A heptahedron (pl.: heptahedra) is a polyhedron having seven sides, or faces.
A heptahedron can take a large number of different basic forms, or topologies. The most familiar are the hexagonal pyramid and the pentagonal prism. Also notable is the tetrahemihexahedron, which can be seen as a tessellation of the real projective plane. No heptahedra are regular.
There are 34 topologically distinct convex heptahedra, excluding mirror images. [2] (Two polyhedra are "topologically distinct" if they have intrinsically different arrangements of faces and vertices, such that it is impossible to distort one into the other simply by changing the lengths of edges or the angles between edges or faces.)
An example of each type is depicted below, along with the number of sides on each of the faces. The images are ordered by descending number of six-sided faces (if any), followed by descending number of five-sided faces (if any), and so on.
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Six topologically distinct concave heptahedra (excluding mirror images) can be formed by combining two tetrahedra in various configurations. The third, fourth and fifth of these have a face with collinear adjacent edges, and the sixth has a face that is not simply connected.[ citation needed ] | |
  13 topologically distinct heptahedra (excluding mirror images) can be formed by cutting notches out of the edges of a triangular prism or square pyramid. Two examples are shown. |   A variety of non-simply-connected heptahedra are possible. Two examples are shown.[ citation needed ] |
One particularly interesting example is the Szilassi polyhedron, a Toroidal polyhedron with 7 non-convex six sided faces. [3]
In geometry, a dodecahedron or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three regular star dodecahedra, which are constructed as stellations of the convex form. All of these have icosahedral symmetry, order 120.
In geometry, every polyhedron is associated with a second dual structure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. Such dual figures remain combinatorial or abstract polyhedra, but not all can also be constructed as geometric polyhedra. Starting with any given polyhedron, the dual of its dual is the original polyhedron.
In geometry, a polyhedron is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices.
In elementary geometry, a polytope is a geometric object with flat sides (faces). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions n as an n-dimensional polytope or n-polytope. For example, a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope. In this context, "flat sides" means that the sides of a (k + 1)-polytope consist of k-polytopes that may have (k – 1)-polytopes in common.
In geometry, a 4-polytope is a four-dimensional polytope. It is a connected and closed figure, composed of lower-dimensional polytopal elements: vertices, edges, faces (polygons), and cells (polyhedra). Each face is shared by exactly two cells. The 4-polytopes were discovered by the Swiss mathematician Ludwig Schläfli before 1853.
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent. It is commonly denoted by .
A hexahedron or sexahedron is any polyhedron with six faces. A cube, for example, is a regular hexahedron with all its faces square, and three squares around each vertex.
A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In classical contexts, many different equivalent definitions are used; a common one is that the faces are congruent regular polygons which are assembled in the same way around each vertex.
In geometry, the Szilassi polyhedron is a nonconvex polyhedron, topologically a torus, with seven hexagonal faces.
A noble polyhedron is one which is isohedral and isogonal. They were first studied in any depth by Edmund Hess and Max Brückner in the late 19th century, and later by Branko Grünbaum.
In geometry, the Császár polyhedron is a nonconvex toroidal polyhedron with 14 triangular faces.
In geometry, a toroidal polyhedron is a polyhedron which is also a toroid, having a topological genus of 1 or greater. Notable examples include the Császár and Szilassi polyhedra.
In geometry, a (globally) projective polyhedron is a tessellation of the real projective plane. These are projective analogs of spherical polyhedra – tessellations of the sphere – and toroidal polyhedra – tessellations of the toroids.
In geometry, an enneahedron is a polyhedron with nine faces. There are 2606 types of convex enneahedron, each having a different pattern of vertex, edge, and face connections. None of them are regular.
In geometry, the density of a star polyhedron is a generalization of the concept of winding number from two dimensions to higher dimensions, representing the number of windings of the polyhedron around the center of symmetry of the polyhedron. It can be determined by passing a ray from the center to infinity, passing only through the facets of the polytope and not through any lower dimensional features, and counting how many facets it passes through. For polyhedra for which this count does not depend on the choice of the ray, and for which the central point is not itself on any facet, the density is given by this count of crossed facets.
A hendecahedron is a polyhedron with 11 faces. There are numerous topologically distinct forms of a hendecahedron, for example the decagonal pyramid, and enneagonal prism.
A tridecahedron, or triskaidecahedron, is a polyhedron with thirteen faces. There are numerous topologically distinct forms of a tridecahedron, for example the dodecagonal pyramid and hendecagonal prism. However, a tridecahedron cannot be a regular polyhedron, because there is no regular polygon that can form a regular tridecahedron, and there are only five known regular polyhedra.
Lajos Szilassi was a professor of mathematics at the University of Szeged who worked in projective and non-Euclidean geometry, applying his research to computer generated solutions of geometric problems.
A enneadecahedron is a polyhedron with 19 faces. No enneadecahedron is regular; hence, the name is ambiguous.
Adventures Among the Toroids: A study of orientable polyhedra with regular faces is a book on toroidal polyhedra that have regular polygons as their faces. It was written, hand-lettered, and illustrated by mathematician Bonnie Stewart, and self-published under the imprint "Number One Tall Search Book" in 1970. Stewart put out a second edition, again hand-lettered and self-published, in 1980. Although out of print, the Basic Library List Committee of the Mathematical Association of America has recommended its inclusion in undergraduate mathematics libraries.
