In
geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, a bitruncation is an operation on
regular polytope
In mathematics, a regular polytope is a polytope whose symmetry group acts transitive group action, transitively on its flag (geometry), flags, thus giving it the highest degree of symmetry. In particular, all its elements or -faces (for all , w ...
s. The original
edges are lost completely and the original
faces
The face is the front of the head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may affect the ...
remain as smaller copies of themselves.
Bitruncated regular polytopes can be represented by an extended
Schläfli symbol
In geometry, the Schläfli symbol is a notation of the form \ that defines List of regular polytopes and compounds, regular polytopes and tessellations.
The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, wh ...
notation or
In regular polyhedra and tilings
For regular
polyhedra
In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices. The term "polyhedron" may refer either to a solid figure or to its boundary su ...
(i.e. regular 3-polytopes), a ''bitruncated'' form is the truncated
dual
Dual or Duals may refer to:
Paired/two things
* Dual (mathematics), a notion of paired concepts that mirror one another
** Dual (category theory), a formalization of mathematical duality
*** see more cases in :Duality theories
* Dual number, a nu ...
. For example, a bitruncated
cube
A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...
is a
truncated octahedron
In geometry, the truncated octahedron is the Archimedean solid that arises from a regular octahedron by removing six pyramids, one at each of the octahedron's vertices. The truncated octahedron has 14 faces (8 regular hexagon, hexagons and 6 Squa ...
.
In regular 4-polytopes and honeycombs
For a regular
4-polytope
In geometry, a 4-polytope (sometimes also called a polychoron, polycell, or polyhedroid) is a four-dimensional polytope. It is a connected and closed figure, composed of lower-dimensional polytopal elements: Vertex (geometry), vertices, Edge (geo ...
, a ''bitruncated'' form is a dual-symmetric operator. A bitruncated 4-polytope is the same as the bitruncated dual, and will have double the symmetry if the original 4-polytope is
self-dual.
A regular polytope (or
honeycomb
A honeycomb is a mass of Triangular prismatic honeycomb#Hexagonal prismatic honeycomb, hexagonal prismatic cells built from beeswax by honey bees in their beehive, nests to contain their brood (eggs, larvae, and pupae) and stores of honey and pol ...
) will have its cells bitruncated into truncated cells, and the vertices are replaced by truncated cells.
Self-dual 4-polytope/honeycombs
An interesting result of this operation is that self-dual 4-polytope (and honeycombs) remain
cell-transitive
In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congruen ...
after bitruncation. There are 5 such forms corresponding to the five truncated regular polyhedra: t. Two are honeycombs on the
3-sphere
In mathematics, a hypersphere or 3-sphere is a 4-dimensional analogue of a sphere, and is the 3-dimensional n-sphere, ''n''-sphere. In 4-dimensional Euclidean space, it is the set of points equidistant from a fixed central point. The interior o ...
, one a honeycomb in Euclidean 3-space, and two are honeycombs in hyperbolic 3-space.
See also
*
uniform polyhedron
In geometry, a uniform polyhedron has regular polygons as Face (geometry), faces and is vertex-transitive—there is an isometry mapping any vertex onto any other. It follows that all vertices are congruence (geometry), congruent. Uniform po ...
*
uniform 4-polytope
In geometry, a uniform 4-polytope (or uniform polychoron) is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedron, uniform polyhedra, and faces are regular polygons.
There are 47 non-Prism (geometry), prism ...
*
Rectification (geometry)
In Euclidean geometry, rectification, also known as critical truncation or complete-truncation, is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points. The resulting p ...
*
Truncation (geometry)
In geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new Facet (geometry), facet in place of each vertex. The term originates from Kepler's names for the Archimedean solids.
Uniform truncation
In ge ...
References
*
Coxeter, H.S.M. ''
Regular Polytopes
''Regular Polytopes'' is a geometry book on regular polytopes written by Harold Scott MacDonald Coxeter. It was originally published by Methuen in 1947 and by Pitman Publishing in 1948, with a second edition published by Macmillan in 1963 and a th ...
'', (3rd edition, 1973), Dover edition, (pp. 145–154 Chapter 8: Truncation)
*
Norman Johnson ''Uniform Polytopes'', Manuscript (1991)
**
N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. Dissertation, University of Toronto, 1966
*
John H. Conway,
Heidi Burgiel
''Heidi'' (; ) is a work of children's fiction published between 1880 and 1881 by Swiss author Johanna Spyri, originally published in two parts as ''Heidi: Her Years of Wandering and Learning'' () and ''Heidi: How She Used What She Learned'' () ...
,
Chaim Goodman-Strauss
Chaim Goodman-Strauss (born June 22, 1967 in Austin, Texas) is an American mathematician who works in convex geometry, especially aperiodic tiling. He retired from the faculty of the University of Arkansas and currently serves as outreach mathem ...
, ''The Symmetries of Things'' 2008, (Chapter 26)
External links
*
{{Polyhedron_operators
Polytopes