The quarter cubic honeycomb, quarter cubic cellulation or bitruncated alternated cubic honeycomb is a space-filling

tessellation A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety ...

(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 ...

) in

Euclidean 3-space In geometry, a three-dimensional space (3D space, 3-space or, rarely, tri-dimensional space) is a mathematical space in which three values (''coordinates'') are required to determine the position of a point. Most commonly, it is the three-dim ...

. It is composed of

tetrahedra In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tet ...

and

truncated tetrahedra In geometry, the truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 equilateral triangle faces, 12 vertices and 18 edges (of two types). It can be constructed by truncating all 4 vertices of a regular tetrahedron. ...

in a ratio of 1:1. It is called "quarter-cubic" because its symmetry unit – the minimal block from which the pattern is developed by reflections – is four times that of the

cubic honeycomb The cubic honeycomb or cubic cellulation is the only proper regular space-filling tessellation (or honeycomb (geometry), honeycomb) in Euclidean 3-space made up of cube, cubic cells. It has 4 cubes around every edge, and 8 cubes around each verte ...

. It is

vertex-transitive In geometry, a polytope (e.g. a polygon or polyhedron) or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of face i ...

with 6

and 2

around each vertex. It is one of the 28

convex uniform honeycomb In geometry, a convex uniform honeycomb is a uniform polytope, uniform tessellation which fills three-dimensional Euclidean space with non-overlapping convex polyhedron, convex uniform polyhedron, uniform polyhedral cells. Twenty-eight such honey ...

s. The faces of this honeycomb's cells form four families of parallel planes, each with a 3.6.3.6 tiling. Its

vertex figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a general -polytope is sliced off. Definitions Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connected ed ...

is an isosceles

antiprism In geometry, an antiprism or is a polyhedron composed of two Parallel (geometry), parallel Euclidean group, direct copies (not mirror images) of an polygon, connected by an alternating band of triangles. They are represented by the Conway po ...

: two

equilateral triangle An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the ...

s joined by six

isosceles triangle In geometry, an isosceles triangle () is a triangle that has two Edge (geometry), sides of equal length and two angles of equal measure. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at le ...

John Horton Conway John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician. He was active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many b ...

calls this honeycomb a truncated tetrahedrille, and its dual

oblate cubille In Christianity (specifically the Roman Catholic, Orthodox, Lutheran, Anglican and Methodist traditions), an oblate is a person associated with a Benedictine monastery or convent who is specifically dedicated to God and service. Oblates are in ...

. The vertices and edges represent a

Kagome lattice In geometry, the trihexagonal tiling is one of 11 uniform tilings of the Euclidean plane by regular polygons. See in particular Theorem 2.1.3, p. 59 (classification of uniform tilings); Figure 2.1.5, p.63 (illustration of this tiling), Theorem 2. ...

in three dimensions, which is the

pyrochlore Pyrochlore () is a mineral group of the niobium end member of the pyrochlore supergroup. Pyrochlore is also a term for the crystal structure ''F''dm. The name is from the Greek , ''fire'', and , ''green'' because it typically turns green on ignit ...

lattice.

Construction

The quarter cubic honeycomb can be constructed in slab layers of truncated tetrahedra and tetrahedral cells, seen as two

trihexagonal tiling In geometry, the trihexagonal tiling is one of 11 uniform tilings of the Euclidean plane by regular polygons. See in particular Theorem 2.1.3, p. 59 (classification of uniform tilings); Figure 2.1.5, p.63 (illustration of this tiling), Theorem 2 ...

s. Two tetrahedra are stacked by a vertex and a

central inversion In geometry, a point reflection (also called a point inversion or central inversion) is a geometric transformation of affine space in which every point (geometry), point is reflected across a designated inversion center, which remains Fixed p ...

. In each

, half of the triangles belong to tetrahedra, and half belong to truncated tetrahedra. These slab layers must be stacked with tetrahedra triangles to truncated tetrahedral triangles to construct the uniform ''quarter cubic honeycomb''. Slab layers of hexagonal prisms and triangular prisms can be alternated for ''elongated'' honeycombs, but these are also not uniform.

Symmetry

Cells can be shown in two different symmetries. The reflection generated form represented by its Coxeter-Dynkin diagram has two colors of truncated cuboctahedra. The symmetry can be doubled by relating the pairs of ringed and unringed nodes of the Coxeter-Dynkin diagram, which can be shown with one colored tetrahedral and truncated tetrahedral cells.

Related polyhedra

This honeycomb is one of five distinct uniform honeycombs
6-1 cases, skipping one with zero marks constructed by the

_3

Coxeter group In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean ref ...

. The symmetry can be multiplied by the symmetry of rings in the

Coxeter–Dynkin diagram In geometry, a Harold Scott MacDonald Coxeter, Coxeter–Eugene Dynkin, Dynkin diagram (or Coxeter diagram, Coxeter graph) is a Graph (discrete mathematics), graph with numerically labeled edges (called branches) representing a Coxeter group or ...

s: The Quarter cubic honeycomb is related to a matrix of 3-dimensional honeycombs: q

References

* 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 ...

, (2008) ''The Symmetries of Things'', (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, Architectonic and Catoptric tessellations, p 292-298, includes all the nonprismatic forms) * George Olshevsky, ''Uniform Panoploid Tetracombs'', Manuscript (2006) ''(Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)'' *

Branko Grünbaum Branko Grünbaum (; 2 October 1929 – 14 September 2018) was a Croatian-born mathematician of Jewish descentGeombinatorics Alexander Soifer is a Russian-born American mathematician and mathematics author. Soifer obtained his Ph.D. in 1973 and has been a professor of mathematics at the University of Colorado since 1979. He was visiting fellow at Princeton University ...

4(1994), 49 - 56. * Norman Johnson ''Uniform Polytopes'', Manuscript (1991) * * * Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,

** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', ath. Zeit. 46 (1940) 380-407, MR 2,10(1.9 Uniform space-fillings) * A. Andreini, ''Sulle reti di poliedri regolari e semiregolari e sulle corrispondenti reti correlative'' (On the regular and semiregular nets of polyhedra and on the corresponding correlative nets), Mem. Società Italiana della Scienze, Ser.3, 14 (1905) 75–129. * D. M. Y. Sommerville, ''An Introduction to the Geometry of n Dimensions.'' New York, E. P. Dutton, 1930. 196 pp. (Dover Publications edition, 1958) Chapter X: The Regular Polytopes *
Uniform Honeycombs in 3-Space: 15-Batatoh
{{Honeycombs 3-honeycombs

Construction

Symmetry

Related polyhedra

See also

References