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Fig 1. Construction of the first isogonic center, X(13). When no angle of the triangle exceeds 120°, this point is the Fermat point. In Euclidean geometry, the Fermat point of a triangle, also called the Torricelli point or Fermat–Torricelli point, is a point such that the sum of the three distances from each of the three vertices of the triangle to the point is the smallest possible [1] or ...
The regular tetrahedron is self-dual, meaning its dual is another regular tetrahedron. The compound figure comprising two such dual tetrahedra form a stellated octahedron or stella octangula . Its interior is an octahedron , and correspondingly, a regular octahedron is the result of cutting off, from a regular tetrahedron, four regular ...
Other names for the same shape are isotetrahedron, [2] sphenoid, [3] bisphenoid, [3] isosceles tetrahedron, [4] equifacial tetrahedron, [5] almost regular tetrahedron, [6] and tetramonohedron. [ 7 ] All the solid angles and vertex figures of a disphenoid are the same, and the sum of the face angles at each vertex is equal to two right angles .
By definition, this isotopic property is common to the duals of the uniform polytopes. An isotopic 2-dimensional figure is isotoxal, i.e. edge-transitive. An isotopic 3-dimensional figure is isohedral, i.e. face-transitive. An isotopic 4-dimensional figure is isochoric, i.e. cell-transitive.
Isogonal, a mathematical term meaning "having similar angles", may refer to: Isogonal figure or polygon, polyhedron, polytope or tiling; Isogonal trajectory, in curve theory; Isogonal conjugate, in triangle geometry
The tetrahedron is self-dual (i.e. its dual is another tetrahedron). The cube and the octahedron form a dual pair. The dodecahedron and the icosahedron form a dual pair. If a polyhedron has Schläfli symbol {p, q}, then its dual has the symbol {q, p}. Indeed, every combinatorial property of one Platonic solid can be interpreted as another ...
A regular tetrahedron, an example of a solid with full tetrahedral symmetry. A regular tetrahedron has 12 rotational (or orientation-preserving) symmetries, and a symmetry order of 24 including transformations that combine a reflection and a rotation.
Tetrahedron: 6: Trigonal prism: 8: Cube: 10: Pentagonal prism: 12: D 2d pseudo-octahedron (dual of snub disphenoid) 14: Dual of triaugmented triangular prism (K 5 associahedron) 16: Square truncated trapezohedron: 18: Dual of edge-contracted icosahedron 20: Dodecahedron