Theoretical and Natural Science
- The Open Access Proceedings Series for Conferences
Theoretical and Natural Science
Vol. 10, 17 November 2023
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Analytical study of spirograph using parametric equations and pattern design by parametric-graphic interactions
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Abstract
Perfectly symmetric curves always have a high degree of aesthetic value. While it is difficult to draw them by hand, Spirograph is an old and popular drawing toy that produces fascinating symmetric patterns. In playing with drawing software, the author explores types of Spirograph patterns, the type identification method, and the parametric equations to express Spirograph mathematically. This paper also discusses the key parameters of a Spirograph pattern and how they affect the pattern’ shape. Finally, Spirograph pattern design is carried out by analyzing the features of a random pattern and estimating its different parameters. These results to some extent demystify and predict the seemingly infinite Spirograph patterns, as the corresponding parameters of a given Spirograph pattern can be found so that the similar image can be drawn by hand through a physical Spirograph set.
Keywords
spirograph, trochoid, rotation, parametric equation, parameters
References
1. Deck, K. M. (1999). Spirograph® Math. Humanistic Mathematics Network Journal, 1(19), 7.
2. Cavanaugh, W. E. (1975). The Spirograph and the Greatest Common Factor. The Mathematics Teacher, 68(2), 162-163.
3. Stender, P. SPIROGRAPH–A TOY AS A MATHEMATICAL PROBLEM. Including the Highly Gifted and Creative Students–Current Ideas and Future Directions, 347.
4. Lessley, M., & Beale, P. (2012, July). Projecting Mathematical Curves with Laser Light. In Proceedings of Bridges 2012: Mathematics, Music, Art, Architecture, Culture (pp. 557-560).
5. Saunders, R., & Gero, J. S. (2001). A curious design agent. In CAADRIA (Vol. 1, pp. 345-350).
6. Etemad, K., Carpendale, S., & Samavati, F. (2014, August). Spirograph inspired visualization of ecological networks. In Proceedings of the Workshop on Computational Aesthetics (pp. 81-91).
7. Konkar, R. (2022). Flattening the curve... of Spirographs. Recreational Mathematics Magazine, 9(16), 1-20.
8. Wang, B., Geng, Y., & Chu, J. (2019, November). Generation and application of hypocycloid and astroid. In Journal of Physics: Conference Series (Vol. 1345, No. 3, p. 032085). IOP Publishing.
9. Shell-Gellasch, A. (2015). The Spirograph and Mathematical Models from 19th-Century Germany. Math Horizons, 22(4), 22-25.
10. Whitaker, R. J. (2001). Harmonographs. ii. circular design. American Journal of Physics, 69(2), 174-183.
11. Ippolito, D. (1999). The mathematics of the spirograph. The Mathematics Teacher, 92(4), 354-358.
12. Hall, L. M. (1992). Trochoids, roses, and thorns—beyond the spirograph. The College Mathematics Journal, 23(1), 20-35.
13. Flores, A., Prichard, M. K., Bezuk, N., & Moody, M. (1992). Activities: Mathematical Connections with a Spirograph. The Mathematics Teacher, 85(2), 129-137.
14. Lin, Y., & Vuillemot, R. (2013, October). Spirograph designs for ambient display of tweets. In IEEE VIS 2013. IEEE.
15. Gerofsky, S., Gomez, F., Rappaport, D., & Toussaint, G. (2009, July). Spirograph patterns and circular representations of rhythm: Exploring number theory concepts through visual, tangible and audible representations. In Proceedings of Bridges 2009: Mathematics, Music, Art, Architecture, Culture (pp. 279-286).
16. Kang, R., Johnson, S., Lambert, E., & Davidson, C. (2020). Doing Mathematics with Spirograph. Mathematics Teacher: Learning and Teaching PK-12, 113(11), 895-903.
17. Sayers, N. (2018, July). The Art and Mathematics of Cycling: Using Old Bicycles to Draw Spirograph Patterns. In Proceedings of Bridges 2018: Mathematics, Art, Music, Architecture, Education, Culture (pp. 455-458).
Data Availability
The datasets used and/or analyzed during the current study will be available from the authors upon reasonable request.
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- Volume Title
- Proceedings of the 2023 International Conference on Mathematical Physics and Computational Simulation
- ISBN (Print)
- 978-1-83558-131-5
- ISBN (Online)
- 978-1-83558-132-2
- Published Date
- 17 November 2023
- Series
- Theoretical and Natural Science
- ISSN (Print)
- 2753-8818
- ISSN (Online)
- 2753-8826
- DOI
- 10.54254/2753-8818/10/20230311
- Copyright
- 17 November 2023
- Open Access
- This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited