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Given a planar graph G, its geometric dual G^* is constructed by placing a vertex in each region of G (including the exterior region) and, if two regions have an edge x in ...

Geometry is the study of figures in a space of a given number of dimensions and of a given type. The most common types of geometry are plane geometry (dealing with objects ...

Goldberg polyhedra are convex polyhedra first described by Goldberg (1937) and classified in more detail by Hart (2013) for which each face is a regular pentagon or regular ...

The bound for the number of colors which are sufficient for map coloring on a surface of genus g, gamma(g)=|_1/2(7+sqrt(48g+1))_| is the best possible, where |_x_| is the ...

On the surface of a sphere, attempt separation of variables in spherical coordinates by writing F(theta,phi)=Theta(theta)Phi(phi), (1) then the Helmholtz differential ...

Points, also called polar reciprocals, which are transformed into each other through inversion about a given inversion circle C (or inversion sphere). The points P and P^' ...

Inversion is the process of transforming points P to a corresponding set of points P^' known as their inverse points. Two points P and P^' are said to be inverses with ...

Some elements of a group G acting on a space X may fix a point x. These group elements form a subgroup called the isotropy group, defined by G_x={g in G:gx=x}. For example, ...

The Kummer surfaces are a family of quartic surfaces given by the algebraic equation (x^2+y^2+z^2-mu^2w^2)^2-lambdapqrs=0, (1) where lambda=(3mu^2-1)/(3-mu^2), (2) p, q, r, ...

The level set of a differentiable function f:R^n->R corresponding to a real value c is the set of points {(x_1,...,x_n) in R^n:f(x_1,...,x_n)=c}. For example, the level set ...