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An open ball excludes the sphere itself, while a closed ball includes the sphere: a closed ball is the union of the open ball and the sphere, and a sphere is the boundary of a (closed or open) ball. The distinction between ball and sphere has not always been maintained and especially older mathematical references talk about a sphere as a solid.
Since spherical geometry violates the parallel postulate, there exists no such triangle on the surface of a sphere. The sum of the angles of a triangle on a sphere is 180°(1 + 4f), where f is the fraction of the sphere's surface that is enclosed by the triangle.
A convex regular polyhedron has all of three related spheres (other polyhedra lack at least one kind) which share its centre: An insphere, tangent to all faces. An intersphere or midsphere, tangent to all edges. A circumsphere, tangent to all vertices.
Geodesic polyhedra are constructed by subdividing faces of simpler polyhedra, and then projecting the new vertices onto the surface of a sphere. A geodesic polyhedron has straight edges and flat faces that approximate a sphere, but it can also be made as a spherical polyhedron (a tessellation on a sphere) with true geodesic curved edges on the ...
In geometry, a spherical polyhedron or spherical tiling is a tiling of the sphere in which the surface is divided or partitioned by great arcs into bounded regions called spherical polygons. A polyhedron whose vertices are equidistant from its center can be conveniently studied by projecting its edges onto the sphere to obtain a corresponding ...
Every convex polyhedron has a combinatorially equivalent polyhedron, the canonical polyhedron, that does have a midsphere, centered at the centroid of the points of tangency of its edges. Numerical approximation algorithms can construct the canonical polyhedron, but its coordinates cannot be represented exactly as a closed-form expression .
However, there exist many finite spaces, such as the 3-sphere and 3-torus, that have no edges. Mathematically, these spaces are referred to as being compact without boundary. The term compact means that it is finite in extent ("bounded") and complete. The term "without boundary" means that the space has no edges.
In the mathematical field of graph theory, a rhombicosidodecahedral graph is the graph of vertices and edges of the rhombicosidodecahedron, one of the Archimedean solids. It has 60 vertices and 120 edges, and is a quartic graph Archimedean graph. [5] Square centered Schlegel diagram