Mathematical Modeling of the Origami Navel Shell

Authors

  • Gurmeher Kaur Chapel Hill High School
  • Crystal Soong Chapel Hill High School
  • Professor Jasleen Kaur Mentor, University of North Carolina Chapel Hill

DOI:

https://doi.org/10.47611/jsrhs.v10i3.1573

Keywords:

Mathematical Modeling, Origami, Curve Fitting, Nautilus Shell, Geometric Model, Spiral

Abstract

The well-known Nautilus shell has been modeled extensively both by mathematicians and origamists. However, there is wide disagreement on the best-fitting mathematical model — partly because there is significant variability across different Nautilus Shells found in nature, and no single model can describe all of them well. Origami structures, however, have precise repeatable folding instructions, and do not exhibit such variability. Ironically, no known mathematical models exist for these structures. In this research, we mathematically model a prominent origami design, the Navel Shell by Tomoko Fuse, believed to be based on the Nautilus.

We use first-principles geometric and trigonometric constructs for developing a non-smooth Geometric Model of the ideal origami spiral. We then search for the best-fitting parametric smooth spiral approximation, by formulating the fitting problem as a minimization problem over four unknowns. We write a Python computer program for searching the space numerically. Our evaluations show that: (i) the Smooth spiral is an excellent fit for the Geometric Model; (ii) our models for Origami Navel Shell are different from prior mathematical models for the Nautilus shell, but they come close to a recent model for a rare species of Nautilus; (iii) the Geometric Model explains the outer edges of origami images quite well and helps identify construction errors in the inner edges; and (iv) the Smooth Model helps understand how well the ideal Navel Shell matches different variants of the Nautilus species. We hope our research lays the foundation for further mathematical modeling of origami structures. 

Downloads

Download data is not yet available.

References or Bibliography

American Institute of Physics. "Origami Helps Scientists Solve Problems." ScienceDaily. ScienceDaily, Feb 2002, https://www.sciencedaily.com/releases/2002/02/020219080203.htm. Accessed 7 Dec 2020.

Asiel, Fatma. “Origami Applications in the Past and Present.” Bibalex, 5 Dec 2017, https://www.bibalex.org/SCIplanet/en/Article/Details?id=10309. Accessed 9 Dec 2020.

Avila, A., Magleby, S. P., Lang, R. J., and Howell, L. L.: Origami fold states: concept and design tool, Mech. Sci., 10, 91–105, 2019.

Bartlett, Christopher. “Nautilus Spirals and the Meta-Golden Ratio Chi.” Nexus Network Journal, vol. 20, no. 3, 2018, p. 279.

Borcherds, M. “GeoGebra Classic.” 2001, https://www.geogebra.org/classic.

Brown, Dan. The Da Vinci Code. 2003, USA. New York: Doubleday.

Chu, Jennifer. “Here Comes the Sun. ” MIT News, 2012, https://news.mit.edu/2012/sunflower-concentrated-solar-0111. Accessed 5 Dec 2020.

Dai, H., Pears, N., Smith, W. et al. “Statistical Modeling of Craniofacial Shape and Texture”. International Journal of Computer Vision, 128, 547–571, 2020.

Demaine, Erik D.; O'Rourke, Joseph Geometric folding algorithms. 2007, Cambridge: Cambridge University Press. doi:10.1017/CBO9780511735172.Demaine, Erik D.; O'Rourke, Joseph Geometric folding algorithms. 2007, Cambridge: Cambridge University Press. ISBN 978-0-521-85757-4. MR 2354878.

Demaine, Erik. “Erik Demaine's Folding and Unfolding Page.” Erik Demaine, 2020, http://erikdemaine.org/folding/. Accessed 4 Dec 2020.

Delvin, Keith. “The Man of Numbers: in Search of Leonardo Fibonacci.” The Man of Numbers, 2010, https://www.maa.org/external_archive/devlin/Fibonacci.pdf. accessed 6 Dec 2020.

Devlin, Keith. The Myth That Will Not Go Away. 2007, https://www.maa.org/external_archive/devlin/devangle.html. Accessed 6 Dec 2020.

Du Sautoy, M. The Secrets of the Nautilus Shell - The Code. Episode 1, BBC Two, 2011. https://www.youtube.com/watch?v=Ysw3iM3ENQA. Accessed 6 Dec 2020.

Falbo, Clement. The Golden Ratio—A Contrary Viewpoint. The College Mathematics Journal. 36, 2005.

Fletcher, R. Proportion and the Living World. Parabola 13(1): 36-51, 1988.

Fuse, Tomoko. Spiral Origami Art Design. Viereck Verlag, 2012.

Geretschläger, Robert. Geometric Origami. 2008, UK: Arbelos. ISBN 978-0-9555477-1-3.

Happy Folding. Navel Shell (Tomoko Fuse) Instructions. https://www.happyfolding.com/instructions-fuse-navel_shell. Accessed 5 Dec 2020.

Hart, George W. “Replicator Constructions.” Replicator Constructions, 2010, http://www.georgehart.com/rp/replicator/replicator.html. Accessed 6 Dec 2020.

Impens, C. Debunking golden ratio shells, 1 & 2. Chris Impens @ Valvas, 2016, http://ci47.blogspot.be/2016/08/debunking-golden-ratio-shells-1.html. Accessed 5 Dec 2020.

Lang, Robert J. "From Flapping Birds to Space Telescopes: The Modern Science of Origami". Usenix Conference, Boston, MA, 2008.

Livio, M. The Golden Ratio: The Story of PHI, the World’s Most Astonishing Number. 2003, Crown, New York: Broadway.

long_quach. “Nautilus.” Neorigami, 2013, https://neorigami.com/neo/index.php/en/geometric-a-abstract/item/6916-nautilus. Accessed 6 Dec 2020.

Malkevitch, Joseph. Mathematics and Art. Feature Column, American Mathematical Society, 2003, http://www.ams.org/publicoutreach/feature-column/fcarc-art1 . Accessed 5 Dec 2020.

Maor, Eli, and Eugen Jost. “Twisted Math and Beautiful Geometry.” American Scientist, Volume 102, Number 2, 2014.

Math Tourist. “Sea Shell Spirals.” The Mathematical Tourist, 2020, http://mathtourist.blogspot.com/2020/06/. Accessed 5 Dec 2020.

McMahon, T. A. and J. T. Bonner. On Size and Life. 1983, New York: Scientific American Books, 1st Edition.

Meisner, Gary. “Is the Nautilus Shell Spiral a Golden Spiral?” GoldenNumber.net, 2014, https://www.goldennumber.net/nautilus-spiral-golden-ratio/. Accessed 5 Dec 2020.

Meisner, Gary. “Spirals and the Golden Ratio.” The Golden Number, 2012, https://www.goldennumber.net/spirals/. Accessed 6 Dec 2020.

Mukhopadhyay, Utpal. “Logarithmic Spiral-A Splendid Curve.” Resonance. 9. P. 39-45. 10.1007/BF02834971, 2004.

Quarteroni, A., Saleri, F., Gervasio, P., “Scientific computing with MATLAB and Octave”. 4th edition, 2014, Springer-Verlag Berlin Heidelberg.

Rogers, Kara. “Scientific Modeling.” Britannica, 2012, https://www.britannica.com/science/scientific-modeling. Accessed 7 Dec 2020.

Rusczyk, Richard. Art of Problem Solving Introduction to Geometry. vol. 2, Aops, Incorporated, 2013.

Ryan, M. Geometry for Dummies. Hoboken: Wiley, 2016.

Sharp, John. “Spirals and the Golden Section.” Nexus Network Journal, vol. 4, no. 1, p. 59-82, 2002.

Smithsonian Ocean Portal website. https://ocean.si.edu/ocean-life/invertebrates/couple-nautiluses. Accessed 8 Dec 2020.

StackExchange. “Distance between a point and a spiral.” September 2013, https://math.stackexchange.com/questions/175106/distance-between-point-and-a-spiral. Accessed 9 Dec 2020.

Strang, Gilbert, and Edwin Herman. “10.3: Polar Coordinates.” LibreTexts, 2019, https://math.libretexts.org/Courses/University_of_California_Davis/UCD_Mat_21C%3A_Multivariate_Calculus/10%3A_Parametric_Equations_and_Polar_Coordinates/10.3%3A_Polar_Coordinates. Accessed 5 Dec 2020.

Strogatz, S. Me, Myself and Math, Proportion Control. 2012, https://opinionator.blogs.nytimes.com/2012/09/24/proportion-control/. Accessed 5 Dec 2020.

Thompson, D’Arcy. On Growth and Form: The Complete Revised Edition. 1992, New York: Dover Publications.

UCSF Department of Radiology & Biomedical Imaging. “Benefits of Imaging using Radiation.” UCSF, 2015, https://radiology.ucsf.edu/patient-care/patient-safety/radiation-safety/benefits. Accessed 7 Dec 2020.

Wikipedia. “Spiral.” Wikipedia, 2020, https://en.wikipedia.org/wiki/Spiral. Accessed 5 Dec 2020.

Wonko. “276: Nautilus.” Setting the Crease, http://www.wonko.info/365origami/?p=1797, 2011. Accessed 6 Dec 2020.

Zodl, Evan. “Nautilus.” EZ Origami, 2016, https://ez-origami.com/original/nautilus/. Accessed 5 Dec 2020.

Published

10-10-2021

How to Cite

Kaur, G., Soong, C. ., & Kaur, J. (2021). Mathematical Modeling of the Origami Navel Shell. Journal of Student Research, 10(3). https://doi.org/10.47611/jsrhs.v10i3.1573

Issue

Section

HS Research Articles