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  2. List of regular polytopes - Wikipedia

    en.wikipedia.org/wiki/List_of_regular_polytopes

    12 3-dimensional "pure" apeirohedra based on the structure of the cubic honeycomb, {4,3,4}. [22] A π petrie dual operator replaces faces with petrie polygons; δ is a dual operator reverses vertices and faces; φ k is a kth facetting operator; η is a halving operator, and σ skewing halving operator.

  3. Archimedean solid - Wikipedia

    en.wikipedia.org/wiki/Archimedean_solid

    The Archimedean solids are a set of thirteen convex polyhedra whose faces are regular polygons, but not all alike, and whose vertices are all symmetric to each other. The solids were named after Archimedes, although he did not claim credit for them. They belong to the class of uniform polyhedra, the polyhedra with regular faces and symmetric ...

  4. Geodesic polyhedron - Wikipedia

    en.wikipedia.org/wiki/Geodesic_polyhedron

    In Magnus Wenninger's Spherical models, polyhedra are given geodesic notation in the form {3,q+} b,c, where {3,q} is the Schläfli symbol for the regular polyhedron with triangular faces, and q-valence vertices. The + symbol indicates the valence of the vertices being increased. b,c represent a subdivision description, with 1,0 representing the ...

  5. List of two-dimensional geometric shapes - Wikipedia

    en.wikipedia.org/wiki/List_of_two-dimensional...

    This is a list of two-dimensional geometric shapes in Euclidean and other geometries. For mathematical objects in more dimensions, see list of mathematical shapes. For a broader scope, see list of shapes.

  6. Polytope - Wikipedia

    en.wikipedia.org/wiki/Polytope

    These bounding sub-polytopes may be referred to as faces, or specifically j-dimensional faces or j-faces. A 0-dimensional face is called a vertex, and consists of a single point. A 1-dimensional face is called an edge, and consists of a line segment. A 2-dimensional face consists of a polygon, and a 3-dimensional face, sometimes called a cell ...

  7. List of mathematical shapes - Wikipedia

    en.wikipedia.org/wiki/List_of_mathematical_shapes

    Tessellations of euclidean and hyperbolic space may also be considered regular polytopes. Note that an 'n'-dimensional polytope actually tessellates a space of one dimension less. For example, the (three-dimensional) platonic solids tessellate the 'two'-dimensional 'surface' of the sphere.

  8. Regular polytope - Wikipedia

    en.wikipedia.org/wiki/Regular_polytope

    In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry.In particular, all its elements or j-faces (for all 0 ≤ j ≤ n, where n is the dimension of the polytope) — cells, faces and so on — are also transitive on the symmetries of the polytope, and are themselves regular polytopes of dimension j≤ n.

  9. Cuboctahedron - Wikipedia

    en.wikipedia.org/wiki/Cuboctahedron

    (In the case of the cuboctahedron, the center is in fact the apex of 6 square and 8 triangular pyramids). This radial equilateral symmetry is a property of only a few uniform polytopes, including the two-dimensional hexagon, the three-dimensional cuboctahedron, and the four-dimensional 24-cell and 8-cell (tesseract). [15]