Boerdijk–Coxeter helix

<h2 id="geometry">Geometry</h2>
<p>The coordinates of vertices of Boerdijk–Coxeter helix composed of tetrahedrons with unit edge length can be written in the form
</p>

(
        r
        cos
        ⁡
        n
        θ
        ,
        r
        sin
        ⁡
        n
        θ
        ,
        n
        h
        )
      
    
    {\displaystyle (r\cos n\theta ,r\sin n\theta ,nh)}

<p>where 
  
    
      
        r
        =
        3
        
          
            3
          
        
        
          /
        
        10
      
    
    {\displaystyle r=3{\sqrt {3}}/10}
  
, 
  
    
      
        θ
        =
        ±
        
          cos
          
            −
            1
          
        
        ⁡
        (
        −
        2
        
          /
        
        3
        )
        ≈
        
          131.81
          
            ∘
          
        
      
    
    {\displaystyle \theta =\pm \cos ^{-1}(-2/3)\approx 131.81^{\circ }}
  
, 
  
    
      
        h
        =
        1
        
          /
        
        
          
            10
          
        
      
    
    {\displaystyle h=1/{\sqrt {10}}}
  
 and 
  
    
      
        n
      
    
    {\displaystyle n}
  
 is an arbitrary integer. The two different values of 
  
    
      
        θ
      
    
    {\displaystyle \theta }
  
 correspond to the two chiral forms. All vertices are located on the cylinder with radius 
  
    
      
        r
      
    
    {\displaystyle r}
  
 along z-axis. Given how the tetrahedra alternate, this gives an <i>apparent</i> twist of 
  
    
      
        2
        θ
        −
        
          
            4
            3
          
        
        π
        ≈
        
          23.62
          
            ∘
          
        
      
    
    {\displaystyle 2\theta -{\frac {4}{3}}\pi \approx 23.62^{\circ }}
  
 every <i>two</i> tetrahedra. There is another inscribed cylinder with radius 
  
    
      
        3
        
          
            2
          
        
        
          /
        
        20
      
    
    {\displaystyle 3{\sqrt {2}}/20}
  
 inside the helix.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a>
</p>
<h2 id="higher-dimensional-geometry">Higher-dimensional geometry</h2>

<p>The <a href="/facts/600-cell/tNBtDUe9">600-cell</a> partitions into <a href="/facts/600-cell/tNBtDUe9">20 rings</a> of 30 <a href="/facts/Tetrahedron/7mkSrl6j">tetrahedra</a>, each a Boerdijk–Coxeter helix.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> When superimposed onto the <a href="/facts/3-sphere/u67nImpY">3-sphere</a> curvature it becomes periodic, with a period of ten vertices, encompassing all 30 cells. The collective of such helices in the 600-cell represent a discrete <a href="/facts/Hopf_fibration/wRKjQMz0">Hopf fibration</a>.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> While in 3 dimensions the edges are helices, in the imposed 3-sphere <a href="/facts/Topology/2EqW47cU">topology</a> they are <a href="/facts/Geodesic/fbWR6l80">geodesics</a> and have no <a href="/facts/Torsion_of_a_curve/qXRnmtjy">torsion</a>. They spiral around each other naturally due to the Hopf fibration.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> The collective of edges forms another discrete Hopf <a href="/facts/600-cell/tNBtDUe9">fibration of 12 rings</a> with 10 vertices each. These correspond to <a href="/facts/120-cell/W4nroBjJ">rings of 10 dodecahedrons</a> in the dual <a href="/facts/120-cell/W4nroBjJ">120-cell</a>.
</p><p>In addition, the <a href="/facts/16-cell/X0YAfobw">16-cell</a> partitions into two <a href="/facts/16-cell/X0YAfobw">8-tetrahedron rings</a>, four edges long, and the <a href="/facts/5-cell/Rh6lwDcc">5-cell</a> partitions into a single degenerate <a href="/facts/5-cell/Rh6lwDcc">5-tetrahedron ring</a>.
</p>

<table><tbody><tr><th>4-polytope</th><th>Rings</th><th>Tetrahedra/ring</th><th>Cycle lengths</th><th>Net</th><th>Projection</th></tr><tr><th><a href="/facts/600-cell/tNBtDUe9">600-cell</a></th><td>20</td><td>30</td><td>30, 103, 152</td><td></td><td></td></tr><tr><th><a href="/facts/16-cell/X0YAfobw">16-cell</a></th><td>2</td><td>8</td><td>8, 8, 42</td><td colspan="2"></td></tr><tr><th><a href="/facts/5-cell/Rh6lwDcc">5-cell</a></th><td>1</td><td>5</td><td>(5, 5), 5</td><td colspan="2"></td></tr></tbody></table>
<h2 id="related-polyhedral-helices">Related polyhedral helices</h2>
<p>Equilateral <a href="/facts/Square_pyramid/wDetDohM">square pyramids</a> can also be chained together as a helix, with two <a href="/facts/Vertex_configuration/wWaBS6t0">vertex configurations</a>, 3.4.3.4 and 3.3.4.3.3.4. This helix exists as finite ring of <a href="/facts/Rectified_600-cell/S2DeX7LG">30 pyramids in a 4-dimensional polytope</a>.
</p>

<p>And equilateral <a href="/facts/Pentagonal_pyramid/tN5CBPjK">pentagonal pyramids</a> can be chained with 3 vertex configurations, 3.3.5, 3.5.3.5, and 3.3.3.5.3.3.5:
</p>

<h2 id="in-architecture">In architecture</h2>
<p>The <a href="/facts/Art_Tower_Mito/5Au8qJ9b">Art Tower Mito</a> is based on a Boerdijk–Coxeter helix.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/600-cell/tNBtDUe9">Clifford parallel cell rings</a></li>
<li><a href="/facts/Toroidal_polyhedron/g6SS84k2">Toroidal polyhedron</a></li>
<li><a href="/facts/Line_group/tcAFjLCy">Line group#Helical symmetry</a></li>
<li><a href="/facts/Skew_apeirogon/l0Y1Kg6c">Skew apeirogon#Helical apeirogons in 3-dimensions</a></li></ul>
<h2 id="notes">Notes</h2>

<ul><li><a href="/facts/H.S.M._Coxeter/D2l5jWGG">Coxeter, H. S. M.</a> (1974). <a href="https://archive.org/details/regularcomplexpo0000coxe"><i>Regular Complex Polytopes</i></a>. Cambridge University Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 052120125X.</li>
<li>Boerdijk, A.H. (1952). "Some remarks concerning close-packing of equal spheres". <i>Philips Res. Rep</i>. 7: 303–313.</li>
<li><a href="/facts/Buckminster_Fuller/fLvx87hc">Fuller, R.Buckminster</a> (1975). Applewhite, E.J. (ed.). <a href="http://www.rwgrayprojects.com/synergetics/toc/toc.html"><i>Synergetics</i></a>. Macmillan.</li>
<li>Pugh, Anthony (1976). "5. Joining polyhedra §5.36 Tetrahelix". <i>Polyhedra: A visual approach</i>. University of California Press. p. 53. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-520-03056-5.</li>
<li>Sadler, Garrett; Fang, Fang; Kovacs, Julio; Klee, Irwin (2013). "Periodic modification of the Boerdijk-Coxeter helix (tetrahelix)". <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/1302.1174v1">1302.1174v1</a> [<a href="https://arxiv.org/archive/math.MG">math.MG</a>].</li>
<li>Lord, E.A.; Ranganathan, S. (2004). <a href="https://ericlord.neocities.org/ericsfiles/pdfs/65.pdf">"The γ-brass structure and the Boerdijk–Coxeter helix"</a> (PDF). <i>Journal of Non-Crystalline Solids</i>. 334–335: 123–5. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2004JNCS..334..121L">2004JNCS..334..121L</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2Fj.jnoncrysol.2003.11.069">10.1016/j.jnoncrysol.2003.11.069</a>.</li>
<li>Zhu, Yihan; He, Jiating; Shang, Cheng; Miao, Xiaohe; Huang, Jianfeng; Liu, Zhipan; Chen, Hongyu; Han, Yu (2014). <a href="https://doi.org/10.1021%2Fja506554j">"Chiral Gold Nanowires with Boerdijk–Coxeter–Bernal Structure"</a>. <i>J. Am. Chem. Soc</i>. 136 (36): 12746–52. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2014JAChS.13612746Z">2014JAChS.13612746Z</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1021%2Fja506554j">10.1021/ja506554j</a>. <a href="/facts/PMID_(identifier)/JlHAvMHt">PMID</a> <a href="https://pubmed.ncbi.nlm.nih.gov/25126894">25126894</a>.</li>
<li>Lord, Eric A.; Mackay, Alan L.; Ranganathan, S. (2006). <a href="https://books.google.com/books?id=s22_x-O8pEoC&pg=PP1">"§4.5 The Boerdijk–Coxeter helix"</a>. <i>New Geometries for New Materials</i>. Cambridge University Press. p. 64. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-86104-5.</li>
<li>Banchoff, Thomas F. (1989). "Geometry of the Hopf Mapping and Pinkall's Tori of Given Conformal Type". In Tangora, Martin (ed.). <i>Computers in geometry and topology</i>. New York and Basel: Marcel Dekker. pp. 57–62.</li>
<li>Banchoff, Thomas F. (2013). "Torus Decompostions of Regular Polytopes in 4-space". In Senechal, Marjorie (ed.). <a href="https://archive.org/details/shapingspaceexpl00sene"><i>Shaping Space</i></a>. Springer New York. pp. <a href="https://archive.org/details/shapingspaceexpl00sene/page/n249">257</a>–266. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-0-387-92714-5_20">10.1007/978-0-387-92714-5_20</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-92713-8.</li>
<li>Sadoc, J.F.; Rivier, N. (1999). "Boerdijk-Coxeter helix and biological helices". <i>The European Physical Journal B</i>. 12 (2): 309–318. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1999EPJB...12..309S">1999EPJB...12..309S</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2Fs100510051009">10.1007/s100510051009</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:92684626">92684626</a>.</li>
<li>Sadoc, Jean-Francois (2001). <a href="https://www.researchgate.net/publication/260046074">"Helices and helix packings derived from the {3,3,5} polytope"</a>. <i><a href="/facts/European_Physical_Journal_E/57aBLWKX">European Physical Journal E</a></i>. 5: 575–582. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2Fs101890170040">10.1007/s101890170040</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:121229939">121229939</a>.</li></ul>

<h2 id="external-links">External links</h2>
<ul><li><a href="http://flickriver.com/photos/fdecomite/5403437189/">Boerdijk-Coxeter helix animation</a></li>
<li><a href="http://www.rwgrayprojects.com/rbfnotes/helix/helix01.html">Tetrahelix Data</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Sadoc & Rivier 1999, p. 314, §4.2.2 The Boerdijk-Coxeter helix and the PPII helix; the helix of tetrahedra occurs in a left- or right-spiraling form, but each form contains both left- and right-spiraling helices of linked edges. - Sadoc, J.F.; Rivier, N. (1999). "Boerdijk-Coxeter helix and biological helices". The European Physical Journal B. 12 (2): 309–318. Bibcode:1999EPJB...12..309S. doi:10.1007/s100510051009. S2CID 92684626. <a href="https://ui.adsabs.harvard.edu/abs/1999EPJB...12..309S" target="_blank">https://ui.adsabs.harvard.edu/abs/1999EPJB...12..309S</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Sadler et al. 2013. - Sadler, Garrett; Fang, Fang; Kovacs, Julio; Klee, Irwin (2013). "Periodic modification of the Boerdijk-Coxeter helix (tetrahelix)". arXiv:1302.1174v1 [math.MG]. <a href="https://arxiv.org/abs/1302.1174v1" target="_blank">https://arxiv.org/abs/1302.1174v1</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Fuller 1975, 930.00 Tetrahelix. - Fuller, R.Buckminster (1975). Applewhite, E.J. (ed.). Synergetics. Macmillan. <a href="http://www.rwgrayprojects.com/synergetics/toc/toc.html" target="_blank">http://www.rwgrayprojects.com/synergetics/toc/toc.html</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>"Tetrahelix Data". <a href="http://www.rwgrayprojects.com/rbfnotes/helix/helix01.html" target="_blank">http://www.rwgrayprojects.com/rbfnotes/helix/helix01.html</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Sadoc 2001, pp. 577–578, §2.5 The 30/11 symmetry: an example of other kind of symmetries. - Sadoc, Jean-Francois (2001). "Helices and helix packings derived from the {3,3,5} polytope". European Physical Journal E. 5: 575–582. doi:10.1007/s101890170040. S2CID 121229939. <a href="https://www.researchgate.net/publication/260046074" target="_blank">https://www.researchgate.net/publication/260046074</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>Banchoff 2013, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the Clifford torus which correspond to Hopf fibrations. - Banchoff, Thomas F. (2013). "Torus Decompostions of Regular Polytopes in 4-space". In Senechal, Marjorie (ed.). Shaping Space. Springer New York. pp. 257–266. doi:10.1007/978-0-387-92714-5_20. ISBN 978-0-387-92713-8. <a href="https://archive.org/details/shapingspaceexpl00sene" target="_blank">https://archive.org/details/shapingspaceexpl00sene</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
<li id="fn:7"><p>Banchoff 1989. - Banchoff, Thomas F. (1989). "Geometry of the Hopf Mapping and Pinkall's Tori of Given Conformal Type". In Tangora, Martin (ed.). Computers in geometry and topology. New York and Basel: Marcel Dekker. pp. 57–62. <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li>
</ol>

Boerdijk–Coxeter helix open-in-new

Boerdijk–Coxeter helix