List of regular polytopes

This page lists the regular polytopes in Euclidean space.

Contents

Two dimensional regular polytopes

The two dimensional convex regular polytopes are regular polygons.

There exist also non-convex regular polytopes in two dimensions, whose Schläfli symbols consist of rational numbers.

  • The pentagram (five-pointed star), with Schläfli symbol {5/2}
  • Two different types of seven-pointed star, with Schläfli symbols {7/2} and {7/3}
  • An eight-pointed star, with Schläfli symbol {8/3}
  • Two different types of nine-pointed star, with Schläfli symbols {9/2} and {9/4}
  • and so on, ad infinitum. In general, for any natural number n, there are n-pointed stars with Schläfli symbols {n/m} for all m such that m < n/2 (strictly speaking {n/m}={n/(n-m)}) and m and n are coprime.

Three dimensional regular polytopes

In three dimensions, the convex regular polytopes (or polyhedra) are the Platonic solids.

  • The tetrahedron, with Schläfli symbol {3,3}, faces are triangles, vertex figures are also triangles.
  • The cube, with Schläfli symbol {4,3}, faces are squares, vertex figures are triangles.
  • The octahedron, with Schläfli symbol {3,4}, faces are triangles, vertex figures are squares.
  • The dodecahedron, with Schläfli symbol {5,3}, faces are pentagons, vertex figures are triangles.
  • The icosahedron, with Schläfli symbol {3,5}, faces are triangles, vertex figures are pentagons.

There exist also non-convex regular polyhedra. These are the Kepler-Poinsot polyhedra.

Four dimensional regular polytopes

The convex regular 4-polytopes are as follows.

  • The 4-dimensional simplex, with Schläfli symbol {3,3,3}, faces and vertex figures are tetrahedra.
  • The 24-cell, with Schläfli symbol {3,4,3}, faces are octahedra, vertex figures are cubes.
  • The 4-dimensional cube, also called a hypercube or tesseract, with Schläfli symbol {4,3,3}, faces are cubes, vertex figures are tetrahedra.
  • The 4-dimensional cross-polytope, with Schläfli symbol {3,3,4}, faces are tetrahedra, vertex figures are octahedra.
  • The 120-cell, with Schläfli symbol {5,3,3}, faces are dodecahedra, vertex figures are tetrahedra.
  • The 600-cell, with Schläfli symbol {3,3,5}, faces are tetrahedra, vertex figures are icosahedra.

There exist also ten non-convex regular polytopes in four dimensions.

Higher dimensional regular polytopes

In dimensions higher than 4, there are only three kinds of convex regular polytopes.

  • n-dimensional simplex, with Schläfli symbol {3,...,3}
  • n-dimensional hypercube, also called a hypercube, with Schläfli symbol {4,3,...,3}
  • n-dimensional cross-polytope, with Schläfli symbol {3,...,3,4}

There are no non-convex regular polytopes in dimensions higher than 4.

External links

Navigation

  • Art and Cultures
    • Art (https://academickids.com/encyclopedia/index.php/Art)
    • Architecture (https://academickids.com/encyclopedia/index.php/Architecture)
    • Cultures (https://www.academickids.com/encyclopedia/index.php/Cultures)
    • Music (https://www.academickids.com/encyclopedia/index.php/Music)
    • Musical Instruments (http://academickids.com/encyclopedia/index.php/List_of_musical_instruments)
  • Biographies (http://www.academickids.com/encyclopedia/index.php/Biographies)
  • Clipart (http://www.academickids.com/encyclopedia/index.php/Clipart)
  • Geography (http://www.academickids.com/encyclopedia/index.php/Geography)
    • Countries of the World (http://www.academickids.com/encyclopedia/index.php/Countries)
    • Maps (http://www.academickids.com/encyclopedia/index.php/Maps)
    • Flags (http://www.academickids.com/encyclopedia/index.php/Flags)
    • Continents (http://www.academickids.com/encyclopedia/index.php/Continents)
  • History (http://www.academickids.com/encyclopedia/index.php/History)
    • Ancient Civilizations (http://www.academickids.com/encyclopedia/index.php/Ancient_Civilizations)
    • Industrial Revolution (http://www.academickids.com/encyclopedia/index.php/Industrial_Revolution)
    • Middle Ages (http://www.academickids.com/encyclopedia/index.php/Middle_Ages)
    • Prehistory (http://www.academickids.com/encyclopedia/index.php/Prehistory)
    • Renaissance (http://www.academickids.com/encyclopedia/index.php/Renaissance)
    • Timelines (http://www.academickids.com/encyclopedia/index.php/Timelines)
    • United States (http://www.academickids.com/encyclopedia/index.php/United_States)
    • Wars (http://www.academickids.com/encyclopedia/index.php/Wars)
    • World History (http://www.academickids.com/encyclopedia/index.php/History_of_the_world)
  • Human Body (http://www.academickids.com/encyclopedia/index.php/Human_Body)
  • Mathematics (http://www.academickids.com/encyclopedia/index.php/Mathematics)
  • Reference (http://www.academickids.com/encyclopedia/index.php/Reference)
  • Science (http://www.academickids.com/encyclopedia/index.php/Science)
    • Animals (http://www.academickids.com/encyclopedia/index.php/Animals)
    • Aviation (http://www.academickids.com/encyclopedia/index.php/Aviation)
    • Dinosaurs (http://www.academickids.com/encyclopedia/index.php/Dinosaurs)
    • Earth (http://www.academickids.com/encyclopedia/index.php/Earth)
    • Inventions (http://www.academickids.com/encyclopedia/index.php/Inventions)
    • Physical Science (http://www.academickids.com/encyclopedia/index.php/Physical_Science)
    • Plants (http://www.academickids.com/encyclopedia/index.php/Plants)
    • Scientists (http://www.academickids.com/encyclopedia/index.php/Scientists)
  • Social Studies (http://www.academickids.com/encyclopedia/index.php/Social_Studies)
    • Anthropology (http://www.academickids.com/encyclopedia/index.php/Anthropology)
    • Economics (http://www.academickids.com/encyclopedia/index.php/Economics)
    • Government (http://www.academickids.com/encyclopedia/index.php/Government)
    • Religion (http://www.academickids.com/encyclopedia/index.php/Religion)
    • Holidays (http://www.academickids.com/encyclopedia/index.php/Holidays)
  • Space and Astronomy
    • Solar System (http://www.academickids.com/encyclopedia/index.php/Solar_System)
    • Planets (http://www.academickids.com/encyclopedia/index.php/Planets)
  • Sports (http://www.academickids.com/encyclopedia/index.php/Sports)
  • Timelines (http://www.academickids.com/encyclopedia/index.php/Timelines)
  • Weather (http://www.academickids.com/encyclopedia/index.php/Weather)
  • US States (http://www.academickids.com/encyclopedia/index.php/US_States)

Information

  • Home Page (http://academickids.com/encyclopedia/index.php)
  • Contact Us (http://www.academickids.com/encyclopedia/index.php/Contactus)

  • Clip Art (http://classroomclipart.com)
Toolbox
Personal tools