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Article

Packing Series of Lenses in a Circle: An Area Converging to 2/3 of the Disc

by
Andrej Hasilik
Institute of Physiological Chemistry, Karl-von-Frisch-Straße 2, 35032 Marburg, Germany
Geometry 2024, 1(1), 16-22; https://doi.org/10.3390/geometry1010003
Submission received: 5 June 2024 / Revised: 30 July 2024 / Accepted: 1 August 2024 / Published: 5 August 2024

Abstract

:
We describe a series of parallel lenses with constant proportions packed in a circle. To construct n lenses, a regular 2(n + 1)-gon is drawn with a central diagonal of 2r length, followed by an array of n parallel diagonals perpendicular to the former. These diagonals and the central angle of the pair of peripherals, the shortest diagonals, are used to construct n rhombi. The rhombi define the shape of lenses tangential to them. To construct the arcs of the lenses, beams perpendicular to the sides of each rhombus are drawn. Four beams radiating from the top and bottom vertices of each rhombus intersect in the centers of a pair of coaxal circles. Thus, the vertical axis of each rhombus coincides with the radical axis of the pair. The intersection of the pair represents the corresponding lens. All n lenses form a tangential sequence along the central diagonal. Their cusps circumscribe the polygon and the lenses themselves. The area covered by the lenses converges to (2/3) πr2.

Graphical Abstract

1. Introduction

We became interested in geometry when Hana Hasilik [1] made her first artefacts in the shape of lentoids. Surprisingly, for the related 2D and 3D shapes that are derived from circles and spheres, mathematicians seem not to have found unique names. Instead, lens (Linse in German) is used for both the cut through as well as the body itself. We examined the packing of lenses in a circle, however, we had not been able to find literature on this topic in textbooks or on the web.
Among coaxal systems defined by D. Pedoe [2] (pp. 16–21), the intersecting type presents circles that yield lenses. In other books dealing extensively with circles [3] (pp. 14–15, pp. 29–30) or with geometric patterns and design examples showing circles with constant diameters, they are found [4] (pp. 61–94). Other shapes than lenses are examined in surveys on packing in circles such as in [5]. In Wolfram Web Resource, a search for “lens packing” reveals 175 entries [6], however, none of them describes the packing of lenses in a circle. The packing of sequences of circles in lenses has been studied by Kucik [7].

2. Materials and Methods

We calculated radii, distances of circle centres and areas of circle intersections using standard equations and operations. For comparison, a few sequences were calculated using a Java script [8]. Graphics and photo work were performed using commercially available applications: Affinity Designer 2, Movavi Photo Editor 6 and LightBurn.

3. Results and Discussion

It can be shown that for any natural number n, lenses with constant proportions can be packed in a circumscript circle with their cuspal axes coinciding with n parallel diagonals of a regular 2(n + 1)-gon. Such lenses are constructed as intersections of sequences of coaxal circle pairs.
Construction of a sequence of n lenses can be performed as follows (Figure 1): (i) Draw a polygon with 2(n + 1) vertices such that a pair of mutually opposite vertices will be on the x-axis. This will also be the line of centers. (ii) Connect the remaining vertices with parallel diagonals that are perpendicular to the axis. (iii) Construct, around the perpendicular diagonals, rhombi with constant proportions such that their angles in the polygon vertices equal 2π/(n + 1), and (iv) in these vertices place beams perpendicular to the sides of the rhombi. v) Find centers of coaxal circle pairs in intersections of the beams with the line of centers (x-axis). The distances between the centers of pairs equal 2tan(α1/2).sin(αi/2). (vi) Draw the circles with radii equal to the distances of the centers from the corresponding vertices and find their intersections. Alternatively, draw just the arcs connecting the vertices to obtain the lenses.
In Figure 1a, at n = 1, the polygon is an encircled square and there is no pair of parallel diagonals and no other circles to intersect. In this limit case the “lens” is nothing other than two circles sharing the radius and the center. When n = 2 we are dealing with 2n − 1 = 3 circles. The construction in Figure 1b–f starts with the 2(n + 1)-gon, which is a hexagon, and results in three circles with two intersections. These lenses are 2r[1 − cos(αi/2] = 1r thick each, where αi is the central angle. Their lengths equal that of the polygon diagonals: 2r[sin(αi/2)]. The combination of the central circle with one of the peripheral ones in Figure 1e corresponds to vesica piscis.
Within any sequence, the ratio of thickness/length is constant and the sum of the thicknesses of all intersection areas (lenses) equals 2r.
In Figure 1f the lenses are highlighted in grey. Their axes coincide with the perpendicular diagonals of the hexagon and with the radical axes of the coaxal pairs of circles. Among the coaxal systems discussed by D. Pedoe [2] (p. 17), these are referred to as intersecting.
In all sequences with n > 1, there are three circles that share one diameter (r). With each increase in n, the distance between the centers increases and the sequence grows by the corresponding number of coaxal pairs of circles with radii R > r.
Sequences of lenses constructed for n = 3 and 4 start with an octagon and a decagon and the results are shown in Figure 2a, Figure 2b and Figure 2c, respectively. Further details are shown in Supplementary Figure S1.
In Figure 2c, the decagon is replaced by a pentagon with two diagonals that intersect each other (highlighted in red). One side of the rhombus in this figure coincides with the axis of the smaller lens. Thus, the ratio of the sizes of the two lenses equals the golden ratio.
The construction of a sequence of four (n = 4) tangent lenses is shown in Figure 3. It begins with drawing a (2n + 1)-gon. This decagon is highlighted in green in Figure 3a. Next, diagonals are drawn that meet in the top and bottom vertices of the decagon at angles α1, thus forming four rhombi (highlighted in blue). To find the centers of the circles, beams perpendicular to the sides of the rhombi are drawn, radiating from the vertices of the polygon (highlighted in red). In Figure 3b, four pairs of coaxal circles are constructed using the previously found centers with radii given by the distances of the centers from the vertices of the polygon. The polygon circumscript circle participates in two coaxal circle pairs, those with the smallest radius. It is referred to as the central circle (highlighted in green) and the intersections with its neighbors represent the peripheral lenses of the sequence that are highlighted in blueish grey in Figure 3b.
The central angle of the smallest lens is given by 2π/(n + 1). Consequently, it takes (n + 1) lenses of that size, each one accommodated in one 2π/(n + 1) sector, to construct full circle. When the circle is constructed using five lenses a decorative structure is obtained (Figure 4a). Similar to other ring-like structures, an endless series of its enlargements may be combined to nested tunnel-resembling images [9] with a decorative character as exemplified by Figure 4b,c.
The inner arcs of rings described here are not to be mistaken for a cyclopentoid. Owing to the ratio of R/r = 5, the maximum distance of the cyclopentoid curve from a circle it can be inscribed in is 0.4 R, whereas the thickness of the present ring is R/ϕ ≈ 0.382 R.
Tangential packing sequences have been described by others. Among recent examples are linear sequences of circles packed in lenses [7] and “shoemaker’s knife”-like sequences of circles that have been described as special cases of Apollonian packing [11]. The packing of lenses shown here formally belongs to the family of packings within a circle.
In Table 1, we present calculations of the radii and distances between the centers as well as areas of the lenses, and sums of the latter. The first and the last lens in these sequences correspond to the intersection of the central circle with its neighbors that are of the same size. Obviously, in any of the sequences, the distance between the centers of the smallest circles is <2. This condition is characteristic of the intersecting coaxal type [2] (p. 18), we see that this distance results from subtracting the thickness of the first lens from 2R1, as is illustrated in Figure 3b. Looking at sequences for n = 1 through 5, the values of the distances are 0, 1, 2 , ϕ, and 3 , respectively. These values equal the lengths of radii of the second smallest lenses in the respective sequences as shown in fifth column in Table 1.
In order to design a larger sequence, for example one consisting of 15 lenses, 29 (2n − 1) circles are due to be constructed: the central cycle, and its two companions of the same size plus thirteen pairs of larger circles. This construction is shown in Supplementary Figure S1.
With increasing n, the cusps of the lenses, actually the vertices of the polygon, nearly complete the circumscript circle. For 12 n-values between 1 and 127, we have calculated the radii of the circles, the distances of their axes from each other as well as the areas of their intersections (Supplementary Table S1). The sums of areas of lens sequences are presented in Table 2. With increasing n, the sum appears to converge to 2/3 of the area of the circumscript circle. As may be expected, the area of the companion sequences of rhombi inscribed within the lenses in the unit circle converges to 0.5 πr2.
Series of lenses such as described here can be considered to represent cuts through bodies of lenses that are commonly referred to as lentoids. Any sequence of n lenses can be rotated around the horizontal or the vertical axis to create 3D surfaces and bodies. A constructive sculpture of a tangential sequence of lentoids has been created by Hana Hasilik [12]. Other attractive geometric constructions, toroids, may be obtained from sequences of packed lenses by revolutions of their orthogonal or rotated graphs.

4. Conclusions

Any member of a by-itself unlimited series of encircled n lenses can be envisioned starting from a regular 2(n + 1) polygon that is inscribed in the central circle of coaxal systems of circles. Within the polygon, the lenses’ axes coincide with n parallel diagonals. A pair of the smallest lenses are constructed as intersections of the central circle with its two doublets, while the remainder, n − 2, is formed by two sets of larger coaxal circles, each. Their intersections contain the diagonals that span the polygon and share the vertices with the latter. Their lens shape is defined by tangential rhombi that, within the sequence, share the central angle of the pair of peripherals, the smallest lenses. The area covered by the lenses in the series converges to 2/3πr2.

Supplementary Materials

The following supporting information can be downloaded at: https://www.mdpi.com/article/10.3390/geometry1010003/s1, Figure S1: Construction of a sequence of 15 tangential lenses using 29 circles. Table S1: Areas of series of circumscribed lenses and circumscribed rhombi.

Funding

This research received no external funding.

Data Availability Statement

The original contributions presented in the study are included in the article and supplementary material, further inquiries can be directed to the corresponding author.

Acknowledgments

Thanks are due to Hana Hasilik, Werner Massa and Geoff Hodbod for critical and constructive comments.

Conflicts of Interest

The author declares no conflicts of interest.

References

  1. Hasilik, H. On Lentoids. Available online: https://www.hana-hasilik.de/en/index.html#lentoide (accessed on 29 July 2024).
  2. 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]
  3. 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).
  4. 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).
  5. Stephenson, K. Circle Packing: A Mathematical Tale. Not. AMS 2003, 50, 1376–1388. [Google Scholar]
  6. MathWorld Search for Keywords “Lens and Packing”. Available online: https://mathworld.wolfram.com/search/?query=lens+packing (accessed on 29 July 2024).
  7. Kucik, J. Lens Sequences. J. Integer Seq. 2020, 23, 1–36. Available online: https://arxiv.org/pdf/0710.3226 (accessed on 29 July 2024).
  8. 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).
  9. MathWorld. Nested Polygon. Available online: https://mathworld.wolfram.com/NestedPolygon.html (accessed on 29 July 2024).
  10. MathWorld. Polygonal Spiral. Available online: https://mathworld.wolfram.com/PolygonalSpiral.html (accessed on 29 July 2024).
  11. Aharonov, D.; Stephenson, K. Geometric Sequences of Discs in the Apollonian Packing. Algebra Anal. 1997, 9, 104–140. [Google Scholar]
  12. Hasilik, H. All in One. 3D Print. Available online: https://www.hana-hasilik.de/en/index.html#3d (accessed on 29 July 2024).
Figure 1. Construction of circles and their intersections. Ab ovo, n = 1, there is just one center with two identical circles, (a) ergo no intersection. The axis (shown as a dashed line) in this and the coaxal axis in the subsequent systems are coincident with the x-axis. In panels (bf) the construction for n = 2 is shown. The radii and the centers are labeled by letters r, R and C. Polygons and the central circle are highlighted in green, cords or diagonals and rhombi in blue, beams perpendicular to the sides of the rhombi and coaxal circles in red, and the intersection areas in grey.
Figure 1. Construction of circles and their intersections. Ab ovo, n = 1, there is just one center with two identical circles, (a) ergo no intersection. The axis (shown as a dashed line) in this and the coaxal axis in the subsequent systems are coincident with the x-axis. In panels (bf) the construction for n = 2 is shown. The radii and the centers are labeled by letters r, R and C. Polygons and the central circle are highlighted in green, cords or diagonals and rhombi in blue, beams perpendicular to the sides of the rhombi and coaxal circles in red, and the intersection areas in grey.
Geometry 01 00003 g001
Figure 2. Construction octagon and decagon sequences of encircled lenses, with the polygons highlighted in green. Three (a) and four lenses (b,c) make up the sequences that are characterized by center angle α1 values of 90° and 72°, respectively. The intersection of diagonals (highlighted in red) shown in (c) indicates the golden ratio.
Figure 2. Construction octagon and decagon sequences of encircled lenses, with the polygons highlighted in green. Three (a) and four lenses (b,c) make up the sequences that are characterized by center angle α1 values of 90° and 72°, respectively. The intersection of diagonals (highlighted in red) shown in (c) indicates the golden ratio.
Geometry 01 00003 g002
Figure 3. Construction of four circumscribed lenses. In panel (a), drawings of a decagon, four rhombi and four pairs of perpendicular beams are shown that determine the centers. Around the latter red circles radii are drawn, determined by pairs of vertices of the decagon as shown in panel (b). One of the circles is shared by two of the pairs and this is located in the center of the system (highlighted in green).
Figure 3. Construction of four circumscribed lenses. In panel (a), drawings of a decagon, four rhombi and four pairs of perpendicular beams are shown that determine the centers. Around the latter red circles radii are drawn, determined by pairs of vertices of the decagon as shown in panel (b). One of the circles is shared by two of the pairs and this is located in the center of the system (highlighted in green).
Geometry 01 00003 g003
Figure 4. Rings consisting of five lenses with golden ratio proportions. (a) A garland made of five lenses in grey. (b) A series of rings diminuted by the golden ratio. (c) A colored lens spiral similar to a polygonal spiral [10]. Five possible spirals are highlighted with rainbow colors.
Figure 4. Rings consisting of five lenses with golden ratio proportions. (a) A garland made of five lenses in grey. (b) A series of rings diminuted by the golden ratio. (c) A colored lens spiral similar to a polygonal spiral [10]. Five possible spirals are highlighted with rainbow colors.
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Table 1. Selected calculations: Radii of circles, distances of centers, lens areas and fractions of lens sequence areas. (For further details see Supplementary Table S1).
Table 1. Selected calculations: Radii of circles, distances of centers, lens areas and fractions of lens sequence areas. (For further details see Supplementary Table S1).
POLYGONNo of
Vertices
Lens
Thickness
PROPORTIONRADIUSDISTANCELENS
AREA
Perpendicular2 × (n + 1)2 h2 h/chord between
the centers
and
diagonal Noα/2°=chord × tan α/4Ratiosin(αi/2):sin(α1/2) % of total
HEXAGON2 × (2 + 1)
16010.577350269111.2283697
212010.577350269111.2283697
Σ = 2 top = 178.2004438
OCTAGON2 × (3 + 1)
1450.585786440.41421356211.414213560.57079633
2900.828427120.4142135621.4142135621.14159265
31350.585786440.41421356211.414213560.57079633
Σ = 2 top = 2 72.6760455
DECAGON2 × (4 + 1)
1360.381966010.32491969611.618033990.30558055
2720.618033990.3249196961.618033992.618033990.80002025
31080.618033990.3249196961.618033992.618033990.80002025
41440.381966010.32491969611.618033990.30558055
Σ = 2 top = ϕ70.3847329
DODE-CAGON2 × (5 + 1)
1300.267949190.26794919211.732050810.18117215
2600.464101610.2679491921.7320508130.54351644
3900.535898380.26794919223.464101620.72468859
41200.464101610.2679491921.7320508130.54351644
51500.267949190.26794919211.732050810.18117215
Σ = 2 top = 3 69.2026628
Table 2. Areas of sequences of lenses in polygons and the inscribed rhombi with their shares of the circumscript unit circle. This is a table.
Table 2. Areas of sequences of lenses in polygons and the inscribed rhombi with their shares of the circumscript unit circle. This is a table.
nNo. of VerticesLenses’ AreaLenses’ Fraction Rhombi Fraction
143.141592651.0000000000.636619772
262.456739400.7820044380.551328895
382.283185310.7267604550.527393088
4102.211201600.7038473290.517125758
5122.174065770.6920266280.511745261
6142.152305410.6851000890.508565079
7162.138427990.6806827710.506526184
11242.113723910.6728192170.502875498
15322.105220770.6701125840.501612598
31642.097090330.6675245840.500401983
631282.095068210.6668809250.500100423
1272562.094563340.6667202170.500025101
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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

AMA Style

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 Style

Hasilik, 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 Style

Hasilik, 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

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