Let T be a regular map of type {p q} on the torus
Theorem. (The Euler characteristic of the torus.) Suppose that G is a toroidal graph and that G has V vertices
We now consider the question of vertices edges
set of edges present. Thus any maximal torus graph havingp vertices may ... xy is an edge of G the two faces of G incident with xy are referred to.
Euler characteristic for other surfaces. Question. Does Euler's formula still hold for the vertices edges
from two minimal graphs and the 2-cell embeddings in the torus from six minimal and faces of G onto the vertices edges
On the other hand if we try to compute the same quantity for the torus we will get V-E+F=0. Regardless of the number of faces edges and vertices we choose
( ) consisting of two new vertices v' and v" and triangular faces v'v"u and v'v"w as well as the triangular faces determined by v' and the edges of r' and
shapes (box cylinder
Each vertex of Gis represented by a point on the torus Each edge inGis represented by a continuous path drawn on the torus connectingthe points corresponding to its vertices These paths do not intersect each other except for the trivial situation where twopaths share a common endpoint
• v is the number of vertices • e is the number of edges • f is the number of faces For a surface it turns out that the Euler characteristic can be expressed solely in terms of the three invariants above Namely: ? = 2 – 2g – c if ? = 1 ? = 2 – g – c if ? = 0 For example if we take the sphere—a closed orientable
The torus Euler characteristic Euler characteristic S a surface G a graph drawn on S so that no edges or vertices cross or overlap all regions (faces) are discs there are V vertices E edges F faces De?nition The Euler characteristic of S is ˜(S) = V E + F Theorem ˜(S) depends only on S and not on G
Figure 1: The torus After the ?rst pair of edges is associated the square looks like a cylinder When the second pair is associated we get the torus Hence the torus can be thought of as the surface of a doughnut 3 Xhas a countable basis The following three surfaces are very important Our goal will be to prove
In this lecture we allow the graphs to have loops and parallel edges Inaddition to the plane (or the sphere) we can draw the graphs on the surfaceof the torus or on more complicated surfaces De nition 1 Asurfaceis a compact connected2-dimensional manifold with-out boundary Intuitive explanation:
The torus is obtained from the square by identifying opposite edges In general a polygon diagram is a polygon whose edges are marked with orientation arrows and colors such that each color occurs exactly twice see figure (6 1) From a polygon diagram we obtain a topological space by gluing edges of the same
Unless I'm very mistaken, the surface of a torus is 2-dimensional, as is the surface of a sphere. The reason being that being on the surface you can only move in 2 dimensions, up or down is not well defined. If I'm wrong, please explain why. My friend got rather upset when I told him this, insisting that the surface of a torus is 3-dimensional.
It lives on the square torus with three punctures, has total absolute curvature • = 12…, two catenoid ends and one planar end. Later the planar end was shown to be deformable into a catenoid end, giving rise to a 3-ended embedded minimal surface for each rectangular torus.
A torus with aspect ratio 3 as the product of a smaller (red) and a bigger (magenta) circle. In geometry, a torus (plural tori) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis coplanar with the circle.
By Lemma2.6the Euler Characteristic of the connected sum of tori isn·XX(T)?2n, and hence is di?erent for all n. Similarly the connected sum of projective planes is never orientable, be-cause the projective plane is not orientable. We have that the Euler Char-acteristic of the connected sum of nprojective planes is n· X(RP2)?2n,by Lemma2.6.