| Hexastix | |
|---|---|
 | |
| Type | Hexagonal prism arrangement |
| Symmetry group | Ia3d |
| Properties | Non-intersecting, homogenous |
Hexastix is a symmetric arrangement of non-intersecting prisms that, when extended infinitely, fill exactly 3/4 of space. The prisms in a hexastix arrangement are all parallel to 4 directions on the body-centered cubic lattice. In The Symmetries of Things , John Horton Conway, Heidi Burgiel, and Chaim Goodman-Strauss named this structure hexastix. [1]
The hexastix arrangement has found use in mathematics, crystallography, reticular chemistry, puzzle design, and art. Michael O'Keeffe and associates define this structure as one of the 6 possible invariant cubic rod packing arrangements. [2] O'Keefe classifies this arrangement as the Γ or Garnet rod packing, [3] and describes it as the densest possible cubic rod packing. Rod packings are used to classify chains of atoms in crystal structures, and in the develop of materials like metal–organic frameworks. [4] It has been proposed that stratum corneum's structure could be modeled using the hexastix cylinder packing geometry. [5] Hexastix geometry has also found use in architecture, being used to construct a 3-story bamboo structure [6] in Ecuador. In recreational mathematics the hexastix arrangement can be found in the design of mechanical burr puzzles. [7] Stewart Coffin has used this geometry in the creation of complex non-rectilinear wooden puzzles. In art, hexastix is used by artist Anduriel Widmark to create complex glass knots. [8] Hexastix is also seen in the sculpture titled "72 Pencils", [9] made by math artist George W. Hart.
Non-intersecting prism arrangements with prime cubic symmetry make up the family "polystix". Related square and triangular prism structures in three and four directions, are named by Conway as tetrastix and "tristix". If the ends of the prisms in a hexastix arrangement are pointed, the directionality modifies the symmetry and the related structure is known as hexastakes [1] . Rod packings with more directions are also possible, as in the quasi-periodic 6 directional rod packing [10] . The Hexahemioctacron is similarly made from hexagonal prisms but unlike hexastix, the prisms are intersecting.
In geometry, a convex uniform honeycomb is a uniform tessellation which fills three-dimensional Euclidean space with non-overlapping convex uniform polyhedral cells.
Crystallography is the branch of science devoted to the study of molecular and crystalline structure and properties. The word crystallography is derived from the Ancient Greek word κρύσταλλος, and γράφειν. In July 2012, the United Nations recognised the importance of the science of crystallography by proclaiming 2014 the International Year of Crystallography.
In crystallography, crystal structure is a description of ordered arrangement of atoms, ions, or molecules in a crystalline material. Ordered structures occur from intrinsic nature of constituent particles to form symmetric patterns that repeat along the principal directions of three-dimensional space in matter.
In crystallography, a crystal system is a set of point groups. A lattice system is a set of Bravais lattices. Space groups are classified into crystal systems according to their point groups, and into lattice systems according to their Bravais lattices. Crystal systems that have space groups assigned to a common lattice system are combined into a crystal family.
In mathematics, physics and chemistry, a space group is the symmetry group of a repeating pattern in space, usually in three dimensions. The elements of a space group are the rigid transformations of the pattern that leave it unchanged. In three dimensions, space groups are classified into 219 distinct types, or 230 types if chiral copies are considered distinct. Space groups are discrete cocompact groups of isometries of an oriented Euclidean space in any number of dimensions. In dimensions other than 3, they are sometimes called Bieberbach groups.
In geometry, the Weaire–Phelan structure is a three-dimensional structure representing an idealised foam of equal-sized bubbles, with two different shapes. In 1993, Denis Weaire and Robert Phelan found that this structure was a better solution of the Kelvin problem of tiling space by equal volume cells of minimum surface area than the previous best-known solution, the Kelvin structure.
In geometry, the truncated hexagonal tiling is a semiregular tiling of the Euclidean plane. There are 2 dodecagons (12-sides) and one triangle on each vertex.
In geometry, the truncated square tiling is a semiregular tiling by regular polygons of the Euclidean plane with one square and two octagons on each vertex. This is the only edge-to-edge tiling by regular convex polygons which contains an octagon. It has Schläfli symbol of t{4,4}.
In geometry, the elongated triangular tiling is a semiregular tiling of the Euclidean plane. There are three triangles and two squares on each vertex. It is named as a triangular tiling elongated by rows of squares, and given Schläfli symbol {3,6}:e.
The cubic honeycomb or cubic cellulation is the only proper regular space-filling tessellation in Euclidean 3-space made up of cubic cells. It has 4 cubes around every edge, and 8 cubes around each vertex. Its vertex figure is a regular octahedron. It is a self-dual tessellation with Schläfli symbol {4,3,4}. John Horton Conway called this honeycomb a cubille.
The tetrahedral-octahedral honeycomb, alternated cubic honeycomb is a quasiregular space-filling tessellation in Euclidean 3-space. It is composed of alternating regular octahedra and tetrahedra in a ratio of 1:2.
In geometry, a honeycomb is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions. Its dimension can be clarified as n-honeycomb for a honeycomb of n-dimensional space.
In geometry, an octahedral prism is a convex uniform 4-polytope. This 4-polytope has 10 polyhedral cells: 2 octahedra connected by 8 triangular prisms.
A crystallographic database is a database specifically designed to store information about the structure of molecules and crystals. Crystals are solids having, in all three dimensions of space, a regularly repeating arrangement of atoms, ions, or molecules. They are characterized by symmetry, morphology, and directionally dependent physical properties. A crystal structure describes the arrangement of atoms, ions, or molecules in a crystal.
In crystallography, a periodic graph or crystal net is a three-dimensional periodic graph, i.e., a three-dimensional Euclidean graph whose vertices or nodes are points in three-dimensional Euclidean space, and whose edges are line segments connecting pairs of vertices, periodic in three linearly independent axial directions. There is usually an implicit assumption that the set of vertices are uniformly discrete, i.e., that there is a fixed minimum distance between any two vertices. The vertices may represent positions of atoms or complexes or clusters of atoms such as single-metal ions, molecular building blocks, or secondary building units, while each edge represents a chemical bond or a polymeric ligand.
The triakis truncated tetrahedral honeycomb is a space-filling tessellation in Euclidean 3-space made up of triakis truncated tetrahedra. It was discovered in 1914.
In geometry, the triakis truncated tetrahedron is a convex polyhedron made from 4 hexagons and 12 isosceles triangles. It can be used to tessellate three-dimensional space, making the triakis truncated tetrahedral honeycomb.
In geometry, John Horton Conway defines architectonic and catoptric tessellations as the uniform tessellations of Euclidean 3-space with prime space groups and their duals, as three-dimensional analogue of the Platonic, Archimedean, and Catalan tiling of the plane. The singular vertex figure of an architectonic tessellation is the dual of the cell of the corresponding catoptric tessellation, and vice versa. The cubille is the only Platonic (regular) tessellation of 3-space, and is self-dual. There are other uniform honeycombs constructed as gyrations or prismatic stacks which are excluded from these categories.
In geometry, it is possible to fill 3/4 of the volume of three-dimensional Euclidean space by three sets of infinitely-long square prisms aligned with the three coordinate axes, leaving cubical voids; John Horton Conway, Heidi Burgiel and Chaim Goodman-Strauss have named this structure tetrastix.