Packing Series of Lenses in a Circle: An Area Converging to 2/3 of the Disc
Abstract
:1. Introduction
2. Materials and Methods
3. Results and Discussion
4. Conclusions
Supplementary Materials
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
- Hasilik, H. On Lentoids. Available online: https://www.hana-hasilik.de/en/index.html#lentoide (accessed on 29 July 2024).
- Pedoe, D. Circles. In International Series of Monographs on Pure and Applied Mathematics; Sneddon, I.N., Ed.; Pergamon Press: London, UK, 1957; Volume 2. [Google Scholar]
- Pedoe, D. Circles: A Mathematical View. The Mathematical Association of America, USA. 1995. Available online: https://books.google.de/books?hl=de&lr=&id=rlbQTxbutA4C&oi=fnd&pg=PR7&dq=pedoe+Daniel+circles&ots=ke_ZvHJwO8&sig=JHT7PceoQopwjgjSUWonoU8O5sI#v=onepage&q=pedoe%20Daniel%20circles&f=false (accessed on 29 July 2024).
- Strick, H.K. Mathematik ist wunderwunderschön. In Noch mehr Anregungen zum Anschauen und Erforschen für Menschen Zwischen 9 und 99 Jahren; 2. korrigierte und ergänzte Auflage; Springer: Berlin/Heidelberg, Germany, 2021; Available online: https://link.springer.com/book/10.1007/978-3-662-59060-7 (accessed on 31 May 2024).
- Stephenson, K. Circle Packing: A Mathematical Tale. Not. AMS 2003, 50, 1376–1388. [Google Scholar]
- MathWorld Search for Keywords “Lens and Packing”. Available online: https://mathworld.wolfram.com/search/?query=lens+packing (accessed on 29 July 2024).
- Kucik, J. Lens Sequences. J. Integer Seq. 2020, 23, 1–36. Available online: https://arxiv.org/pdf/0710.3226 (accessed on 29 July 2024).
- Bislin, W. Schnittfläche Zweier KREISE (JavaScript). Available online: http://walter.bislins.ch/blog/index.asp?page=Schnittfl%E4che+zweier+Kreise+%28JavaScript%29 (accessed on 29 July 2024).
- MathWorld. Nested Polygon. Available online: https://mathworld.wolfram.com/NestedPolygon.html (accessed on 29 July 2024).
- MathWorld. Polygonal Spiral. Available online: https://mathworld.wolfram.com/PolygonalSpiral.html (accessed on 29 July 2024).
- Aharonov, D.; Stephenson, K. Geometric Sequences of Discs in the Apollonian Packing. Algebra Anal. 1997, 9, 104–140. [Google Scholar]
- Hasilik, H. All in One. 3D Print. Available online: https://www.hana-hasilik.de/en/index.html#3d (accessed on 29 July 2024).
POLYGON | No of Vertices | Lens Thickness | PROPORTION | RADIUS | DISTANCE | LENS AREA |
---|---|---|---|---|---|---|
Perpendicular | 2 × (n + 1) | 2 h | 2 h/chord | between the centers | and | |
diagonal No | α/2°=chord × tan α/4 | Ratio | sin(αi/2):sin(α1/2) | % of total | ||
HEXAGON | 2 × (2 + 1) | |||||
1 | 60 | 1 | 0.577350269 | 1 | 1 | 1.2283697 |
2 | 120 | 1 | 0.577350269 | 1 | 1 | 1.2283697 |
Σ = 2 | top = 1 | 78.2004438 | ||||
OCTAGON | 2 × (3 + 1) | |||||
1 | 45 | 0.58578644 | 0.414213562 | 1 | 1.41421356 | 0.57079633 |
2 | 90 | 0.82842712 | 0.414213562 | 1.41421356 | 2 | 1.14159265 |
3 | 135 | 0.58578644 | 0.414213562 | 1 | 1.41421356 | 0.57079633 |
Σ = 2 | top = | 72.6760455 | ||||
DECAGON | 2 × (4 + 1) | |||||
1 | 36 | 0.38196601 | 0.324919696 | 1 | 1.61803399 | 0.30558055 |
2 | 72 | 0.61803399 | 0.324919696 | 1.61803399 | 2.61803399 | 0.80002025 |
3 | 108 | 0.61803399 | 0.324919696 | 1.61803399 | 2.61803399 | 0.80002025 |
4 | 144 | 0.38196601 | 0.324919696 | 1 | 1.61803399 | 0.30558055 |
Σ = 2 | top = ϕ | 70.3847329 | ||||
DODE-CAGON | 2 × (5 + 1) | |||||
1 | 30 | 0.26794919 | 0.267949192 | 1 | 1.73205081 | 0.18117215 |
2 | 60 | 0.46410161 | 0.267949192 | 1.73205081 | 3 | 0.54351644 |
3 | 90 | 0.53589838 | 0.267949192 | 2 | 3.46410162 | 0.72468859 |
4 | 120 | 0.46410161 | 0.267949192 | 1.73205081 | 3 | 0.54351644 |
5 | 150 | 0.26794919 | 0.267949192 | 1 | 1.73205081 | 0.18117215 |
Σ = 2 | top = | 69.2026628 |
n | No. of Vertices | Lenses’ Area | Lenses’ Fraction | Rhombi Fraction |
---|---|---|---|---|
1 | 4 | 3.14159265 | 1.000000000 | 0.636619772 |
2 | 6 | 2.45673940 | 0.782004438 | 0.551328895 |
3 | 8 | 2.28318531 | 0.726760455 | 0.527393088 |
4 | 10 | 2.21120160 | 0.703847329 | 0.517125758 |
5 | 12 | 2.17406577 | 0.692026628 | 0.511745261 |
6 | 14 | 2.15230541 | 0.685100089 | 0.508565079 |
7 | 16 | 2.13842799 | 0.680682771 | 0.506526184 |
11 | 24 | 2.11372391 | 0.672819217 | 0.502875498 |
15 | 32 | 2.10522077 | 0.670112584 | 0.501612598 |
31 | 64 | 2.09709033 | 0.667524584 | 0.500401983 |
63 | 128 | 2.09506821 | 0.666880925 | 0.500100423 |
127 | 256 | 2.09456334 | 0.666720217 | 0.500025101 |
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Hasilik, A. Packing Series of Lenses in a Circle: An Area Converging to 2/3 of the Disc. Geometry 2024, 1, 16-22. https://doi.org/10.3390/geometry1010003
Hasilik A. Packing Series of Lenses in a Circle: An Area Converging to 2/3 of the Disc. Geometry. 2024; 1(1):16-22. https://doi.org/10.3390/geometry1010003
Chicago/Turabian StyleHasilik, Andrej. 2024. "Packing Series of Lenses in a Circle: An Area Converging to 2/3 of the Disc" Geometry 1, no. 1: 16-22. https://doi.org/10.3390/geometry1010003
APA StyleHasilik, A. (2024). Packing Series of Lenses in a Circle: An Area Converging to 2/3 of the Disc. Geometry, 1(1), 16-22. https://doi.org/10.3390/geometry1010003