geometry project

On Honeycombs and Regular Hexagon Applications

Gabriel Murillo, PhD MVC Mathematics

May 3, 2020

Abstract

The regular hexagonal cross-section shape of the honeycomb is ideal for many industrial applications. This is due in part to the inherent “strength” of such a shape, as well as the minimal amount of materials that are needed in constructions using this shape. Among regular tessellating polygons, it can be shown that no shape has a better area to perimeter ratio than the regular hexagon.

0 Proposal

“The cells of the honeycomb are hexagonal because this design needs the minimum amount of wax to hold the maximum amount of honey—almost 21

4 pounds of honey in just 1 square foot

of comb. The bees are so careful with their wax because it takes nearly 16 pounds of honey to produce enough comb for a colony: wax secretion is therefore a very expensive item in the hive’s energy economy.” [3]

This quote represents my interest in learning more about the wisdom and intelligent design demonstrated by honeybees in storing their honey in regular hexagonal prisms. I am primarily interested in the efficiency of this shape as a means for storage. I will explore why the regular hexagon is such an ideal shape for storage using results from the course textbook [1].

1 Application History

Historically, the efficiency of the regular hexagonal prism has been a wonder to many thinkers. For instance, the following was written by the ancient Greek geometer Pappus of Alexandria.

“By virtue of a certain geometrical forethought. . . (bees) know that the hexagon is greater than the square and the triangle and will hold more honey for the same expenditure of material in constructing each.” [5]

Likewise, the naturalist Charles Darwin stated that the honeycomb was “absolutely perfect in economising labour and wax.” [5] Finally, the following was written in The New York Times.

“In recent years, engineers and product designers have increasingly realized something that bees apparently have always known: configuring even a very thin material into a six-sided honeycomb pattern makes it much stronger than it would be in some other shape. This has become important as companies try to minimize their use of materials that are difficult to recycle and to increase their use of recycled paper products.” [2]

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MAT-37 G. Murillo

Figure 1: Sandwich-structured composite “Hon- eycomb” used to fill empty space in packaging. [9]

Figure 2: BMW i3 crash force absorption safety structure [10]

Applications of the use of the regular hexagon can be divided into natural and industrial applications. There are many natural applications of regular hexagons. As mentioned, perhaps the most well known is that of the honeycomb of the honeybee, which can be described as an “architectural masterpiece”. This regular hexagonal prism shape can also be seen in the nest of a paperwasp. Geologic formations can in- clude columnar basalt, which is often in the shape of a regular hexagonal prism. In California, this type of formation is notably seen at the Devil’s Postpile National Monument. On the micro-level, ice crystals are arranged in the regular hexagon shape. Our final natural example is found when one zooms in on the eye of an insect, which is composed of a regular hexagonal lattice of ommatidium.

There are also many industrial applications of the use of regular hexagons or regular hexagonal prisms. As indicated by the above quote from The New York Times and figure 1, many companies use what is known as sandwich-structured composite “honeycomb” to fill empty space in packaging. The New York Times comments further on the strength and recyclability (two attributes not explored in this present paper) of honeycomb composite, made out of regular hexagonal prisms.

“Honeycomb paper’s property of crushing evenly also appeals to the military, which uses paper honeycomb to cushion the blow for equipment that is dropped by parachute. Even objects as heavy as jeeps are fastened to platforms with blocks of honeycomb underneath to absorb the blow of landing. Honeycomb’s ability to cushion a fall comes from its hexagonal pattern. The many walls make the honeycomb’s paper fibers more resistant to crumpling than if they were part of a circle or square. Other shippers may be attracted less by the strength of honeycomb than by its attractiveness as an alternative to Styrofoam, which cannot easily be recycled. About 40 percent of paper honeycomb is made up of recycled materials, and the honeycomb itself is recyclable.” [2]

Figure 2 is another application showcasing the lightweight strength of the honeycomb composite core. There we see the regular hexagonal prism made with an special plastic material, engineered to withstand and absorb the force of an automobile crash. Aircraft engineers have also used the honeycomb core in airplane panels to increase insulation and strength, reduce weight and hence reduce fuel consumption. Finally, grain elevators have also adopted the regular hexagonal prism design as a means of saving considerable amounts of concrete.

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Figure 3: Part of a pure tes- sellation of congruent reg- ular (equilateral) triangles. [6]

Figure 4: Part of a pure tessellation of congruent squares. [7]

Figure 5: Part of a pure tes- sellation of congruent reg- ular hexagons. [8]

2 Application Demonstration

Please note the following discussion on tessellations on page 366 of [1]:

“Tessellations are patterns composed strictly of interlocking and nonoverlapping regular poly- gons. All of the regular polygons of a given number of sides will be congruent. Tessella- tions are commonly used in design, but especially in flooring (tiles and vinyl sheets). A pure tessellation is one formed by using only one regular polygon in the pattern. . . . In the ac- companying pure tessellation, only the regular hexagon appears. In nature, the beehive has compartments that are regular hexagons. The sum of the measures of the adjacent angles must equal 360◦; in this case, 120◦ + 120◦ + 120◦ = 360◦. It would also be possible to form a pure tessellation of congruent squares because the sum of the adjacent angles’ measures would be 90◦+90◦+90◦+90◦ = 360◦ . . . a) Can congruent equilateral triangles be used to form a pure tessellation? . . . ANSWERS a) Yes, because 6×60◦ = 360◦”

From the above discussion, it is evident that three common regular polygons can form pure tessellations, namely the regular (equilateral) triangle, the square and the regular hexagon. Please see figures 3, 4 and 5, respectively, for an example portion of these three pure tessellations.

We can also ask if other regular polygons can also form pure tessellations. Based on the above quote from [1], a necessary condition would be that the “sum of the measures of the adjacent angles must equal 360◦”. In other words, we can say that each interior angle measure (in degrees) of the regular polygon will need to be a (integer) divisor of 360◦. To calculate the measure of each interior angle of a regular polygon, we consult the following Corollary in [1]:

Corollary 2.5.3 “The measure I of each interior angle of a regular polygon or equiangular polygon of n sides is I = (n−2)·180

◦

n ”.

Table 1 provides a list of the interior angle measures (in degrees) and the ratio of 360◦ to the interior angle measure for regular polygons with 3 to 20 sides. We see from this table, that the only regular poly- gons with 3 to 20 sides, that have interior angle measures that are divisors of 360◦ are triangles, squares and hexagons. In fact, it can be shown that even when we extend this table for regular polygons to be between 3 and 1,000 sides, still just only regular triangles, squares and hexagons are the regular polygons that have interior angle measures that are divisors of 360◦. Hence, among regular polygons between 3 and

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Regular Polygon Sides Interior Angle Measure (Degrees) 360◦/(Interior Angle Measure) 3 60.0000 6.0000 4 90.0000 4.0000 5 108.0000 3.3333 6 120.0000 3.0000 7 128.5714 2.8000 8 135.0000 2.6667 9 140.0000 2.5714 10 144.0000 2.5000 11 147.2727 2.4444 12 150.0000 2.4000 13 152.3077 2.3636 14 154.2857 2.3333 15 156.0000 2.3077 16 157.5000 2.2857 17 158.8235 2.2667 18 160.0000 2.2500 19 161.0526 2.2353 20 162.0000 2.2222

Table 1: A list of the interior angle measures (in degrees) and the ratio of 360◦ to the interior angle measure for regular polygons with 3 to 20 sides. The rows for equilateral triangles, squares and regular hexagons are in bold which indicate that these three regular shapes form pure tessellations.

Figure 6: A regular polygon with 20 sides, namely a regular icosagon, visually ap- pears to be similar to a circle. [4]

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Figure 7: The three common regular polygons that can form pure tessellations, each drawn with an area of 1 inch2. The perimeters of the equilateral triangle, square and regular hexagon are ~4.5590 inches, 4 inches and ~3.7224 inches, respectively.

Figure 8: A color photograph of the three common regular polygons in figure 7. Each polygon is over- laid with 14 similar yellow paper circles so as to demonstrate that each polygon has the roughly the same area, namely 1 inch2.

1,000 sides, only regular triangles, squares and hexagons can form pure tessellations. It can be argued, how- ever, that table 1 need not be extended to regular polygons with 1,000 sides, since a regular polygon with just 20 sides, namely a regular icosagon, as seen in figure 6, already visually appears to be similar to a circle.

Now let us consider the ratio of area to perimeter among regular polygons that tessellate. Page 363 of [1] provides a picture proof for the following statement.

Area of an Equilateral Triangle “. . . the area of an equilateral triangle whose sides are of length s is given by A = s

2

4

√ 3.

From figure 3, we can see that a regular hexagon is composed of 6 equilateral triangles. Hence, from this fact, and the above statement, we can determine that the area of a regular hexagon whose sides are of length s is given by A = 3

√ 3

2 s2. Finally, page 345 of [1] provides the formula for the area of a square.

Theorem 8.1.1 “The area A of a square whose sides are each of length s is given by A = s2.

Let us consider figure 7. To simplify matters, this figure shows the three common regular polygons, that can form pure tessellations, each drawn with an area of 1 inch2. The perimeters of the equilateral triangle, square and regular hexagon are ~4.5590 inches, 4 inches and ~3.7224 inches, respectively. Figure 8 is an adaption of figure 7, provided to demonstrate that each polygon has the approximately the same area.

Given the area and perimeter values above, we can calculate the area to perimeter ratios for the three common regular polygons that can form pure tessellations. For the equilateral triangle, we have 1 in

2

4.5590 in ≈ 0.2193 in. For the square we have 1 in

2

4 in = 0.2500 in. Finally for the regular hexagon, we have 1 in2

3.7224 in ≈ 0.2651 in. So we can see that the regular hexagon is the most space efficient of these three common regular polygons. While it cannot form a pure tessellation, it can also be shown that the circle is the most space efficient shape. For a circle, the area to circumference (perimeter) ratio, for a circle with an area of 1 inch2

is 1 in 2

3.5449 in ≈ 0.2821 in. Hence, the physical advantage of using a regular hexagon, or a regular hexagonal prism for storage, should be clear.

There are other ways to demonstrate the space efficiency (maximal area to perimeter ratio of a pure tessellating regular polygon) of a regular hexagon. First, figures 9 and 10 demonstrate, that given 7 randomly placed uniformly shaped cylinders, one can lasso these cylinders and then tighten the lasso. We see that tightening the lasso in a relatively slow and stable fashion, rearranges the cylinders into a regular hexagon shape. Also, as demonstrated in figure 11, one can create a group of 7 bubbles, made from soapy water,

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Figure 9: This is yet another way to demonstrate the space efficiency of a regular hexagon. Given 7 randomly placed uniformly shaped cylinders, one can lasso these cylinders and then tighten the lasso.

Figure 10: Continuing where figure 9 left off, tight- ening the lasso in a relatively slow and stable fash- ion, rearranges the cylinders into a regular hexagon shape.

adjacent to each other. The boundaries of the center bubble seem to form an almost regular hexagon, which is highlighted with the overlaid red lines in this figure.

References

[1] Daniel C. Alexander and Geralyn M. Koeberlein. Elementary Geometry for College Students. Cen- gage Learning, 2015.

[2] Adam Bryant. “Technology; What Packers and Builders Can Learn From the Bees”. In: The New York Times (Oct. 1991), p. 11. URL: https://www.nytimes.com/1991/10/06/business/ technology-what-packers-and-builders-can-learn-from-the-bees.html.

[3] Hattie Ellis. Sweetness and Light: The Mysterious History of the Honeybee. Broadway Books, 2004.

[4] László Németh. File:Regular polygon 20 annotated.svg. Accessed: 2020-04-23. July 2013. URL: https://en.wikipedia.org/wiki/File:Regular_polygon_20_annotated. svg.

[5] Science X Network. “Revealed: Secret of bees’ honeycomb”. In: phys.org (July 2013). URL: https: //phys.org/news/2013-07-revealed-secret-bees-honeycomb.html.

[6] Tom Ruen. File:Vertex type 3-3-3-3-3-3.svg. Accessed: 2020-04-23. June 2015. URL: https:// en.wikipedia.org/wiki/File:Vertex_type_3-3-3-3-3-3.svg.

[7] Tom Ruen. File:Vertex type 4-4-4-4.svg. Accessed: 2020-04-23. June 2015. URL: https://en. wikipedia.org/wiki/File:Vertex_type_4-4-4-4.svg.

[8] Tom Ruen. File:Vertex type 6-6-6.svg. Accessed: 2020-04-23. June 2015. URL: https://en. wikipedia.org/wiki/File:Vertex_type_6-6-6.svg.

[9] User:Amada44. File:Cardboard Honeycomb 9093.jpg. Accessed: 2020-04-30. Aug. 2015. URL: https: //commons.wikimedia.org/wiki/File:Cardboard_Honeycomb_9093.jpg.

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Figure 11: Yet another interesting, and perhaps im- perfect, way to observe the space efficiency of a hexagon is to create a group of 7 bubbles adjacent to each other. The boundaries of the center bubble seem to form an almost regular hexagon, which is highlighted with the overlaid red lines in this figure.

[10] youkeys. File:2013 IAA BMW i3 Honeycomb structure.jpg. Accessed: 2020-04-29. Sept. 2013. URL: https://en.wikipedia.org/wiki/File:2013_IAA_BMW_i3_Honeycomb_ structure.jpg.

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