Editing Tetrahedron (section)

===Fundamental domains===
[[File:Coxeter-Dynkin 3-space groups.png|480px|thumb|For Euclidean 3-space, there are 3 simple and related Goursat tetrahedra. They can be seen as points on and within a cube.]]An irregular tetrahedron which is the [[fundamental domain]]{{Sfn|Coxeter|1973|p=63|loc=§4.3 Rotation groups in two dimensions; notion of a ''fundamental region''}} of a [[Coxeter group|symmetry group]] is an example of a [[Goursat tetrahedron]]. The Goursat tetrahedra generate all the regular polyhedra (and many other uniform polyhedra) by mirror reflections, a process referred to as [[Wythoff construction|Wythoff's kaleidoscopic construction]].

For polyhedra, Wythoff's construction arranges three mirrors at angles to each other, as in a [[kaleidoscope]]. Unlike a cylindrical kaleidoscope, Wythoff's mirrors are located at three faces of a Goursat tetrahedron such that all three mirrors intersect at a single point. (The [[Coxeter-Dynkin diagram]] of the generated polyhedron contains three ''nodes'' representing the three mirrors. The dihedral angle between each pair of mirrors is encoded in the diagram, as well as the location of a single ''generating point'' which is multiplied by mirror reflections into the vertices of the polyhedron.)

Among the Goursat tetrahedra which generate 3-dimensional [[Honeycomb (geometry)|honeycombs]] we can recognize an orthoscheme (the characteristic tetrahedron of the cube), a double orthoscheme (the characteristic tetrahedron of the cube face-bonded to its mirror image), and the space-filling disphenoid illustrated [[#Disphenoid|above]].{{Sfn|Coxeter|1973|pp=71–72|loc=§4.7 Characteristic tetrahedra}} The disphenoid is the double orthoscheme face-bonded to its mirror image (a quadruple orthoscheme). Thus all three of these Goursat tetrahedra, and all the polyhedra they generate by reflections, can be [[Dissection into orthoschemes|dissected into characteristic tetrahedra of the cube]].
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