_8-polytope` · `Uniform_9-polytope` · `Uniform_k_21_polytope` · `Uniform_polyhedron` · `Uniform_polytope` · `Vertex_(geometry)` · `Victor_Klee` · `Zero-dimensional_space` ## Overview In geometry, a **polytope** is a flat-sided object that generalizes the polygon (a 2-polytope) and the polyhedron (a 3-polytope) to any dimension; the general term is an *n*-polytope. The five regular convex polyhedra shown here were known to the ancient Greeks, associated with the classical elements in Plato's *Timaeus*, and proved to number exactly five in Book XIII of Euclid's *Elements*. The word *polytope* comes from the German *Polytop*, coined by Reinhold Hoppe and brought into English by Alicia Boole Stott; Ludwig Schläfli's 19th-century work extended the theory to higher dimensions and gave us the Schläfli symbol. Today polytopes are central to convex geometry, combinatorics, computer graphics, and optimization. ## How a polytope is built A convex polytope has two equivalent descriptions. The **vertex (V) form** is the convex hull of finitely many points — the smallest convex region containing them. The **half-space (H) form** is a bounded intersection of finitely many half-spaces, $P = \{x \in \mathbb{R}^n : Ax \le b\}$. The **Minkowski–Weyl theorem** proves these define the same bounded objects. A **face** is where the polytope meets a supporting hyperplane; faces range from 0-dimensional **vertices** and 1-dimensional **edges** up to the $(n-1)$-dimensional **facets**. Every convex 3-polytope obeys **Euler's formula** $V - E + F = 2$. | Solid | Schläfli {p,q} | V | E | F | Face | |---|---|---|---|---|---| | Tetrahedron | {3,3} | 4 | 6 | 4 | triangle | | Cube | {4,3} | 8 | 12 | 6 | square | | Octahedron | {3,4} | 6 | 12 | 8 | triangle | | Dodecahedron | {5,3} | 20 | 30 | 12 | pentagon | | Icosahedron | {3,5} | 12 | 30 | 20 | triangle | Here $\{p, q\}$ means *p*-sided faces with *q* of them meeting at each vertex. Regularity forces $(p-2)(q-2) < 4$, whose only solutions with $p, q \ge 3$ are these five. Dual pairs swap $p \leftrightarrow q$: cube ↔ octahedron, dodecahedron ↔ icosahedron, with the tetrahedron self-dual. ## Controls -> what each maps to | Control | Maps to | Range / values | Meaning | |---|---|---|---| | Explode Faces | Outward displacement of each face from the center | 0 – 1.4 | 0 = closed assembled solid; larger values separate the facets into an exploded view that reveals face count and shape | | Tetrahedron | Select solid $\{3,3\}$ | button | 4 triangles, 4 vertices, 6 edges; self-dual | | Cube | Select solid $\{4,3\}$ | button | 6 squares, 8 vertices, 12 edges; dual of the octahedron | | Octahedron | Select solid $\{3,4\}$ | button | 8 triangles, 6 vertices, 12 edges; dual of the cube | | Dodecahedron | Select solid $\{5,3\}$ | button | 12 pentagons, 20 vertices, 30 edges; dual of the icosahedron | | Icosahedron | Select solid $\{3,5\}$ | button | 20 triangles, 12 vertices, 30 edges; dual of the dodecahedron | ## Learning objective After playing, a learner can name the five Platonic solids, predict each one's vertex, edge, and face counts, confirm Euler's formula $V - E + F = 2$, and explain why exactly five regular convex polyhedra can exist. ## Limits and connections The sim shows only the regular convex 3-polytopes — not star (non-convex) polytopes, semiregular solids, or higher-dimensional ones such as the four-dimensional tesseract. Yet the same vertex–edge–face data is exactly what a [[Graphics_processing_unit|GPU]] stores as a mesh and what [[Combinatorial_optimization|linear and integer programming]] exploit: the simplex method walks the vertices of a feasible polytope toward an optimum, which must exist because a continuous objective on a closed, bounded polytope attains its extremes (extreme value theorem). Building the convex hull that wraps a point cloud is itself a core computational-geometry task, solvable in $O(n \log n)$ time in two and three dimensions. ## Poster & source <div class="microsim-fallback">  <p><em>Live microsim · <a href="https://wikitube-3d-microsims.netlify.app/Polytope.html">open full</a> · source: Microsims for Dissemination/ALGORITHM_microsims/Polytope.html</em></p> </div>  *Built to the [[WT!Three_js_Microsim_Master_Class|three.js Master Class]].*  ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Polytope) : [Wikitube](https://en.wikitube.io/wiki/Polytope) ## Previous hub tags Tree parent: [[Graph_theory]]. Legacy hubs: `ALGORITHM`. --- *Sources: 2 legacy notes. Minted wave 1, 2026-07-30 (v1.6 order).*