Polyhedron - Wikiparency

# Polyhedron ## Microsim (three.js) <div class="microsim-player">  <p class="wt-pending"><strong>Microsim staged, not yet on the CDN.</strong> <code>Polyhedron.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> *A [[Polytope|polytope]] you can hold in your hand: in the ALGORITHM hub the polyhedron is the working solid of computational geometry — the triangulated mesh a [[Graphics_processing_unit|GPU]] rasterizes to pixels, and the feasible region a [[Combinatorial_optimization|linear program]] searches for an optimum.* > A polyhedron is a three-dimensional shape whose surface is built entirely from flat polygon **faces**, joined along straight **edges** that meet at sharp corners called **vertices**. Everyday examples include the cube, the triangular pyramid, and the highly symmetric Platonic solids. Because a polyhedron is captured completely by *which* vertices, edges, and faces touch one another, it is the object that 3D graphics, CAD, and optimization software actually store and compute with. This microsim draws one in an interactive 3D canvas: drag to orbit it in space, and use the on-screen controls to swap between solids and watch how their faces, edges, and vertices fit together. ## About this microsim The sim shows a single polyhedron in a 3D canvas that you inspect by **mouse-orbit** — click and drag to spin it to any angle, so hidden faces rotate into view and you can trace an edge all the way around. The **on-screen controls** let you choose which polyhedron is displayed, cycling through solids such as the five Platonic solids. Because you *see* the solid before any formula is stated, the abstract triple $(V, E, F)$ becomes concrete: orbit a cube and count its 8 corners, 12 edges, and 6 faces, then switch to the octahedron and watch those roles swap. Rotation is the whole point — it turns a flat silhouette back into an unambiguous solid whose structure you can read off directly. ## Related microsims - [[Polytope]] — the $n$-dimensional generalization of a polyhedron - [[Graphics_processing_unit]] — hardware that rasterizes polyhedral meshes into pixels - Raster graphics — turning polygon faces into the pixels on screen - [[Combinatorial_optimization]] — linear programs whose feasible region is a convex polyhedron - Bounded set — a bounded polyhedron is exactly a polytope - [[Expected_value]] — related ALGORITHM microsim - Field of view — related ALGORITHM microsim ## Links (Wikipedia order)  `4-polytope` · `A_History_of_Folding_in_Mathematics` · `Abstract_polytope` · `Academic_Press` · `Affine_space` · `Albrecht_Dürer` · `Aleksandr_Aleksandrov_(mathematician)` · `Algebraic_variety` · [[Algorithm]] · `Ancient_Egypt` · `Ancient_Greek` · `Andrey_Kolmogorov` · `Antiprism` · `Apex_(geometry)` · `Archimedean_solid` · `Archimedes` · [[Architecture]] · `Archive_for_History_of_Exact_Sciences` · `Arthur_Cayley` · `Associahedron` · `Augustin-Louis_Cauchy` · `Bartel_Leendert_van_der_Waerden` · `Base_(geometry)` · `Bernhard_Riemann` · `Bipyramid` · `Branko_Grünbaum` · `Carolyn_Eisele` · [[Cartesian_coordinate_system]] · `Cartography` · `Cauchy's_theorem_(geometry)` · `Centroid` · `Charles_Sanders_Peirce` · `Chirality_(mathematics)` · `Christopher_Zeeman` · `Christos_Papadimitriou` · `Classical_element` · `Classification_of_manifolds` · `Combinatorics` · `Commentarii_Mathematici_Helvetici` · `Commutative_algebra` · `Complex_number` · `Complex_polytope` · `Complex_reflection_group` · `Computational_geometry` · `Computer_graphics` · `Configuration_(polytope)` · `Connected_space` · `Convex_Polytopes` · `Convex_hull` · `Convex_polygon` · `Convex_polytope` · `Conway_polyhedron_notation` · `Craig_S._Kaplan` · `Császár_polyhedron` · `Cube` · `Cuboctahedron` · `Cuboid` · `Cyclic_group` · `Cylinder` · `David_A._Cox` · `David_Mount` · `David_Richeson` · `Degeneracy_(mathematics)` · `Delone_set` · `Deltahedron` · `Deltoidal_icositetrahedron` · `Diagonal` · `Digon` · `Dihedral_angle` · `Dihedron` · `Dimension` · `Discrete_&_Computational_Geometry` · `Discrete_Mathematics_(journal)` · `Disdyakis_dodecahedron` · `Disk_(mathematics)` · `Dissection_problem` · `Divergence_theorem` · `Dodecahedron` · `Dot_product` · `Dual_polyhedron` · `Duality_(order_theory)` · `Dymaxion_map` · `Edge_(geometry)` · `Egon_Schulte` · `Egyptian_pyramids` · `Ehrhart_polynomial` · `Elemente_der_Mathematik` · `Elongated_square_gyrobicupola` · `Empty_set` · `Encyclopedia_of_Mathematics` · `Enrico_Betti` · `Eric_W._Weisstein` · `Erik_Demaine` · `Ernst_Haeckel` · `Ernst_Steinitz` · `Etruscan_civilization` · `Euclid` · `Euclid's_Elements` · `Euclidean_space` · `Euler's_Gem` · [[Euler_characteristic]] · `Evgraf_Fedorov` · `Face_(geometry)` · `Flexible_polyhedron` · `Francesco_Maurolico` · `Frustum` · `Gauss–Bonnet_theorem` · `Geminus` · `Genus_(mathematics)` · `Geodesic` · `Geoffrey_Colin_Shephard` · `Geometric_Folding_Algorithms` · `Geometric_design` · [[Geometry]] · `George_W._Hart` · `Goldberg_polyhedron` · `Graph_(discrete_mathematics)` · `Graph_of_a_polytope` · [[Graph_theory]] · `Greek_language` · `Greenland` · `Gyrobifastigium` · `H.S.M._Coxeter` · `HIV` · `Hal_Schenck` · `Half-space_(geometry)` · `Henri_Poincaré` · `Hexahedron` · `Hilbert's_problems` · `Hilbert_space` · `Honeycomb_(geometry)` · `Hosohedron` · `Hyperbolic_space` · `Icosahedral_symmetry` · `Icosahedron` · `Imre_Lakatos` · `Incidence_geometry` · `Integer` · `Integral_polytope` · `Involution_(mathematics)` · `Isogonal_figure` · `Jacopo_de'_Barbari` · `Jean_Taylor` · `Jeffrey_Lagarias` · [[Johannes_Kepler]] · `John_B._Little_(mathematician)` · `Joseph_O'Rourke_(professor)` · `Joseph_S._B._Mitchell` · `Judith_V._Field` · `János_Pach` · `Kepler–Poinsot_polyhedron` · `Klein_bottle` · `Kunstformen_der_Natur` · `Leonardo_da_Vinci` · `Leonhard_Euler` · `Linear_equation` · `Linear_programming` · `List_of_books_about_polyhedra` · `Liu_Hui` · `Louis_Poinsot` · `Luca_Pacioli` · `Ludwig_Schläfli` · `Manifold` · `Marjorie_Senechal` · `Marquetry` · `Martin_Gardner` · `Martin_Grötschel` · `MathWorld` · `Mathematical_Models_(Cundy_and_Rollett)` · `Max_Dehn` · `Metric_space` · `Micha_Sharir` · `Michiel_Hazewinkel` · `Midsphere` · `Mikhail_Lavrentyev` · `Moscow_Mathematical_Papyrus` · `Möbius_strip` · `Net_(polyhedron)` · `Noble_polyhedron` · `Notices_of_the_American_Mathematical_Society` · `Numeral_prefix` · `Octahedral_symmetry` · `Octahedron` · `Old_Babylonian_Empire` · `Orientability` · `Orthogonal_polyhedron` · `Otfried_Cheong` · `Paolo_Uccello` · `Pappus_of_Alexandria` · `Parallelepiped` · `Parallelohedron` · `Partially_ordered_set` · `Pat_Hanrahan` · `Pentagonal_icositetrahedron` · `Pentagram` · `Pentakis_dodecahedron` · `Perspective_(graphical)` · `Peter_McMullen` · `Piero_della_Francesca` · `Planar_graph` · `Plato` · `Platonic_solid` · `Point_groups_in_three_dimensions` · `Polygon` · `Polygon_mesh` · `Polyhedral_combinatorics` · `Polyhedral_skeletal_electron_pair_theory` · `Polyhedral_symbol` · `Polyhedron_model` · `Polyomino` · [[Polytope]] · `Portrait_of_Luca_Pacioli` · `Prism_(geometry)` · `Proofs_and_Refutations` · `Pyramid_(geometry)` · `Pythagoras` · `Real_number` · `Real_projective_plane` · `Rectangular_cuboid` · `Rectilinear_polygon` · `Reflection_(mathematics)` · `Reflection_symmetry` · `Regular_Polytopes_(book)` · `Regular_dodecahedron` · `Regular_icosahedron` · `Regular_octahedron` · `Regular_polyhedron` · `Regular_tetrahedron` · `Renaissance` · `René_Descartes` · `Rhombic_dodecahedron` · `Rhombic_triacontahedron` · `Richard_P._Stanley` · `Right_angle` · `Robin_Hartshorne` · `Rotation` · `Rotation_(mathematics)` · `SIAM_Journal_on_Computing` · `Scientific_American` · `Scripta_Mathematica` · `Sergei_Tabachnikov` · `Seven_Bridges_of_Königsberg` · `Simple_polygon` · `Siobhan_Roberts` · `Small_stellated_dodecahedron` · `Soma_cube` · `South_America` · `Spherical_polyhedron` · `Springer_Science+Business_Media` · `Star_polygon` · `Stars_(M._C._Escher)` · `Steinitz's_theorem` · `Surface_(mathematics)` · `Surface_area` · `Symmetry` · `Symmetry_group` · `Szilassi_polyhedron` · `Tangent` · [[Tessellation]] · `Tetradecahedron` · `Tetrahedral-octahedral_honeycomb` · `Tetrahedral_symmetry` · `Tetrahedron` · `Tetrahemihexahedron` · `Tetrakis_hexahedron` · `The_American_Mathematical_Monthly` · `The_Mathematical_Gazette` · `Theaetetus_(mathematician)` · `Timaeus_(dialogue)` · `Tohoku_Mathematical_Journal` · `Topology` · `Toric_variety` · `Toroid` · `Toroidal_polyhedron` · `Torus` · `Trapezohedron` · `Triakis_tetrahedron` · `Triangular_orthobicupola` · `Triangular_prism` · `Truncated_icosahedron` · `Truncated_octahedron` · `Truncated_tetrahedron` · `Uniform_polyhedron` · `Unit_vector` · `Vertex_(geometry)` · `Vertex_connectivity` · `Vertex_figure` · `Virus` · `Visibility_(geometry)` · `Volume` · `Warring_States_period` · [[Wayback_Machine]] · `Weaire–Phelan_structure` · `Wikisource` · `Wilbur_Knorr` · `William_Thurston` · `Winfield,_Kansas` · `Wire-frame_model` · `World_Scientific` ## Overview Polyhedra have been studied since antiquity. Euclid's *Elements* (Book XIII) constructs the five regular convex solids and proves no others exist; Plato had earlier tied them to the classical elements, giving the *Platonic solids* their name. Around 1750 Leonhard Euler noted that the vertex, edge, and face counts of any convex polyhedron obey a fixed relation — now a founding theorem of topology and graph theory. Today polyhedra underpin 3D rendering, computer-aided design, finite-element meshing, and linear optimization, where the solutions of a system of linear inequalities form a convex polyhedron. ## How it works A **convex polyhedron** can be described two ways that computation moves between: as the *convex hull* of a finite set of points (its vertices), or as the intersection of finitely many half-spaces $\{x : Ax \le b\}$. A bounded polyhedron of this kind is called a **polytope**. Its combinatorics are governed by **Euler's formula**, $V - E + F = 2,$ which holds for every polyhedron topologically equivalent to a sphere; more generally $V - E + F = 2 - 2g$ for genus $g$. A polyhedron is **regular** when all faces are congruent regular $p$-gons with $q$ meeting at each vertex, written with the Schläfli symbol $\{p, q\}$. Requiring the angles at a vertex to sum to less than $360^\circ$ forces $\frac{1}{p} + \frac{1}{q} > \frac{1}{2},$ whose only integer solutions ($p,q \ge 3$) give exactly the five Platonic solids. From $\{p,q\}$ the counts follow mechanically: $E = 2\big/\!\left(\tfrac{2}{p} + \tfrac{2}{q} - 1\right)$, then $V = 2E/q$ and $F = 2E/p$. | Solid | $\{p,q\}$ | $F$ | $E$ | $V$ | $V-E+F$ | |---|---|---|---|---|---| | Tetrahedron | {3,3} | 4 | 6 | 4 | 2 | | Cube | {4,3} | 6 | 12 | 8 | 2 | | Octahedron | {3,4} | 8 | 12 | 6 | 2 | | Dodecahedron | {5,3} | 12 | 30 | 20 | 2 | | Icosahedron | {3,5} | 20 | 30 | 12 | 2 | Dual pairs (cube ↔ octahedron, dodecahedron ↔ icosahedron, tetrahedron self-dual) swap the roles of $V$ and $F$ while fixing $E$ — visible in the table and on screen. In software the same structure is stored as a **polygon mesh** (a vertex list plus faces that index it) or a half-edge structure for $O(1)$ local traversal. ## Controls -> what each maps to | Control | Maps to | Range / values | Meaning | |---|---|---|---| | Mouse-orbit (click-drag on canvas) | Camera orientation around the solid | Drag in any direction | Rotates the viewpoint so every face, edge, and vertex can be brought into view | | On-screen controls | Which polyhedron is displayed | Discrete set of solids | Switches the shown solid (e.g. among the Platonic solids) to compare structure | ## Learning objective After playing, a learner can orbit an unfamiliar solid, count its vertices, edges, and faces, predict that they satisfy $V - E + F = 2$, and explain why only five regular convex polyhedra exist. ## Limits and connections The sim emphasizes convex, mostly regular solids; it does not depict non-convex, star (Kepler–Poinsot), or higher-genus polyhedra, for which $V - E + F \ne 2$. The same vertex/edge/face representation scales up to the general [[Polytope|polytope]] in any dimension and underlies both graphics pipelines and the geometry of linear programming. ## Poster & source <div class="microsim-fallback">  <p><em>Live microsim · <a href="https://wikitube-3d-microsims.netlify.app/geometry/Polyhedron.html">open full</a> · source: Microsims for Dissemination/ALGORITHM_microsims/Polyhedron.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/Polyhedron) : [Wikitube](https://en.wikitube.io/wiki/Polyhedron) ## Previous hub tags Tree parent: [[Graph_theory]]. Legacy hubs: `ALGORITHM`. --- *Sources: 2 legacy notes. Minted wave 1, 2026-07-30 (v1.6 order).*