Hexastix

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.

Applications

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. O'Keefe classifies this arrangement as the ''Γ'' or Garnet rod packing, 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. It has been proposed that stratum corneum's structure could be modeled using the hexastix cylinder packing geometry. Hexastix geometry has also found use in architecture, being used to construct a 3-story bamboo structure in Ecuador.
In recreational mathematics the hexastix arrangement can be found in the design of mechanical burr puzzles. 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. Hexastix is also seen in the sculpture titled "72 Pencils", made by math artist George W. Hart.

Related structures

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. Rod packings with more directions are also possible, as in the quasi-periodic 6 directional rod packing. The Hexahemioctacron is similarly made from hexagonal prisms but unlike hexastix, the prisms are intersecting.

See also

*Tetrastix
*Stick puzzle

References

{{reflist, refs=

{{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
, 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, bibcode = 1977AcCrA..33..914O

{{citation
, last1 = Coffin , first1 = Stewart , author1-link = Stewart Coffin
, isbn = 0198532075
, publisher = Oxford University Press
, title = The Puzzling World of Polyhedral Dissections
, year = 1990

{{cite web
, last1 = George , first1 = Hart , author1-link = George W. Hart
, title=72 Pencils
, url=https://www.georgehart.com/sculpture/pencils.html , publisher=George Hart
, access-date=15 December 2021

{{cite web , title=Wild Child Village , url=https://www.precht.at/wild-child-village/ , website=Precht Architects , access-date=25 January 2022

{{cite journal , last1=Norlén , first1=L , last2=Al-Amoudi , first2=A , title=Stratum corneum keratin structure, function, and formation: the cubic rod-packing and membrane templating model. , journal=The Journal of Investigative Dermatology , date=October 2004 , volume=123 , issue=4 , pages=715–32 , doi=10.1111/j.0022-202X.2004.23213.x , pmid=15373777 , doi-access=free

{{cite book , last1=Ogawa , first1=Tohru , last2=Teshima , first2=Yoshinori , last3=Watanabe , first3=Yoshinori , chapter=Geometry and Crystallography of Self-Supporting Rod Structures , title=Katachi ∪ Symmetry , date=1996 , pages=239–246 , doi=10.1007/978-4-431-68407-7_26 , isbn=978-4-431-68409-1 , chapter-url=https://link.springer.com/chapter/10.1007%2F978-4-431-68407-7_26 , access-date=26 January 2022

{{cite journal , last1=O'Keeffe , first1=M. , last2=Plévert , first2=J. , last3=Teshima , first3=Y. , last4=Watanabe , first4=Y. , last5=Ogama , first5=T. , title=The invariant cubic rod (cylinder) packings: symmetries and coordinates , journal=Acta Crystallographica Section A: Foundations of Crystallography , date=1 January 2001 , volume=57 , issue=1 , pages=110–111 , doi=10.1107/S010876730001151X, pmid=11124509 , doi-access=free

{{cite journal , last1=Rosi , first1=Nathaniel L. , last2=Kim , first2=Jaheon , last3=Eddaoudi , first3=Mohamed , last4=Chen , first4=Banglin , last5=O'Keeffe , first5=Michael , last6=Yaghi , first6=Omar M. , title=Rod Packings and Metal−Organic Frameworks Constructed from Rod-Shaped Secondary Building Units , journal=Journal of the American Chemical Society , date=1 February 2005 , volume=127 , issue=5 , pages=1504–1518 , doi=10.1021/JA045123O, pmid=15686384 , bibcode=2005JAChS.127.1504R

{{cite book , last1=Widmark , first1=Anduriel , title=BRIDGES : mathematics, art, music, architecture, culture. , date=2021 , publisher=TESSELLATIONS PUBLISHING , location=PHOENIX , isbn=978-1-938664-39-7 , pages=293–296 , url=http://archive.bridgesmathart.org/2021/bridges2021-293.html

{{cite book , last1=Widmark , first1=Anduriel , title= Polystix Sculpture Design Revisited. , date=2022 , publisher=TESSELLATIONS PUBLISHING , location=PHOENIX , isbn=978-1-938664-42-7 , pages=379–382 , url=http://archive.bridgesmathart.org/2022/bridges2022-379.html

{{cite book , last1=Widmark , first1=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

Cubes