Orthogonal projection of onto the -eigenspace of yields thus the twelve vertices of the icosahedron.Ī second straightforward construction of the icosahedron uses representation theory of the alternating group acting by direct isometries on the icosahedron.Īccording to specific rules defined in the book The fifty nine icosahedra, 59 stellations were identified for the regular icosahedron. System of six equiangular lines in intersecting pairwise at a common acute angle of. Of the six coordinate axes in forms thus a This implies that hasĮigenvalues, both with multiplicity 3 since is symmetric and of trace 0.Īn Euclidean structure on the quotient space which is isomorphic to since the kernel ( Mnemonic: the matrix encodes the angles, acute for a positive entry, obtuse otherwise, between five cyclically consecutive vertices adjacent to a first vertex of an icosahedron centered at the origin.)Ī straightforward computation yields. In order to construct such an equiangular system, we start with the matrix Conversely, supposing the existence of a regular icosahedron, lines defined by its six pairs of opposite vertices form an equiangular system. Lines with an Euclidean sphere centered at their common intersection yields the twelve vertices of a regular icosahedron as can easily be checked. Indeed, intersecting such a system of equiangular The following construction of the icoshaedron avoids tedious computations in the number field necessary in more elementary approaches.Įxistence of the icosahedron amounts to the existence of six equiangular lines in. The five octahedra defining any given icosahedron form a regular polyhedral compound, as do the two icosahedra that can be defined in this way from any given octahedron.Ĭonstruction by a system of equiangular lines This is done by first placing vectors along the octahedron's edges such that each face is bounded by a cycle, then similarly partitioning each edge into the golden mean along the direction of its vector. The 12 edges of a regular octahedron can be partitioned in the golden ratio so that the resulting vertices define a regular icosahedron. Note that these vertices form five sets of three mutually centered, mutually orthogonal golden rectangles. Where is the golden ratio (also written τ).
The following Cartesian coordinates define the vertices of an icosahedron with edge-length 2, centered at the origin: The surface area and the volume of a regular icosahedron of edge length are: Where (also called ) is the golden ratio. While the midradius, which touches the middle of each edge, is If the edge length of a regular icosahedron is, the radius of a circumscribed sphere (one that touches the icosahedron at all vertices) isĪnd the radius of an inscribed sphere ( tangent to each of the icosahedron's faces) is 4 Construction by a system of equiangular lines.
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