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 ...

, it is possible to fill 3/4 of the volume of three-dimensional

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...

by three sets of infinitely-long square

prism PRISM is a code name for a program under which the United States National Security Agency (NSA) collects internet communications from various U.S. internet companies. The program is also known by the SIGAD . PRISM collects stored internet ...

s aligned with the three coordinate axes, leaving cubical voids;

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 ...

, Heidi Burgiel and

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 ...

have named this structure tetrastix.

Applications

The motivation for some of the early studies of this structure was its applications in the

crystallography 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 (; "clear ice, rock-crystal"), and (; "to write"). In J ...

of crystal structures formed by rod-shaped molecules. Shrinking the square cross-sections of the prisms slightly causes the remaining space, consisting of the cubical voids, to become linked up into a single polyhedral set, bounded by axis-parallel faces. Polyhedra constructed in this way from finitely many prisms provide examples of axis-parallel polyhedra with

n

vertices and faces that require

\Omega(n^)

pieces when subdivided into convex pieces; they have been called Thurston polyhedra, after

William Thurston William Paul Thurston (October 30, 1946August 21, 2012) was an American mathematician. He was a pioneer in the field of low-dimensional topology and was awarded the Fields Medal in 1982 for his contributions to the study of 3-manifolds. Thurst ...

, who suggested using these shapes for this lower bound application. Like the

Schönhardt polyhedron In geometry, a Schönhardt polyhedron is a polyhedron with the same combinatorial structure as a regular octahedron, but with dihedral angles that are non-convex along three disjoint edges. Because it has no interior diagonals, it cannot be ...

, these polyhedra have no

triangulation In trigonometry and geometry, triangulation is the process of determining the location of a point by forming triangles to the point from known points. Applications In surveying Specifically in surveying, triangulation involves only angle m ...

into tetrahedra unless additional vertices are introduced. Anduriel Widmark has used the tetrastix and hexastix structures as the basis for artworks made from glass rods, fused to form tangled knots.

Related structures

The space occupied by the union of the prisms can be divided into the prisms of the tetrastix structure in two different ways. If the prisms are divided into unit cubes, offset by half a unit from the integer grid aligned with the prism sides, then these cubes together with the unit cube voids of the tetrastix structure form a tiling of space by cubes, combinatorially equivalent to the

Weaire–Phelan structure 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 solutio ...

for tiling space with unit volumes of low surface area. The tetrastix and Weaire–Phelan structures have the same group of symmetries., "Understanding the Irish Bubbles", p. 351. Although this cube tiling includes some cubes (the ones filling the voids of the tetrastix) that do not meet face-to-face with any other cube, results of

Oskar Perron Oskar Perron (7 May 1880 – 22 February 1975) was a German mathematician. He was a professor at the University of Heidelberg from 1914 to 1922 and at the University of Munich from 1922 to 1951. He made numerous contributions to differentia ...

Keller's conjecture In geometry, Keller's conjecture is the conjecture that in any Tessellation, tiling of -dimensional Euclidean space by identical hypercubes, there are two hypercubes that share an entire -dimensional face with each other. For instance, in any til ...

prove that (like the cubes within each prism of the tetrastix) every tiling of three-dimensional space by unit cubes must include an infinite column of cubes that all meet face-to-face. Similar constructions to the tetrastix are possible with triangular and hexagonal prisms, in four directions, called by Conway et al. "tristix" and hexastix.

References

{{reflist, refs= {{citation , last1 = Carrigan , first1 = Braxton , last2 = Bezdek , first2 = András , contribution = Tiling polyhedra with tetrahedra , contribution-url = https://2012.cccg.ca/papers/paper55.pdf , pages = 217–222 , title = Proceedings of the 24th Canadian Conference on Computational Geometry, CCCG 2012, Charlottetown, Prince Edward Island, Canada, August 8-10, 2012 , year = 2012 {{citation , last1 = Conway , first1 = John H. , author1-link = John Horton Conway , last2 = Burgiel , first2 = Heidi , last3 = Goodman-Strauss , first3 = Chaim , author3-link = Chaim Goodman-Strauss , contribution = Polystix , isbn = 978-1-56881-220-5 , mr = 2410150 , pages = 346–348 , publisher = A K Peters , location = Wellesley, Massachusetts , title = The Symmetries of Things , title-link = The Symmetries of Things , contribution-url = https://books.google.com/books?id=Drj1CwAAQBAJ&pg=PA346 , year = 2008 {{citation , last = Widmark , first = Anduriel , date = April 2020 , doi = 10.1080/17513472.2020.1734517 , issue = 1–2 , journal = Journal of Mathematics and the Arts , pages = 167–169 , title = Stixhexaknot: A symmetric cylinder arrangement of knotted glass , volume = 14 {{cite book , last = Widmark , first = Anduriel , title = Polystix Adventures: An Artist’s Guide Through the Geometry of Hexastix and Beyond , date = 2024 , publisher = Anduriel Widmark , location = Denver, CO , isbn = 978-1-304-51803-3 {{Citation , last1=Perron , first1=Oskar , author-link = Oskar Perron , title=Über lückenlose Ausfüllung des

n

-dimensionalen Raumes durch kongruente Würfel , doi=10.1007/BF01181421 , mr = 0003041 , year=1940 , journal=

Mathematische Zeitschrift ''Mathematische Zeitschrift'' ( German for ''Mathematical Journal'') is a mathematical journal for pure and applied mathematics published by Springer Verlag. History The journal was founded in 1917, with its first issue appearing in 1918. It wa ...

, volume=46 , pages=1–26, s2cid=186236462 , ref=none ; {{Citation , last1=Perron , first1=Oskar , author-mask=2 , title=Über lückenlose Ausfüllung des

n

-dimensionalen Raumes durch kongruente Würfel. II , doi=10.1007/BF01181436 , mr = 0006068 , year=1940 , journal=

, volume=46 , pages=161–180, s2cid=123877436 , ref=none {{citation , last1 = Paterson , first1 = Mike , author1-link = Mike Paterson , last2 = Yao , first2 = F. Frances , author2-link = Frances Yao , editor-last = Mehlhorn , editor-first = Kurt , editor-link = Kurt Mehlhorn , contribution = Binary partitions with applications to hidden surface removal and solid modelling , doi = 10.1145/73833.73836 , pages = 23–32 , publisher = ACM , location = New York , title = Proceedings of the Fifth Annual Symposium on Computational Geometry, Saarbrücken, Germany, June 5-7, 1989 , year = 1989, url = http://wrap.warwick.ac.uk/60834/12/WRAP_cs-rr-139.pdf {{citation , last1 = O'Keeffe , first1 = M. , last2 = Andersson , first2 = Sten , date = November 1977 , doi = 10.1107/s0567739477002228 , issue = 6 , journal = Acta Crystallographica Section A , pages = 914–923 , title = Rod packings and crystal chemistry , volume = 33 {{citation , last = Holden , first = Alan , page = 161 , publisher = Columbia University Press , location = New York , title = Shapes, Space, and Symmetry , year = 1971; reprinted by Dover, 1991 Cubes

Applications

Related structures

See also

References