Search results
Results from the WOW.Com Content Network
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.
The genus of a compact surface is defined as the genus of the corresponding closed surface. [ 2 ] This classification follows almost immediately from the classification of closed surfaces: removing an open disc from a closed surface yields a compact surface with a circle for boundary component, and removing k open discs yields a compact 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 ...
A torus. The next case is a Riemann surface of genus =, such as a torus /, where is a two-dimensional lattice (a group isomorphic to ). Its genus is one: its first singular homology group is freely generated by two loops, as shown in the illustration at the right.
Poloidal direction (red arrow) and toroidal direction (blue arrow) A torus of revolution in 3-space can be parametrized as: [2] (,) = (+ ) (,) = (+ ) (,) = . using angular coordinates , [,), representing rotation around the tube and rotation around the torus' axis of revolution, respectively, where the major radius is the distance from the center of the tube to the center of ...
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.)
Get AOL Mail for FREE! Manage your email like never before with travel, photo & document views. Personalize your inbox with themes & tabs. You've Got Mail!
A radially symmetric 7-colored torus – regions of the same colour wrap around along dotted lines An 8-coloured double torus (genus-two surface) – opposite circles are linked by handles A 6-colored Klein bottle, the only exception to the Heawood conjecture