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Polystix Blog Bibliography  

Figure 130: Polystix.  

As we near the end of this topical survey of polystix modeling, I want to emphasize the open-ended nature of these modeling projects, and that problem-solving does not always require a straight forward path or defined ending (divergent versus a convergent approach).  

Polystix modeling projects offer many opportunities for understanding and designing space. Symmetric arrangements made with linear elements can be expanded in infinite directions, both geometric and atheistic. Developing familiarity with polystix can be a fun way to learn some mathematics and sculpture design, and is also useful in the development and understanding of: tensegrity, geodesics, kinematics, deployable structures, complex symmetric weaving, linked spirals, connected lattice paths, periodic nets, links, and knots, material design, liquid crystals, rectilinear chemistry, minimal surfaces, and advanced puzzle solving and design, to name just a few possible applications. 

I plan on taking a short hiatus from this blog to focus on some other projects. However, I may return to update the blog with novel discoveries and introduce new questions, as I develop level specific lesson plans and new polystix art works. The best place to keep up with my related research and art work is on social media, instagram.com/andurielstudios, and my website.  

I value all of the feed-back from interested readers and appreciate those of you that have stuck along this far. I hope this polystix blog and papers, [10], [11], help promote an appreciation and enjoyment of polystix and the possibilities within mathematics and the arts.  

Polystix Blog Bibliography 

[1] A. Bakker. https://www.antonbakker.com/artist-statement

[2] A. Ericson. https://alejandroerickson.com/2011/09/27/090-how-to-make-hexastix-detailed-photo-instructions.html.

[3] A. Ericson. https://www.youtube.com/watch?v=2WtyMP1n5Js&t=1s

[4] A. Hizume. http://www.starcage.org

[5] A. Holden. Shapes, Space, and Symmetery, Columbia Univ Pr, 1971. 

[6] A. H. Schoen. https://schoengeometry.com.

[7] A. Schoen. Infinite periodic minimal surfaces without self-intersections. No. C-98. 1970. https://ntrs.nasa.gov/citations/19700020472

[8] A. H. Schoen. Shapes of Soap Films': Triply-Periodic Minimal Surfaces part 1, https://www.youtube.com/watch?v=JulrXPr19hs

[9] A. Holden. Orderly Tangles: Cloverleafs, Gordian Knots and Regular Polylinks, 1983.

[10] A. Widmark. https://archive.bridgesmathart.org/2021/bridges2021-293.pdf.  

[11] A. Widmark. https://archive.bridgesmathart.org/2022/bridges2022-379.pdf.  

[12] A. Widmark. https://www.tandfonline.com/doi/abs/10.1080/17513472.2020.1734517?journalCode=tmaa20

[13] A. Widmark. http://gallery.bridgesmathart.org/exhibitions/2015-bridges-conference/anduriel-widmark.

[14] A. Widmark. http://gallery.bridgesmathart.org/exhibitions/2016-bridges-conference/anduriel-widmark.  

[15] A. Widmark. http://gallery.bridgesmathart.org/exhibitions/2016-joint-mathematics-meetings/anduriel-widmark.

[16] A. Widmark. http://gallery.bridgesmathart.org/exhibitions/2017-bridges-conference/anduriel-widmark.

[17] A. Widmark. http://gallery.bridgesmathart.org/exhibitions/2017-joint-mathematics-meetings/anduriel-widmark.  

[18] A. Widmark. http://gallery.bridgesmathart.org/exhibitions/2018-bridges-conference/anduriel-widmark.

[19] A. Widmark. http://gallery.bridgesmathart.org/exhibitions/2018-joint-mathematics-meetings/anduriel-widmark.

[20] A. Widmark. http://gallery.bridgesmathart.org/exhibitions/2019-bridges-conference/anduriel-widmark.

[21] A. Widmark. http://gallery.bridgesmathart.org/exhibitions/2019-joint-mathematics-meetings/anduriel-s-widmark.

[22] A. Widmark. http://gallery.bridgesmathart.org/exhibitions/2020-bridges-conference/anduriel-widmark.

[23] A. Widmark. http://gallery.bridgesmathart.org/exhibitions/2020-joint-mathematics-meetings/anduriel-widmark.  

[24] A. Widmark. http://gallery.bridgesmathart.org/exhibitions/2021-bridges-conference/anduriel-widmark

[25] A. Widmark. http://gallery.bridgesmathart.org/exhibitions/2021-joint-mathematics-meetings/anduriel-widmark.

[26] A. Widmark. http://gallery.bridgesmathart.org/exhibitions/2022-bridges-conference/anduriel-widmark.

[27] A. Widmark. http://gallery.bridgesmathart.org/exhibitions/2022-bridges-conference-short-film-festival/anduriel-widmark

[28] A. Widmark. http://gallery.bridgesmathart.org/exhibitions/2022-joint-mathematics-meetings/anduriel-widmark.  

[29] A. Widmark. https://www.youtube.com/watch?v=EGT0SbrDL-0.  

[30] A. Widmark. https://www.youtube.com/watch?v=3hsMPOvyECw&t=53s

[31] A. Widmark. https://www.youtube.com/watch?v=XilncX8FLXQ.

[32] A. Widmark. https://www.youtube.com/watch?v=_QZiHaZEt8U.

[33] A. Widmark. https://www.andurielstudios.com/

[34] C. Séquin. Tubular Sculptures, Bridges Proceedings 2009: Pages 87–96. 

[35] C. Heyring. https://patents.google.com/patent/US8909505.

[36] E. J. Janse van Rensburg, and A. Rechnitzer. Minimal knotted polygons in cubic lattices, Journal of Statistical Mechanics: Theory and Experiment, Volume 2011

[37] E. Veit . A Cubic Archimedean Screw. Phil. Trans. R. Soc. 1996, A.354 2071–2075.

[38] G. Hart. https://archive.bridgesmathart.org/2005/bridges2005-449.html#gsc.tab=0, 2005. 

[39] G. Hart. https://archive.bridgesmathart.org/2011/bridges2011-357.html#gsc.tab=0, 2011. 

[40] G. Hart. 72 Pencils, https://www.georgehart.com/sculpture/pencils.html

[41] G. Wyvill. Nova plexus, http://www.cs.otago.ac.nz/graphics/Geoff/Geoff.html

[42] https://en.wikipedia.org/wiki/Tetrastix.

[43] https://en.wikipedia.org/wiki/Hexastix.  

[44] J. Conway, H. Burguel, C. Goodman-Strauss. The Symmetries of Things, CRC Press, Boca Raton, FL, 2008. 

[45] J. G. Parkhouse. and A. Kelly. The Regular Packing of Fibres in Three Dimensions. Proc. R. Soc. 1998, A.4541889–1909. 

[46] J. and J. Kosticks. http://www.kosticks.com

[47] J. Mallos http://weaveanything.blogspot.com .

[48] J. R. Branfield. Geoboard Geometery. The Mathematical Gazzette, vol. 54, no. 390, 1970, pp. 359–361. 

[49] K. Snelson. http://kennethsnelson.net/tensegrity/ .

[50] L. Norlén, and A. Al-Amoudi. Stratum corneum keratin structure, function, and formation: the cubic rod-packing and membrane templating model, J Invest Dermatol, Oct;123(4):715-32. 2004.  

[51] M. Evans, V. Robins,  and S. T Hyde. Ideal Geometry of Periodic Entanglements, Proc. R. Soc. 2015.

[52] M. Oster, M. A. Dias, T.D. Wolff, and M. E. Evans. Reentrant tensegrity: A three-periodic, chiral, tensegrity structure that is auxetic, SCIENCE ADVANCES, Vol 7, Issue 50, 2021. 

[53] M. O'Keeffe and S. Andersson, Rod Packings and Crystal Chemistry, Acta Cryst, 1977. A33, 914-923. 

[54] M. O'Keeffe, Cubic Cylinder Packings, Acta Cryst. 1992. A48, 879-884. 

[55] M. O'Keeffe, J. Plévert, Y. Teshima, Y. Watanabe and T. Ogama, The invariant cubic rod (cylinder) packings: symmetries and coordinates, Acta Cryst. 2001. A57, 110-111. 

[56] M. O'Keeffe, J. Plévert and T. Ogawa, Homogeneous cubic cylinder packings revisited, Acta Cryst. 2002. A58, 125-132. 

[57] M. O'Keeffe, M. M. J. Treacy. Isogonal Weavings on the Sphere Acta Cryst. 2020. A76, 661-621.  

[58] M. O'Keeffe, B. Hyde, Crystal Structures: Patterns and Symmetry, Dover, 2020. 

[59] M. Weber, https://theinnerframe.org/2015/09/30/avoiding-collisions-spirals-i/.  

[60] M. Weber, https://theinnerframe.org/2016/10/24/cylinders-and-triangles/. 

[61] O. V._Deventer. https://i.materialise.com/forum/t/bamboozle-spirals-xl-by-oskar/1606

[62] Precht Architects. "Wild Child Village", https://www.precht.at/wild-child-village/

[63] P. Saludjian* and  F. Reiss-Husson, Structure of the body-centered cubic phase of lipid systems. Proc Natl Acad Sci U S A.  77(12), 1980. 

[64] P. Gailiunas. Rods, Helices and Polyhedra, Journal of Mathematics and the Arts, 2021. 15:3-4, 218-231. 

[65] P. J. Pearce  https://pjpearcedesign.com/index.php/ethos/morphology/.

[66] P. Tucker. Moorish Fretwork Revisited, Bridges Conference Proceedings, 2012. 

[67] Rahman, Md, M.Said, Suhana, & S, Balamurugan. Blue phase liquid crystal: Strategies for phase stabilization and device development. Science and Technology of Advanced Materials. 16. 2015.

[68] R. Lang. https://langorigami.com/article/polypolyhedra/.

[69] S. Andersson, M. Jacob, The Nature of Mathematics and the Mathematics of Nature, 1998. 

[70] S. Coffin. Geometric Puzzle Design, A K Peters Ltd, Massachusetts, 2007. https://johnrausch.com/PuzzlingWorld/default.htm. 

[71] S. Bozoki http://old.sztaki.hu/~bozoki/bozoki_en.html

[72] T. Ogawa, Y. Teshima & Y Watanabe ,Geometry and Crystallography of Self-Supporting Rod Structures, Katachi ∪Symmetry pp 239–246, 1996. 

[73] T. Verhoeff and K. Verhoeff. “From Chain-link Fence to Space-Spanning Mathematical Structures.” Bridges Conference Proceedings, Coimbra, Portugal, Jul. 27–31, 2011, pp. 73–80. 

[74] T. Verhoeff and K. Verhoeff. “The Mathematics of Mitering and its Artful Application.” Bridges Conference Proceedings, Leeuwarden, the Netherlands, Jul. 24–29, 2008, pp. 225–234. 

[75] W. Cutler. https://www.billcutlerpuzzles.com.

[76] Y. Teshima, Y. Watanabe, T. Ogawa. A New Structure of Cylinder Packing, JCDCG pp 351-361, 2001. 

[77] Y. Teshima, R. Higashida, T. Matsumoto. Cylinder packing with five directions, Acta Cryst. A70, C447 2017. 

[78] Y. Teshima, T. Matsumoto. Space Group of heterogeneous cylinder packings with six <110> directions. Glass physics and chemisty 38, 41-48, 2012. 

[79] Z. Abel. http://zacharyabel.com

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