User:Richard Pinch/sandbox-11

Honeycomb

"A symmetrical subdivision of a three-dimensional manifold into a number of polyhedral cells all alike, each rotation that is a symmetry of a cell being also a symmetry of the entire configuration."

A regular honeycomb is described by a Schläfli symbol $\{p,q,r\}$ denoting polyhedral cells that are Platonic solids $\{p,q\}$, such that every face $\{p\}$ belongs to just two cells, and every edge to $r$ cells.

References

• H.S.M. Coxeter "Twisted honeycombs", Conference Board of the Mathematical Sciences. Regional Conference Series in Mathematics. No.4. American Mathematical Society (1970) ISBN 0-8218-1653-5 Zbl 0217.46502
• H.S.M. Coxeter, "Regular complex polytopes" , Cambridge Univ. Press (1974) ISBN 0-521-20125-X Zbl 0732.51002

Schläfli symbol

A symbol encoding classes of polygons, polyhedra, polytopes and tessellations.

The symbol $\{p\}$ denotes a regular $p$-gon; the symbol $\{p,q\}$ a polyhedron with faces which are regular $p$-gons, $q$ of which meet at each vertex. The Platonic solids correspond to:

• tetrahedron: $\{3,3\}$;
• cube: $\{4,3\}$;
• octahedron: $\{3,4\}$;
• dodecahedron: $\{5,3\}$;
• icosahedron: $\{3,5\}$.

There are three plane tessellations: $\{3,6\}$, $\{4,4\}$, $\{6,3\}$. The dual solid or tessellation to $\{p,q\}$ is $\{q,p\}$.

The symbol $\{p,q,r\}$ denotes a polytope in four dimensions or a honeycomb.

The plane symbol may be extended to $\left\lbrace\frac{p}{q}\right\rbrace$ (where $p,q$ are coprime) denoting a $p$-gram or star polygon: a figure inscribed in a regular $p$-gon by joining every $q$-th vertex.

References

• H.S.M. Coxeter, "Regular complex polytopes" , Cambridge Univ. Press (1974) ISBN 0-521-20125-X Zbl 0732.51002

Glide

glide reflection

An indirect (orientation-reversing) Euclidean isometry. In the plane, given a line $\ell$, a glide with axis $\ell$ is the composite of a translation in the direction of $\ell$ and reflection in $\ell$ as mirror. In space, given a plane $\Pi$, a glide is the composite of a translation parallel to $\Pi$ and reflection in $\Pi$.

The indirect isometries of the Euclidean plane are all glide reflections (including reflections as a special case). The indirect isometries of Euclidean space are either glide reflections or rotatory reflections (including reflections as a special case).

References

• H. S. M. Coxeter, "The Beauty of Geometry: Twelve Essays" Dover (1999) ISBN 0486409198 Zbl 0941.51001
• E.G. Rees, "Notes on Geometry" Springer (1983)) ISBN 3-540-12053-X Zbl 0498.51001

Rotatory reflection

rotatory inversion

An indirect (orientation-reversing) isometry of Euclidean space. Given a plane $\Pi$ and a line $\ell$ perpendicular to $\Pi$, a rotatory reflection is the composite of a rotation with $\ell$ as axis and reflection in $\Pi$.

A rotatory inversion: given a line $\ell$ and a point $P$ on $\ell$, the composite of a rotation with $\ell$ as axis and central inversion (or reflection) in the point $P$.

Every rotatory reflection can be expressed as a rotatory inversion, and conversely.

The indirect isometries of Euclidean space are either rotatory reflections or glide reflections (including reflections as a special case).

References

• H. S. M. Coxeter, "The Beauty of Geometry: Twelve Essays" Dover (1999) ISBN 0486409198 Zbl 0941.51001
• E.G. Rees, "Notes on Geometry" Springer (1983)) ISBN 3-540-12053-X Zbl 0498.51001

Central inversion

central symmetry, reflection in a point

An isometry of a Euclidean space with respect to a centre $O$. The image of point $A$ is that point $A'$ on the line $\overline{OA}$ such that $A'O = OA$. In the Euclidean plane, this is a rotation by a half-turn about the point $O$.

References

• H. S. M. Coxeter, "The Beauty of Geometry: Twelve Essays" Dover (1999) ISBN 0486409198 Zbl 0941.51001
• E.G. Rees, "Notes on Geometry" Springer (1983)) ISBN 3-540-12053-X Zbl 0498.51001