{"id":2807,"date":"2019-07-29T09:43:39","date_gmt":"2019-07-29T14:43:39","guid":{"rendered":"http:\/\/www.fanagrams.com\/blog\/?p=2807"},"modified":"2019-07-29T10:31:21","modified_gmt":"2019-07-29T15:31:21","slug":"the-elegance-of-the-hexagon-2","status":"publish","type":"post","link":"https:\/\/www.fanagrams.com\/blog\/2019\/07\/the-elegance-of-the-hexagon-2\/","title":{"rendered":"The Elegance of the Hexagon"},"content":{"rendered":"

Not that there was any pitched battle for my loyalties, but I have to say that the hexagon is my favorite geometric shape.\u00a0 I first became enamored in biochemistry when introduced to the basic building block of the carbon ring, illustrated as a hexagon of carbon atoms with other molecules hanging off of them.\u00a0 In medical school I spent hours of quality time with glucose, and even though the molecule can be folded and twisted, I appreciated its simplified depiction as an equilateral hexagon.<\/p>\n

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The next step of my hexagon journey was Charles Darwin and the conundrum of the honeycomb.\u00a0 The precise hexagonal structure of the honeycomb has fascinated naturalists for thousands of years.<\/p>\n

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In 36 BC the Greek mathematician Marcus Ternetius Varro suggested that a tile of hexagons produce the smallest total perimeter of wax compared to a square or a triangle, the two other shapes can make an unbroken tiled surface.\u00a0 The bottom line is that hexagons minimize the burden of wax production.\u00a0 This theory became known as the \u201choneybee conjecture.\u201d\u00a0 Though widely accepted, the theory was not proved mathematically until 1999, some two thousand years later.\u00a0 Darwin proposed that economy of wax was the engine driving the natural selection advantage for\u00a0 those bees who could make a hexagon.<\/p>\n

While this hypothesis explained why bees do what they do, Darwin faced the bigger challenge of explaining how they do it.\u00a0 The biggest hurdle was the church.\u00a0 Theologians had made the honeycomb one of the poster children for intelligent design.\u00a0 William Paley, an influential naturalist in Darwin\u2019s time, provided a detailed analogy to a watch, stating that there could be no watch without the guiding hand of a watchmaker.\u00a0 Paley believed\u00a0some sort of governing intelligence was the only explanation for a design that flummoxed human engineers.<\/p>\n

I can understand this line of thought.\u00a0 As I try to construct a quilt in a pattern of interlocking hexagons, I think of the bee doing the same thing\u00a0in the dark<\/em>, working alongside other bees to construct a three dimensional hive<\/em>.\u00a0 I\u2019ve got a ruler and precise cutting tools but frankly,\u00a0it\u2019s a bitch<\/em>\u00a0to get precise 120 degree angles so that the three corners snug in\u00a0together. \u00a0If one angle is off kilter, the mistake is compounded in the next, to the point of ripping the whole thing out midst a string of expletives.\u00a0 But the uneducated bee, with its mere smidge of a brain, does it effortlessly.\u00a0 Oh and by the way, the hive is tilted just so from the horizontal to prevent the honey from dripping out.<\/p>\n

The cornerstone of Darwin\u2019s theory of natural selection is that there is no agenda to natural selection, no plan to make a watch or a honeycomb.\u00a0 Form simply evolves in response to survival pressures.\u00a0 If Darwin could not show that bees could change over time and become more efficient, his entire theory would crumble in the face of religious skeptics who believed \u201cas it was in the beginning, now is and ever shall be, world without end.\u201d<\/p>\n

The hexagonal comb was my gateway into the relentless curiosity of Darwin.\u00a0 One summer I thought I might read the Origin of the Species<\/em>.\u00a0 After a few chapters, I abandoned the project.\u00a0\u00a0 The writing is so dense, filled with long descriptions crying out for a diagram or even better a YouTube video.\u00a0 But leaf through and you\u2019ll find a vivid demonstration of an insatiable curiosity at work.<\/p>\n

Darwin starts his description of the honeycomb by commenting that only a \u201cdull man\u201d could not be impressed by the hexagonal skills of the bee whose geometry skills outshine even the most accomplished craftsmen.\u00a0 He then blithely brushes aside thoughts of intelligent design and states, \u201cBut the difficulty is not nearly so great as it first appears; all this beautiful work can be shown, I think, to follow from a few very simple instincts.\u201d<\/p>\n

The next ten pages are a recitation of his extensive empirical research, which included a vast correspondence with other naturalists studying the honeycomb.\u00a0 Darwin was looking for a transitional structure, a simple comb that would demonstrate the starting point of its evolution.\u00a0 And thank God, a colleague sent Darwin a honeycomb from a Mexican bee that made circular, not hexagonal, cells.\u00a0 No special skills were needed to explain this\u00a0geometry.\u00a0 The bee could simply rotate around a single point.\u00a0 However, the wasted spaced between circles put this bee at a survival disadvantage.\u00a0 Its hive was inefficient.<\/p>\n

Darwin communicated with Francois Huber, another honeycomb expert.\u00a0 Huber was blind, but still managed to make meticulous observations of honeycomb construction with the assistance of his patient wife.\u00a0 He discovered Darwin’s “hive” bees start cell construction with a circular blob of wax which they then excavate.\u00a0 As the two adjoining cells compress against each other they form a straight edge.<\/p>\n

Problem solved.\u00a0 Based on the evidence that simpler forms of hives exist, Darwin wrote \u201cHence we may safely conclude that if we could slightly modify the instincts already possessed by the [Mexican bee], this bee would make a structure as wonderfully perfect as the hive-bee.\u201d<\/p>\n

Darwin then described his own experiments where he inserted wax of different colors and configurations into a hive and observed the bees.\u00a0 I like to imagine\u00a0him\u00a0pitter-patter-puttering around his beehives, magnifying glass in hand and enduring the occasional sting, all in an effort to make his theory of evolution bullet-proof.\u00a0 He was simultaneously studying such things as earthworms, barnacles, the dispersal of seeds and the webbed feet of upland ducks.\u00a0 I smile to think of the mailman delivering yet another unusual package to his home.<\/p>\n

One of the Darwin\u2019s charms is that his experiments are simple and clever and could be duplicated by amateur naturalists in their own backyards.\u00a0 In fact, I think this would make a noteworthy science project for the motivated grade school student.\u00a0 Perhaps not start by putting colored wax into a beehive \u2013 I can\u2019t imagine how many times Darwin got stung \u2013 but rather earthworms.\u00a0 Darwin played them music to test their hearing.<\/p>\n

Once sensitized to the hexagon, I began to see them everywhere.\u00a0\u00a0The compound eyes of the dragonfly are hexagonal, the sea turtle shell has hexagonal plates, both presumably driven by the efficient packing of the hexagon.<\/p>\n

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Cell phone companies use hexagonal shaped cells to efficiently map the location of phone towers so that there will be no gaps in coverage.\u00a0 Settlers of Catan also uses hexagonal tiling.<\/p>\n

There are some very practical advantages to the hexagon.\u00a0 Consider the pencil.\u00a0 It typically has a hexagonal shape, perhaps to provide a better grip, or better packing in a box or maybe to keep it from rolling off the table. (Personally, I would like hexagonal raspberries.\u00a0\u00a0 My floor is stained with raspberries that have rolled off the table and gotten squished.)<\/p>\n

The\u00a0hexagonal heads of nuts and bolts are the perfect compromise between round and square to provide the necessary torque for the wrench.\u00a0 An eight-sided nut would be too circular and the wrench would fall off.\u00a0 A square shape would provide inadequate surface area for the torque.\u00a0 \u00a0\u00a0A quick perusal of our garage revealed a variety of hexagonal gas caps and knobs.\u00a0 The bathroom plumbing at my local library provided another example!<\/p>\n

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Chicken wire is hexagonal.\u00a0 A rectangular configuration loses its spunk if one wire fails – chickens can easily escape by squeezing through the vertical slit.\u00a0 In a hexagonal structure the wires are wrapped around two separate strands, if one fails, the fence retains its integrity, at least for chickens.<\/p>\n

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The overall shape of a snowflake is hexagonal, many crystals are hexagonal, (including the graphite in the pencil), all driven by the underlying chemistry of attraction and repulsion.\u00a0\u00a0Our physical world boasts some impressive hexagons.\u00a0 The Giant\u2019s Causeway in Ireland consists of a lava field tiled in hexagons.\u00a0 The shrinking of the cooling lava created tension across the field leading to cracks.\u00a0 According to fracture mechanics, a hexagonal shape of cracks releases the most tension.<\/p>\n

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A similar configuration of basalt piles exists in California, called the Devil\u2019s Postpile. \u00a0The imperfection of the hexagons is related to the variable rate of cooling across the field.<\/p>\n

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Darwin must have known about the ubiquity of hexagons.\u00a0 Certainly he\u2019d stared into the compound eyes of a dragonfly as he pondered the evolution of the eye.\u00a0 He constantly received animal specimens in the mail and would likely have a collection of hexagonal-plated turtle shells.\u00a0 \u00a0Ireland\u2019s Giant\u2019s Causeway became widely known in 1692.\u00a0 Chicken wire was invented in 1844.\u00a0 Darwin, ever curious, must have also noticed the practical hexagonal design of nuts and bolts.\u00a0\u00a0 Perhaps the hexagon was bringing it all together for him, from micro to macro, the surprising overlap of the physical, chemical and natural world.<\/p>\n

Darwin died 100 years too early to appreciate the\u00a0vast\u00a0hexagon on the northern pole of Saturn, identified during the 1981 Voyager mission.\u00a0 This hexagon is 18,000 miles wide.\u00a0 I can imagine Darwin\u2019s cosmic curiosity with this huge cloud swirling around a churning storm at its center, producing the exact same geometry as his humble bee.\u00a0 \u201cWhat wild wind is at work here?\u201d he comments as he taps his hexagonal pencil filled with hexagonal graphite crystals on his desk.<\/p>\n

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 <\/p>\n

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Not that there was any pitched battle for my loyalties, but I have to say that the hexagon is my favorite geometric shape.\u00a0 I first became enamored in biochemistry when introduced to the basic building block of the carbon ring, … Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":true,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","enabled":false},"version":2}},"categories":[33],"tags":[],"class_list":["post-2807","post","type-post","status-publish","format-standard","hentry","category-nature"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/p7iVc8-Jh","post_mailing_queue_ids":[],"_links":{"self":[{"href":"https:\/\/www.fanagrams.com\/blog\/wp-json\/wp\/v2\/posts\/2807","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.fanagrams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.fanagrams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.fanagrams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.fanagrams.com\/blog\/wp-json\/wp\/v2\/comments?post=2807"}],"version-history":[{"count":5,"href":"https:\/\/www.fanagrams.com\/blog\/wp-json\/wp\/v2\/posts\/2807\/revisions"}],"predecessor-version":[{"id":2813,"href":"https:\/\/www.fanagrams.com\/blog\/wp-json\/wp\/v2\/posts\/2807\/revisions\/2813"}],"wp:attachment":[{"href":"https:\/\/www.fanagrams.com\/blog\/wp-json\/wp\/v2\/media?parent=2807"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.fanagrams.com\/blog\/wp-json\/wp\/v2\/categories?post=2807"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.fanagrams.com\/blog\/wp-json\/wp\/v2\/tags?post=2807"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}