Bitruncated
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In
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a bitruncation is an operation on regular polytopes. It represents a
truncation In mathematics and computer science, truncation is limiting the number of digits right of the decimal point. Truncation and floor function Truncation of positive real numbers can be done using the floor function. Given a number x \in \mathbb ...
beyond
rectification Rectification has the following technical meanings: Mathematics * Rectification (geometry), truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points * Rectifiable curve, in mathematics * Recti ...
. The original
edges Edge or EDGE may refer to: Technology Computing * Edge computing, a network load-balancing system * Edge device, an entry point to a computer network * Adobe Edge, a graphical development application * Microsoft Edge, a web browser developed by ...
are lost completely and the original faces 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 regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to more ...
notation or


In regular polyhedra and tilings

For regular polyhedra (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 (grammatical ...
. For example, a bitruncated
cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only r ...
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: vertices, edges, faces (polygons), an ...
, 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 In mathematics, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a Injective function, one-to-one fashion, often (but not always) by means of an Involution (mathematics), involutio ...
. A regular polytope (or honeycomb) 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 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 3-sphere is a higher-dimensional analogue of a sphere. It may be embedded in 4-dimensional Euclidean space as the set of points equidistant from a fixed central point. Analogous to how the boundary of a ball in three dimensi ...
, 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 faces and is vertex-transitive (i.e., there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent. Uniform polyhedra may be regular (if also fa ...
* uniform 4-polytope *
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 Edge (geometry), edges, and cutting off its Vertex (geometry), vertices ...
* Truncation (geometry)


References

* Coxeter, H.S.M. '' Regular Polytopes'', (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 John Horton Conway (26 December 1937 – 11 April 2020) was an English people, English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to ...
,
Heidi Burgiel ''Heidi'' (; ) is a work of children's fiction published in 1881 by Swiss author Johanna Spyri, originally published in two parts as ''Heidi: Her Years of Wandering and Learning'' (german: Heidis Lehr- und Wanderjahre) and ''Heidi: How She Used ...
, Chaim Goodman-Strauss, ''The Symmetries of Things'' 2008, (Chapter 26)


External links

* {{Polyhedron_operators Polytopes