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2. Genus (mathematics) - Wikipedia

   en.wikipedia.org/wiki/Genus_(mathematics)

   The non-orientable genus, demigenus, or Euler genus of a connected, non-orientable closed surface is a positive integer representing the number of cross-caps attached to a sphere. Alternatively, it can be defined for a closed surface in terms of the Euler characteristic χ, via the relationship χ = 2 − k , where k is the non-orientable genus.

3. Torus - Wikipedia

   en.wikipedia.org/wiki/Torus

   As examples, a genus zero surface (without boundary) is the two-sphere while a genus one surface (without boundary) is the ordinary torus. The surfaces of higher genus are sometimes called n-holed tori (or, rarely, n-fold tori). The terms double torus and triple torus are also occasionally used.

4. Fundamental polygon - Wikipedia

   en.wikipedia.org/wiki/Fundamental_polygon

   In the case of genus one, a fundamental convex polygon is sought for the action by translation of Λ = Z a ⊕ Z b on R 2 = C where a and b are linearly independent over R. (After performing a real linear transformation on R 2, it can be assumed if necessary that Λ = Z 2 = Z + Z i; for a genus one Riemann surface it can be taken to have the form Λ = Z 2 = Z + Z ω, with Im ω > 0.)

5. Toroidal polyhedron - Wikipedia

   en.wikipedia.org/wiki/Toroidal_polyhedron

   A polyhedral torus can be constructed to approximate a torus surface, from a net of quadrilateral faces, like this 6x4 example. In geometry, a toroidal polyhedron is a polyhedron which is also a toroid (a g-holed torus), having a topological genus (g) of 1 or greater. Notable examples include the Császár and Szilassi polyhedra.

6. Genus g surface - Wikipedia

   en.wikipedia.org/wiki/Genus_g_surface

   In mathematics, a genus g surface (also known as a g-torus or g-holed torus) is a surface formed by the connected sum of g distinct tori: the interior of a disk is removed from each of g distinct tori and the boundaries of the g many disks are identified (glued together), forming a g-torus. The genus of such a surface is g. A genus g surface is ...

7. Regular map (graph theory) - Wikipedia

   en.wikipedia.org/wiki/Regular_map_(graph_theory)

   The Dyck map is a regular map of 12 octagons on a genus-3 surface. Its underlying graph, the Dyck graph , can also form a regular map of 16 hexagons in a torus. The following is a complete list of regular maps in surfaces of positive Euler characteristic , χ: the sphere and the projective plane.

8. Mapping class group - Wikipedia

   en.wikipedia.org/wiki/Mapping_class_group

   This is because the surface N has a unique class of one-sided curves such that, when N is cut open along such a curve C, the resulting surface is a torus with a disk removed. As an unoriented surface, its mapping class group is GL ⁡ ( 2 , Z ) {\displaystyle \operatorname {GL} (2,\mathbb {Z} )} .

9. 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.