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The self-intersection of a one-sided surface. The word "cross-cap" is sometimes also written without the hyphen as the single word "crosscap." The cross-cap can be thought of ...

The set of roots of a polynomial f(x,y,z)=0. An algebraic surface is said to be of degree n=max(i+j+k), where n is the maximum sum of powers of all terms ...

A class of illusion in which an object which is physically unrealizable is apparently depicted. More than 100 papers have been written about impossible figures (Kulpa 1987), ...

Every planar graph (i.e., graph with graph genus 0) has an embedding on a torus. In contrast, toroidal graphs are embeddable on the torus, but not in the plane, i.e., they ...

Betti numbers are topological objects which were proved to be invariants by Poincaré, and used by him to extend the polyhedral formula to higher dimensional spaces. ...

A graph H is a minor of a graph G if a copy of H can be obtained from G via repeated edge deletion and/or edge contraction. The Kuratowski reduction theorem states that any ...

A graph with projective plane crossing number equal to 0 may be said to be projective planar. Examples of projective planar graphs with graph crossing number >=2 include the ...

A paradox also known as the surprise examination paradox or prediction paradox. A prisoner is told that he will be hanged on some day between Monday and Friday, but that he ...

The fundamental group of an arcwise-connected set X is the group formed by the sets of equivalence classes of the set of all loops, i.e., paths with initial and final points ...

Lovász (1970) conjectured that every connected vertex-transitive graph is traceable (Gould, p. 33). This conjecture was subsequently verified for several special orders and ...