Search results
Results from the WOW.Com Content Network
The regular dodecahedron is a polyhedron with twelve pentagonal faces, thirty edges, and twenty vertices. [1] It is one of the Platonic solids, a set of polyhedrons in which the faces are regular polygons that are congruent and the same number of faces meet at a vertex. [2] This set of polyhedrons is named after Plato.
In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edges congruent), and the same number of faces meet at each vertex. There are only five such polyhedra:
Three mutually perpendicular golden ratio rectangles, with edges connecting their corners, form a regular icosahedron. Another way to construct it is by putting two points on each surface of a cube. In each face, draw a segment line between the midpoints of two opposite edges and locate two points with the golden ratio distance from each midpoint.
A regular dihedron, {n, 2} [13] (2-hedron) in three-dimensional Euclidean space can be considered a degenerate prism consisting of two (planar) n-sided polygons connected "back-to-back", so that the resulting object has no depth, analogously to how a digon can be constructed with two line segments.
Tracing around an n-gon in general, the sum of the exterior angles (the total amount one rotates at the vertices) can be any integer multiple d of 360°, e.g. 720° for a pentagram and 0° for an angular "eight" or antiparallelogram, where d is the density or turning number of the polygon.
The surface area and the volume of the truncated icosahedron of edge length are: [2] = (+ +) = +. The sphericity of a polyhedron describes how closely a polyhedron resembles a sphere. It can be defined as the ratio of the surface area of a sphere with the same volume to the polyhedron's surface area, from which the value is between 0 and 1.
If the regular tetrahedron has edge length 𝒍 = 2, its characteristic tetrahedron's six edges have lengths , , around its exterior right-triangle face (the edges opposite the characteristic angles 𝟀, 𝝉, 𝟁), [b] plus , , (edges that are the characteristic radii of the regular tetrahedron). The 3-edge path along orthogonal edges of the ...
[2]: p. 1 They could also construct half of a given angle, a square whose area is twice that of another square, a square having the same area as a given polygon, and regular polygons of 3, 4, or 5 sides [2]: p. xi (or one with twice the number of sides of a given polygon [2]: pp. 49–50 ).