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Tessellations of euclidean and hyperbolic space may also be considered regular polytopes. Note that an 'n'-dimensional polytope actually tessellates a space of one dimension less. For example, the (three-dimensional) platonic solids tessellate the 'two'-dimensional 'surface' of the sphere.
Another type of sphere arises from a 4-ball, whose three-dimensional surface is the 3-sphere: points equidistant to the origin of the euclidean space R 4. If a point has coordinates, P ( x , y , z , w ) , then x 2 + y 2 + z 2 + w 2 = 1 characterizes those points on the unit 3-sphere centered at the origin.
Regular polyhedron. Platonic solid: . Tetrahedron, Cube, Octahedron, Dodecahedron, Icosahedron; Regular spherical polyhedron. Dihedron, Hosohedron; Kepler–Poinsot ...
Gaussian curve with a two-dimensional domain. Many shapes have metaphorical names, i.e., their names are metaphors: these shapes are named after a most common object that has it. For example, "U-shape" is a shape that resembles the letter U, a bell-shaped curve has the shape of the vertical cross section of a bell, etc.
Among the most common 3-dimensional shapes are polyhedra, which are shapes with flat faces; ellipsoids, which are egg-shaped or sphere-shaped objects; cylinders; and cones. If an object falls into one of these categories exactly or even approximately, we can use it to describe the shape of the object.
A solid figure is the region of 3D space bounded by a two-dimensional closed surface; for example, a solid ball consists of a sphere and its interior. Solid geometry deals with the measurements of volumes of various solids, including pyramids , prisms (and other polyhedrons ), cubes , cylinders , cones (and truncated cones ).