Incenter centers known to the ancient Greeks, and the only one that does not in general lie on the Euler line. It is the first listed center, X(1), in Clark Kimberling's Encyclopedia of Triangle Centers, and the identity element of the multiplicative group of triangle centers. For polygons with more than three sides, the incenter only exists for tangential polygons—those that have an incircle that is tangent to each side of the polygon. In this case the incenter is the center of this circle and is equally distant from all sides. It is a theorem in Euclidean geometry that the three
Regular polyhedron valleys, enabling them to construct three infinite folded surfaces which they called regular skew polyhedra. Coxeter offered a modified Schläfli symbol {l,m|n} for these figures, with {l,m} implying the vertex figure, with "m" regular "l"-gons around a vertex. The "n" defines "n"-gonal "holes". Their vertex figures are regular skew polygons, vertices zig-zagging between two planes. Finite regular skew polyhedra exist in 4-space. These finite regular skew polyhedra in 4-space can be seen as a subset of the faces of uniform 4-polytopes. They have planar regular polygon faces, but regular skew polygon vertex figures. Two dual solutions are related to the