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This Is Why … I’m Writing

Let’s face it, physics has an image problem. If we were to play a little word association game, what are the first words that comes to mind when I say “physics”?

  • boring
  • hard
  • math-y (i.e. see points 1 and 2!)

Or, if we play an image association game, what are the first things you picture when I say “physics”?

But here’s the thing – physics is so much more than cosmology and quantum computers, and you don’t have to be ‘Sheldon-Cooper smart’ to appreciate it. Fundamental physics principles help us to understand everything from how the heart pumps blood through the circulation system to why soap films form spheres when you blow air into them.

This is why I write: I love that a surprisingly small set of fundamental physical principles can explain the ‘why’ of just about any everyday phenomenon. I want to share that with anyone who will listen. The world is full of incredible examples of physics in action and you don’t need a PhD in quantum mechanics to appreciate them! And, although I feel a little late to the party in starting this blog now, I’m taking inspiration from something I saw at #SciCommTO 2020:

by Michelle Rial

This is Why … Igloos Are An Incredible Example of Human Ingenuity

Our amazing crew at Royal City Science has been brainstorming ideas for travelling exhibits that we can take to schools, festivals, Farmers’ Markets, etc. to share our passion for all things STEAM (science, technology, engineering, arts, and mathematics), while we fundraise to build a permanent home in Guelph. A blog post from the NFB recently caught my eye and sparked a little bit of an obsession with the idea of reproducing our own replica igloo because, as I dug into the details, I was truly blown away by the incredible brilliance of the design. As June is National Indigenous History Month, it seems timely to share what I’ve learned so far.

Can’t you just picture it? We arrive at a school gymnasium and throw 70-100 perfectly shaped pre-cut plastic blocks on the floor for the students to puzzle over and work together to assemble just so. And then they take turns to climb inside, to marvel at the stable and aesthetically beautiful structure the Inuit are masters at assembling with nothing more than snow and a knife. The Arctic is an unforgiving landscape, with no trees to provide timber for the typical structures built in warmer regions. Shelter is essential for survival and the Inuit succeed in this harsh landscape through their ingenuity, patience, and understanding of the natural world around them, collaborating with nature rather than trying to conquer.

If you live in a region that regularly gets snow in the winter, then you have possibly tried to assemble your own igloo at some point. Naively you cut rectangular blocks and position them in a ring, but it probably isn’t long before you realize that this simple approach isn’t going to work. First of all, you need the walls to slope inwards to form that classic dome shape, so your blocks need to be shorter on the inside edge than on the outside to a precise degree, as shown in the sketch I’ve made below from the NFB summary of their project.

The next thing to notice is that, starting at Block #1 in this sketch, the first layer of blocks is actually cut to form a ramp or slope that is maintained all the way up to the top of the structure in a continuous spiral. Picture a screw thread superimposed on the dome. Why? When you place the next block up the ramp, part of its weight is supported by the block beside it that you just positioned, giving it more stability. This is particularly important as you near the top where the blocks are angled quite significantly inwards. I mean the genius of this design is truly breathtaking.

The dome, as well as the related arch, have been used in building structures for thousands of years. A domed roof is a fantastic way to enclose a relatively big area without any internal supports – the curved shape does a great job of spreading forces applied at the top all around to the whole structure, making it incredibly stable. This is also why an egg is so strong, even when mumma hen is sitting on it!

So maybe now you can see why I want to make this happen and I reached out to some brilliant colleagues in the Department of Physics at the University of Guelph to explore the idea further. My first stop was Steve Wilson, our fabulous workshop supervisor. He was (understandably) a little worried about the NFB blog description that “it was too complicated to pre-plan all the angles”; they measured and cut each piece by hand to fit for their exhibit. Fair enough, so my next stop was Bernie Nickel, a retired faculty member in our department who loves a puzzle, and in no time at all he came back with a mathematical model that seems to work remarkably well.

The images below compare Bernie’s models with the shape of the igloo shown in the NFB documentary, traced out on the screen and assuming that camera and screen capture distortions aren’t a big issue here. The blue symbols are the tracing marks from the film. The red curve in the first image is a catenary arch; the black is a catenoid igloo. Bernie wasn’t satisfied with either of these, so the second image shows his refined model of a catenoid igloo at the base, with a spherical cap placed on top. He is now working on a scale model with cardboard to see how it works and I am so excited to see the end product! Stay tuned!

Image Citations

  1. opening image of part of an igloo with the Northern Lights dancing overhead – by Ross Burgener, originally posted to Flickr, licensed under the Creative Commons Attribution 2.0 Generic license
  2. Child inside a homemade snow structure – from Shutterstock (ID #1917164666)
  3. Sketch of blocks used to assemble an igloo, created by the author based on the documentary footage from the NFB
  4. The interior of the Pantheon in Rome, Italy – from Shutterstock (ID # 190788227)
  5. Images provided by Bernie Nickel, personal communication, comparing his mathematical models with the shape of the igloo featured in the NFB documentary cited here

References

  • How to build an igloo by Laurence Desrosiers-Guité and Dan Thornhill, NFB Education.
  • How to build an igloo – Documentary film by Douglas Wilkinson, filmed in 1949 with support of the NFB. Please note this message from the NFB before viewing: this is an archival film that makes use of the word “Eskimo,” an outdated and offensive term. While the origin of the word is a matter of some contention, it is no longer used in Canada. The term was formally rejected by the Inuit Circumpolar Council in 1980 and has subsequently not been in use at the NFB for decades. This film is therefore a time-capsule of a bygone era, presented in its original version. The NFB apologizes for the offence caused.
  • Mathematics of Igloo Design – by Bernie Nickel, personal communication
  • The Igloo, by Charlotte and David Yue, Houghton Mifflin Company, Boston, USA, 1988

This Is Why … I have a love/hate relationship with #IWD

Last month, the world celebrated the 7th annual International Day of Women and Girls in Science (IDWGS). I was honoured to be featured by my institution for my work in fostering the natural curiosity of young girls through our fun video series, among other initiatives. I LOVE talking and doing science with kids, so getting the opportunity to share this passion more widely is fantastic. But today, on International Women’s Day (IWD), I find myself struggling to ‘celebrate’. I HATE that, in 2022, we still need an international day to shine a light on the lack of gender equality in so many fields, including science.

  • Text reads: International Day of Women and Girls in Science, February 11
Image: cartoon of a young woman with long brown hair and glasses, holding a flask containing a blue solution and bubbles coming out. There is a rack of test tubes on the table in front of her with different coloured samples. She wears a white lab coat and holds a piece of white paper.

A little over 30 years ago, I graduated from high school in Toronto and headed off to university to study science. My strongest marks by far were in physics and math, but all my teachers ever focused on when giving career advice was medicine or nursing: no one suggested I pursue physics or engineering. Being a little contrary, I decided to give the applied physics program a go, telling myself I could always switch out if it proved too difficult. After all, girls didn’t do physics or engineering 30 years ago, it was almost exclusively a boys’ club back then.

Fast forward 30 years and here’s the incredibly depressing thing – the statistics haven’t changed. Physics, math, and engineering are still boys’ clubs.

How significant is this stagnant imbalance? Comparing StatsCan census data on post-secondary qualifications by field of study, we find that the percentage breakdowns by gender in every STEM (science, technology, engineering, and math) category have barely changed from 1996 to 2016:

Bar graphs from census data from 1996 and 2016 in the following categories for field of study: 
math & physical sciences
engineering
health-related
biological science, agriculture
The male/female ratios are largely unchanged from 1996 to 2016.
All data here from archival Stats Can census data

More recent data from colleagues at the University of Guelph (2019) demonstrate that we lose women from physics and engineering at just about every step of their secondary schooling in Ontario:

Image of a leaky pipeline with female:male ratios dropping continuously:
Grade 10 science (50/50)
Grade 11 physics (40/60)
Grade 12 physics (34/66)
1st year physics & engineering (20/80)
physics & engineering grads (20/80)
licensed engineers (11/89)
Image from the “Closing the Gender Gap” report from the Ontario Network of Women in Engineering, 2019

It’s not like we’ve had our collective heads stuck in the sand for three decades. There are scores of non-profit organizations in Canada that have worked for 20 or 30 years to increase the number of students pursuing the STEM disciplines at the post-secondary level. These organizations all recognize the ongoing lack of women in certain disciplines, and countless programs have sprung up in response.

Some progress has been made, as seen in tracking what happens when we ask children to ‘draw a scientist’: the percentage of girls drawing a female scientist has increased substantially over a 50-year period. But we still have a lot of work to do, as both girls and boys are more likely to draw a male scientist as they get older:

Why does this matter? Because men don’t have a monopoly on curiosity and wonder. Spend some time in a kindergarten classroom and you’ll see that the girls are just as inquisitive about how the world works as the boys. Somewhere along the way, the boys get the message that physics and engineering are the fields in which to explore this curiosity and girls find themselves heading towards health-related fields. I’m furious that we still find ourselves here, 30 years later. The definition of insanity is doing the same thing over and over again and expecting a different result. Something must change.

And let’s deal with the elephant in the room right now: yes, girls are just as capable of understanding physics as boys are. Data from standardized exams in the UK as recently as 2018 show that 30% of the girls writing the A-level physics exam earned the highest two grades possible compared with 29.5% of the boys who wrote the same exam. But just like Canadian students, the numbers are hugely skewed: 8300 girls wrote that exam compared to 29,400 boys, i.e. 22% vs. 78%. Sound familiar? I cite the UK data for two reasons: 1) the standardized nation-wide exam gives us large n statistics to look at and 2) the Institute of Physics (IOP) seems to be lightyears ahead of other national professional organizations in championing this issue and aggressively seeking solutions.

The IOP is currently running a suite of gender balance programs, building upon decades of research into the issue. The main conclusion so far seems to be that tinkering around the edges by working with individual teachers or even whole science departments in specific schools doesn’t move the needle. However, effective change has been achieved in pilot projects that holistically tackle the culture of schools from early-learning centres through to secondary schools.

Two young girls seated at a lab bench with various electronics, cables, a multimeter, soldering iron, in front of them. The girls are sharing a high five and smiling.

One pilot project worked on improving the confidence of 13 and 14-year-old girls directly and worked with physics teachers on best practices for inclusive teaching techniques, while simultaneously taking a whole-school approach in working with leaders, teachers, and students across all subjects on gender equity. This project saw the number of girls starting A-level physics (at age 16, comparable to our Grade 11) increase by a factor of three over a two-year period.

Change is possible if we are prepared to look in the mirror and recognize that we are failing our students by tinkering around the edges. STEM-themed summer camps and after-school programs for girls are great, as are targeted scholarships for women in postsecondary studies, but they tend to reach the minority who manage to maintain their passion for the discipline despite the structural biases working against them. Let’s follow the lead of our UK colleagues and recognize that systemic change, from kindergarten to graduate school and beyond, is the only way to truly address this decades-long issue, before another 30 years goes by and my grandchildren are facing the same obstacles. In the words of Marie Curie, progress is neither swift nor easy. But when it comes to gender equity in science, it is long overdue.

A caricature of Marie Curie, holding a flask of a coloured solution. The text reads: I was taught that the way of progress was neither swift nor easy ~ Marie Curie

References:

50 Years of Children Drawing Scientists by Youki Terada in Edutopia, May 2019

Closing the Gender Gap in Engineering and Physics: The Role of High School Physics by Mary Wells, Martin Williams, Eamonn Corrigan, and Valerie Davidson for the Ontario Network of Women in Engineering, December 2018

Women in physics: Why there’s a problem and how we can solve it by Valerie Jamieson in New Scientist, November 2018

Stats Can Census Data

Institute of Physics, UK, Gender Imbalance Initiatives

This is Why … Ducklings Form Single-File Flotillas Behind Mumma

This adorable example of everyday physics took me entirely by surprise. We’ve all seen a sweet little floating parade in the springtime at our local river or pond, with a mumma duck or goose followed closely by a line of babies paddling quickly to keep up. But I never really noticed that the spacings of each duckling is quite consistent down the line. Or that the formation tends to be single file much more than other possible arrangements. Why? It’s all about the wave pattern generated by the duck in the lead position and how the wakes of the other ducklings in line interact with that pattern. Scientists have recently concluded that these sweet little fuzzballs find the exact right position in line every time so that they can surf along with a boost of energy to keep them moving with ease.

Two adult geese followed by three goslings in single file – a video I shot last May, completely unaware of the physics on display!

It honestly makes me smile to think that I have witnessed a fantastic example of wave interference countless times without realizing it. The only reason I now understand what’s happening is because I stumbled across a lovely little paper in the Journal of Fluid Mechanics a month or so ago by a collaboration from the University of Strathclyde (Glasgow, Scotland) & Jiangsu University of Science & Technology (Jiangsu, China). This team set about modelling the fluid dynamics both mathematically and numerically, discovering that these wee fluffs are both cute and smart: when they swim at the ‘sweet spots’ behind mumma, they instinctively choose positions that correspond to dips in the overall wave pattern. For ducklings in positions 1, 2, and 3, riding the crest of a wave (just under their little butts) gives enough of a boost that it is energetically favorable for them to swim there. The model found that ducklings in positions beyond #3 don’t gain in energy, but there is also no energy loss to drag in the water – a delicate dynamic equilibrium is established. Plus, ducklings have been observed to switch up their order in line, which might be an egalitarian approach to sharing the surfing boost among sibs.

It honestly blows my mind a little that math and physics can explain why ducklings swim in single file and can even predict the positions they will choose behind mumma; this article by Yuan and team was a lovely discovery. Their paper also led me to an equally delightful experimental study in which biologist Frank E. Fish from West Chester University measured the mechanical effort exerted by ducklings swimming in formation, with and without the presence of a mumma generating a wake for them to swim in. Dr. Fish imprinted his subjects to a decoy and then had the babies swim behind the decoy in a tank in which the flow rate they swam against was controlled. The decoy was always visible in front of the ducklings, but it could be raised slightly above the water so that it didn’t create a wake.

Sketch of the clever experimental setup, figure 1 from the paper by Dr. Fish, copyright Wiley-Liss Inc.

Instinctively, the ducklings self-organized into single-file formation behind the decoy. Dr. Fish then analyzed video recordings from the side of the clear tank, calculating the arc length of the foot stroke taken by a duckling as a measure of the effort exerted. Mumma’s wake, when she was lowered into the tank, resulted in a reduction in effort for all the ducklings studied – more so for the younger babies (3 days old) than the older ones (14 days old). Consistent with the mathematical modelling from Yuan et al, by swimming in the sweet spots, the ducklings take advantage of wave interference to expend less energy to paddle along behind their mumma.

Even though I wasn’t actually wondering why in this instance, now I know the physics behind the formation! So next spring, when I see this amazing example of fluid dynamics in action, I’ll be smiling at both the spectacle and the science.

References:

Wave-riding and wave-passing by ducklings in formation swimming, Zhi-Ming Yuan et al, Journal Fluid Mechanics (2021) vol 928 R2 doi:10.1017/jfm.2021.820

Kinematics of ducklings swimming in formation: consequences of position, Frank E. Fish, Journal of Experimental Zoology (1995) 273 1-11

Image sources:

Ducklings swimming behind mumma – Shutterstock.com ID 1381730477

Video of Canada geese family – taken by the author, May 2021, Rockwood, ON

Mathematical model results of the wave pattern and the coefficient of drag (CDR) for mumma duck and 6 babies – Figure 4 from Yuan, Z., Chen, M., Jia, L., Ji, C., & Incecik, A. (2021). Wave-riding and wave-passing by ducklings in formation swimming. Journal of Fluid Mechanics, 928, R2. doi:10.1017/jfm.2021.820 (Huge thank you to Cambridge University Press for believing in Open Access publishing!)

Close up of ducklings swimming behind mumma – same Shutterstock image as the opening file with red lines indicating wave crests in the overall pattern added by the author

Sketch of the experimental setup of Dr. Fish, Figure 1 from the Journal of Experimental Zoology (1995) 273: 1-11, Copyright Wiley-Liss Inc.

This is Why … My Zen Space Sometimes Smells Like Cotton Candy

(Alt text – A cotton candy vendor walking under the forest canopy during a morning in Bardiya National Park in Nepal, light filtering through the canopy. By Gaurav Aryal, Shutterstock.com)

Like everyone else, the start of the new school year has been incredibly stressful for me. Going for walks in the arboretum on campus has been a huge coping mechanism, especially on the days when I just want to go home and curl up in a ball with the blankets over my head.

The arboretum is a spectacular sight this time of year, with maples, birches, and beeches showing off in russets, scarlets, and golds. But there’s a quiet little grove in the middle of this amazing space that also has a treat for your nose: if you’re standing in the right spot and inhale deeply, be ready for caramel, fresh-out-of-the-oven cake cooling on the counter, or, in my opinion, cotton candy. The source of this incredible scent is the katsara tree (Cercidiphyllum japonicum), a tree native to Japan and China that was imported to North America in the mid-1800s.

close up of yellowy-green, heartshaped leaves on the katsura tree with blurred greenery in the background, By imacoconut, Shutterstock.com

As the heart-shaped leaves begin to turn from green to yellow to brown every autumn, an aromatic molecule called maltol is released to the air, in concentrations high enough that it can smell like walking past the waffle station at your favourite brunch spot. The scent is so strong and so distinctive that katsuras are known as “kuchenbaum” (cake tree) in German or “arbre à caramel” (caramel tree) in French.

chemical structure of Maltol, also known as 3-Hydroxy-2-methyl-4H-pyran-4-one, from Wikimedia – public domain (File:3-hydroxy-2-methyl-4H-pyran-4-one 200.svg - Wikimedia Commons)

Maltol, chemical formula C6H6O3, occurs naturally in certain foods like coffee and cocoa, getting converted from sugars to this amazing smelling molecule in the baking or roasting process. Maltol is also found in pine needles, the bark of larch trees, and red ginseng. The white crystalline powder with the butterscotch fragrance is extensively used in the food industry as a flavouring agent as well as an additive in perfumes.

When I first noticed the smell a few years ago, I was honestly completely baffled. It was always in the same spot but it wasn’t always noticeable. Some days it would be incredibly strong but then months would go by without the slightest whiff. Looking around for the source of the scent, there are no flowers (or bakeries) in sight. But, with the help of the arboretum signage and Google, the mystery was revealed. And I was greatly relieved to find out that I wasn’t having some kind of stress-induced olfactory hallucination!

With a little digging, I found myself reading about some pretty incredible properties of this deliciously scented molecule. Researchers are exploring its antioxidant capabilities and its potential application in treating a huge array of concerns, from liver disease to osteoarthritis, glaucoma to bacterial resistance to common antibiotics. Among other functions, maltol works to control the amount of metal in the human body, forming stable complexes with metallic ions like Al+3, Fe+3, Zn+2, etc. Some of these complexes have been investigated as possible treatments for anemia, Alzheimer’s, cancer, and diabetes. So it turns out that my favourite spot to destress in the arboretum is full of amazing science, right under my nose.

References

Maltol, a Food Flavoring Agent, Attenuates Acute Alcohol-Induced Oxidative Damage in Mice. Ye Han, Qi Xu, Jiang-ning Hu, Xin-yue Han, Wei Li and Li-chun Zhao Nutrients 20157(1), 682-696

Novel Synthesis of Maltol Capped Copper Nanoparticles and Their Synergistic Antibacterial Activity with Antibiotics. Naqvi et al 2021 Plasmonics https://doi.org/10.1007/s11468-021-01452-3

Maltol prevents the progression of osteoarthritis by targeting PL3K/Akt/NF-kB pathway: in vitro and in vivo studies. Lu et al 2021 J Cell Mol Med25: 499-509 https://doi.org/10.111/jcmm.16104

This is Why … Geese Are Yoga Gurus

(An ashy-headed goose demonstrating its exceptional balancing ability; image from Shutterstock.com)

The other day I spent some time exploring the “arm” at Guelph Lake, curious to see how far up the Speed River I could get in my kayak before bottoming out. (Turns out, not very far. Water shoes would have been a good idea. I definitely need to try this earlier in the season when the water is higher.)

(map of Guelph Lake with Speed River indicated)

It’s a very quiet stretch of water and, as I came around a bend with a magnificent willow holding court on the bank, I paused in my paddling to drift soundlessly past a large flock of Canada geese napping and preening in its shade. Some were standing, some were resting on the ground with their heads tucked around on their backs, and quite a few were balancing nonchalantly on one leg. Why is this a comfortable resting position? If I have a few quiet minutes on a summer afternoon, I don’t feel inclined to strike a tree pose on a riverbank …. what’s up with our feathered friends?

Canada geese are not the only bird to strike such a pose – flamingos and other long-legged aquatic birds such as storks and herons are perhaps more often pictured in this position. A recent report in Functional Ecology estimates that standing on one leg has been observed in approximately one third of the 852 avian species in their study.

In order to stay upright in this position, a two-footed creature (goose, heron, human, etc.) has to shift their weight slightly to the side to get their centre of mass over the single supporting foot, as is evident in the head-on photo of the ashy-headed goose at the top of this article, as well as in this video clip of my ever-helpful daughter, Mara, filmed for our AMASE video series: (sound on)

It takes effort for us to maintain this crucial vertical alignment of our centre of mass and the small area of support afforded us when only one foot is touching the ground. So what’s the advantage for birds to stand this way? There are two possible motivations commonly proposed by biologists: heat regulation and reducing muscle fatigue. However, it’s also possible, based on a recent study reported in Biology Letters, that it’s all about forces and torques (aka mechanics)!

First, let’s talk about the standard theories. When you see a long-legged bird standing on one foot, the retracted leg is often tucked up underneath the body, hidden inside the cozy plumage. This gives rise to the heat regulation theory: when the surrounding air is cold, a lot of body heat is lost through the legs since they are not covered with insulating feathers. Birds do adjust for this by reducing blood flow to their legs when its cold outside, as well as employing a neat heat-exchange trick between the cold blood coming back to the body from the feet and the warm blood heading down from the heart. Biologists have speculated that the one-legged pose is a means of further reducing heat loss during chilly weather by insulating one of the bare legs inside the chest feathers. But this couldn’t be the reason for my Guelph Lake geese friends, as the weather was a balmy 24°C that afternoon. And observations in the literature are mixed as to whether this pose is seen more often at lower ambient temperatures. Plus, the lifted leg is not always hidden inside the plumage; sometimes it’s only partially lifted and sometimes it’s extended back behind the bird. It seems that heat conservation has a role to play, but it can’t be the only reason.

Ducks regulating heat loss by standing on one leg on an icy surface, as well as shielding their bills in their feathers to prevent heat loss there. From Shutterstock.com

The other theory is something you have probably done – shift your weight from one leg to another when you stand for long periods of time. By shifting your weight, you are temporarily giving the muscles in one leg a little break. Seems reasonable that birds might do the same.

(Gif of the Brady Bunch, rapidly switching weight from one leg to the other, from Giphy)

However, there is a big difference here: we don’t lift the resting leg completely off the ground into mid-air when we give it a break. This takes energy and requires more muscle firing in the supporting leg to keep our balance over a smaller supporting area. According to a thorough summary by Dr. Reinhold Necker, professor emeritus from the University of Bochum in Germany, “there is so far no experimental support” that this one-legged bird pose is struck to reduce muscle fatigue. Which brings us to the 2017 paper in Biology Letters.

The study conducted by Young-Hui Chang and Lena H. Ting (Georgia Tech and Emory University, respectively) looked at both the anatomy and the behaviour of the one-legged stance in flamingos. With cadaver samples, the team manipulated the body of the flamingo while it was held in a one-legged pose by clamps, testing the passive stability of this position. The researchers also analyzed the patterns of the pressure exerted by the foot of live birds while they stood on one foot on a force plate, both awake and asleep.

The clever combination of experimental manipulations of cadavers and measurements of exerted pressures on force plates from living subjects led the team to conclude that it may be energetically favourable to stand on one leg due to a passive, gravity-driven body weight support mechanism in the lower joints of these birds.

From the cadaver studies, Chang and Ting observed a balance between the rotation about the knee and hip joints due to gravity and a countermeasure arising from anatomical limits on the range of motion of these joints – i.e. certain bony and cartilaginous structures may act to block the joints against the destabilizing rotations that would otherwise happen. Chang and Ting observed that this anatomical countermeasure only activates when the foot is positioned a little inside the hip joint, as it naturally is when a bird (or Mara, see above) tries to stand on one leg!

The force plate data from the live subjects further confirmed that flamingos are incredibly good at passively balancing on one foot, as the point of highest pressure remained static for long periods of time, even when the birds had their eyes closed. Try standing in tree pose for a few minutes and focus on how much the contact between your foot and the floor is shifting to maintain your balance – I predict that it moves around considerably more than the corresponding observations in flamingos by Chang and Ting.

(Image from Chang and Ting, 2017, Biology Letters 13: 20160948)

As a physicist, even I realize that applying a study in flamingos to understanding an observation in Canada geese is a little suspect. But, as a physicist, I love that an everyday sight such as geese relaxing on a riverbank has led me to discover a free-body diagram in a paper in Biology Letters! Partially thermodynamics, partially mechanics, and wholly fascinating, I now have a slightly better understanding of why geese are yoga gurus. Can’t wait to share this with my first-year biological science students the next time I’m at the front of the room in PHYS*1300!

References:

 Birds standing on one leg: mechanisms and meaning; Dr. Reinhold Necker, http://reinhold-necker.de/seite11a.html

Chang Y-H and Ting L. H. 2017 Mechanical evidence that flamingos can support their body on one leg with little active muscular force Biology Letters 13: 20160948 http://dx.doi.org/10.1098/rsbl.2016.0948

Pavlovic G., Weston M. A., Symonds M. R. E. 2019 Morphology and geography predict the use of heat conservation behaviours across birds Functional Ecology 33(2) 286-296 https://doi.org/10.1111/1365-2435.13233

This is Why … Falling on the Ice May Have Saved My Life

(images from Shutterstock.com unless otherwise noted)

On an icy cold morning at the end of February, I tentatively headed down the slope behind our house, on my way out for some daily exercise. As I stepped off a retaining wall down to the walkway below, my foot went right out from under me and I crashed down, hard. The small of my back came down on the edge of the six-inch high concrete step as the rest of me landed on the lower level − I experienced the most intense pain I have ever felt. And that includes the drug-free labour and delivery of our first child.

There was no one in the house … no one to hear me scream. No matter how I tried to gingerly move, bolts of searing pain shot through me. I knew I was in trouble, so I sheepishly phoned a friend who raced over, took one look at me, and said: “I’m calling 911!” The combination of this fall[1] and that call quite possibly saved my life because, on our way to the local hospital, the incredibly wonderful paramedic said: “I know you’re in a lot of pain and quite stressed right now, but your blood pressure is really high. You need to follow up on that!”

How high? It was consistently reading over 180/100 (mmHg) that morning.  This is WAY above the normal range of 120/80, and solidly in the danger zone of risk of heart attack or stroke. According to the FDA, readings in the 180/110 range or higher are considered “hypertensive crisis” and need immediate medical attention. And here’s the truly scary part in hindsight – I had absolutely no idea. It’s not called ‘the silent killer’ for nothing.

High blood pressure (hypertension[2]) affects almost 1 in 4 Canadian adults according to a 2015 study, with the prevalence increasing dramatically with age. We tend to think of this as a concern for our parents or grandparents, or at least I naively did, but the incidence rate in women in the 40 to 59 age range (i.e. me) is actually 21%. Of the hypertensive women in my age range, about 20% were not aware of their condition either.

Data from Table 3, Hypertension prevalence & awareness in Canada, combined 2012 to 2015 data (statcan.gc.ca)

Why is this an issue? Because hypertension is currently the leading risk factor for death in the world.  High blood pressure means that your heart has to work harder to move blood around your body – like trying to pump air into a bike tire that is already fairly full – so the muscle of your heart thickens and is more prone to suffering a heart attack or failing. The vessels themselves can also narrow and get damaged, making blood flow more difficult.  And it’s not just damage to your heart, as if that isn’t serious enough. The damage to blood vessels happens everywhere, including in the brain. Left unchecked, restricted blood flow can lead to blockages or bursts, i.e. strokes and their debilitating consequences. High blood pressure also raises the risk of developing vascular dementia, the second most common form of dementia. Reading that stopped me cold in my tracks.

For a middle-aged woman, I’m reasonably fit. Our family eats mostly home-cooked meals, avoiding the very high salt content typically found in prepared foods. I walk daily, play with my kids, row/kayak in the summers, snowshoe in the winters; 10,000 steps are pretty standard on a daily basis. But I do have a family history of concern and I should have been paying closer attention. It’s just so easy, when running a busy household and a demanding career, to de-emphasize your own health in the ever-fluctuating list of priorities. I also experienced no symptoms whatsoever, at least I didn’t think I did. It turns out that the symptoms can be pretty vague and easily chalked up to daily stresses. I mean ‘headaches’, ‘fatigue’, … who doesn’t suffer from these regularly, particularly in our current pandemic crisis?

I can guarantee you that I would not have scheduled a routine check-up with my family doctor for many months (years?) to come, given all that is going on in our lives. But my guardian-angel paramedic’s warning on my very first (and hopefully last) ambulance ride struck a chord, particularly since I lost my mother to a massive stroke when she was only 67 years old. Her premature dementia in the final years before her stroke, as well as the stroke itself, are fates I wish to avoid. Fast forward a few months and my family doc now has me on some low dose meds, I’ve amped up my exercise regime, and I have significantly reduced my alcohol intake (yes, during a pandemic, it can be done!).

To all my fellow middle-aged women, consider this a friendly nudge. It’s all too easy to dismiss little warnings as no big deal; I did, even though I should have known better, and it could have cost me my life.

References

High Blood Pressure – Understanding the Silent Killer, U.S. Food and Drug Administration

Blood pressure and hypertension, Jason DeGuire et al Release date: February 20, 2019, Health Reports, StatsCan (statcan.gc.ca)

Padwal RS, Bienek A, McAllister FA, Campbell NR. Epidemiology of hypertension in Canada: an updateCanadian Journal Cardiology 2016; 32(5): 687–694.

Forouzanfar MH, Liu P, Roth GA, et al. Global burden of hypertension and systolic blood pressure of at least 110 to 115 mm Hg, 1990-2015. The Journal of the American Medical Association 2017; 317(2): 165–182.

Connor A. Emdin et alBlood Pressure and Risk of Vascular DementiaStroke, 2016; 47:1429–1435 http://dx.doi.org/10.1161/STROKEAHA.116.012658


[1] It turns out I was INCREDIBLY lucky in my fall – no broken bones, no organ damage, no slipped disks. Some deep muscle bruising and strain that took a couple of weeks of bedrest to begin to heal. More or less full mobility was back in about a month. HUGE thanks to my good friend, Cindy Johnson, for rushing to help!

[2] I’m using the definition of hypertension of a systolic pressure (the top number) greater than 140 mmHg OR a diastolic pressure (the bottom number) greater than 90 mmHg. This is technically Stage 2 hypertension, with Stage 1 usually considered at systolic greater than 130 mmHg or diastolic greater than 80 mmHg.


This Is Why … Rainbows Have Nothing to Hide

A few weeks ago during dinner there was a sudden burst of sunlight in the midst of a rain shower. Knowing that these were ideal rainbow-making conditions, we jumped up from the table and ran out to investigate. Sacrificing our meal was entirely worth it: this photo was taken just a few hundred metres down the road from our house. Once again, I am blown away by the beauty of physics in the world around me.

The bright feature in the photo labeled A is the primary bow, which is always observed with the red band on the outer edge of the arc and the violet band on the inner edge. Light coming from the Sun, behind you, scatters in all directions from the raindrops in front of you. The light doesn’t scatter equally in all directions though, as shown in the following image, modified from a 2016 paper by Alexander Haußmann. In this plot of intensity versus scatter angle, zero degrees represents light that is scattered straight back towards you, while light scattered directly away from you is at an angle of 180°.

Sketch showing the cone of light that scatters back towards the viewer from the raindrops, with 42 degrees as the maximum angle of the cone

(image from Haußmann’s 2016 paper)

We can see that the intensity is pretty steady at small angles, coming straight back towards you, then reaches a peak at 42°. This is the angle where you see the primary rainbow: 42° is the maximum angle of the cone of light that scatters straight back towards you from one bounce inside a raindrop.

At the very edge of this cone of scatter, we see the separation of the different colours of light that come from the Sun: the highest wavelength, red, scatters back at a slightly higher maximum angle (42.2°) than the lower wavelengths, like blue, which has a maximum angle of 41.2°, as we see in the sky with the red on the outside edge and violet on the inside of the primary bow.

Moving along the relative intensity graph from 42° to the next large spike at 51°, there is a region known as Alexander’s band, named after Alexander of Aphrodisias (circa 200 AD), which is marked B in my photo at the top. The band ends at 51°, which is the minimum scatter angle for light to bounce twice inside the drop and then reach our eyes as sketched here. The angle of 51° corresponds to the location of the secondary bow (C).

Alexander’s band is dark because it spans the ‘dead zone’ between the maximum angle for scattering with one bounce and the minimum angle for scattering with two bounces inside the droplet. Since light can’t bounce 1 ½ times, the intensity in this in-between range is quite low and the sky looks darker here than above the secondary bow or below the primary bow. It’s not midnight black though, because light is being scattered from clouds, other droplets, the front surface of droplets, etc.

The secondary bow, with its sharp spike at 51°, will always appear with the red band on the inside and the violet band on the outside, opposite to the order seen in the primary band. It’s not always easy to see this feature though; the intensity is less than that of the primary bow. With each bounce, some of the light continues through the droplet rather than reflecting, which means that the tertiary (and higher order) bows are rarely observed. The added challenge with seeing the tertiary bow is that it appears at about 140°, very much in the forward direction, away from the observer. To see the tertiary or quaternary bows, you have to be facing the Sun with scattering raindrops between you and the Sun, which, shining brightly, tends to wash out faint features like these weak bows. The angular spread of the colours increases with each order of bow, so this smearing out also makes them harder to see. The fifth order bow is predicted to appear just inside the secondary bow, as seen as the little spike in the intensity graph at around 50°.

We have marveled at, and tried to understand, rainbows for millennia. Way back in 1637, Descartes published a chapter in his Discours de la Méthode in which he correctly calculated the angles observed for the primary and secondary bows, although the colours were still a mystery in his day. Newton famously solved this mystery by deducing that white light is made up of all these colours. But it wasn’t until the early 1800s when Thomas Young linked the colour of light to its wavelength that physics was able to explain the observation of “supernumary arcs”, shown here in a cropped piece of my photo, marked as D.

Bright and dark bands inside the primary bow are called supernumerary arcs

These bands of light and dark regions just inside the primary bow are classic interference patterns caused by light scattered at the same angle but travelling different path lengths through a raindrop. Young’s theory correctly predicted the effect of drop size on this feature of the rainbow and provided further support for the wave theory of light.

Once James Clerk Maxwell figured out that light is transmitted in the form of electromagnetic (EM) waves (around 1864), a new theory emerged to understand light scattering from small objects like raindrops, known as Mie scatter after the German physicist Gustav Mie. By breaking the incoming EM wave into a series of partial waves, we then calculate how each scatters from a sphere of a particular size and electrical properties, and then add up the results. Intensive calculations with Mie theory were not possible until we had the help of computers in the latter part of the 20th century, but now millennia of wonder and delight has been explained, matching observations beautifully.

A young girl (my daughter) in a unicorn costume with a rainbow-coloured frozen layered smoothie

Refinements continue today, with tweaks to deal with non-spherical droplets (affectionately known as the ‘hamburger bun’ model), different sizes of droplets within the rain shower, effects of the intensity of rain in different regions of the shower, and more. Science marches on, so grab your rubber boots and let’s dash out, chasing rainbows. My little helper unicorn likes to refresh with a rainbow treat when we get back inside!

References:

Rainbows in nature: recent advances in observation and theory by Alexander Haußmann 2016 Eur. J. Phys. 37 063001

More to rainbows than meets the eye by Simon Davies 2016 IOP Publishing News

The Mysterious Physics of Rainbows by Jon Butterworth, 2017, The Atlantic

A Funny Thing Happened on My Way to the Lecture Hall…

As I strode across campus to meet my second-year physics students in our Electricity and Magnetism class a few years ago, it suddenly struck me that I had the makings of a fantastic opportunity tucked under my arm. My teaching assistant had returned their last assignment of the semester to me, so I quickly formulated my plan. As I entered the room and prepped for our session, I projected a stern and serious air, quite unlike my usual smiley/jokey self. As everyone settled down, I told them that I was very disappointed to hear from our TA that there had been a considerable amount of ‘sharing’ in the assignments that had been submitted, which is completely unacceptable. I told the class that I wanted them to take out a piece of paper immediately and work through the solution to one question again, individually, right then and there.

There was much muttering and sharing of furtive looks around the room, but everyone complied as I projected the question onto the screen with the document camera. One or two students looked at me suspiciously, but I maintained my grave look and told them to get started. After a few minutes, I said to the class: “Please make sure you write your name and the date on the page …. today is April 1st.” As my students slowly looked up at me, realization dawning on their faces, I shouted: “APRIL FOOLS!” It took a good 10 minutes before we were all settled down again and ready to discuss the physics of magnetic materials, but I maintain that it was 10 minutes well spent.

Incentivize Lecture Attendance

Study after study has demonstrated the value of a ‘flipped’ classroom, in which students engage with each other and the instructor in meaningful and deep learning activities (1,2). This little April Fools’ prank was certainly not such an activity, but it is a good example of the value of being a little silly with your students: it incentivizes lecture attendance. Step one for maximizing classroom engagement is to make your session an adventure, an event that students don’t want to miss because they never know what might happen. You care about their learning, you go to great lengths to fine-tune your pedagogy based on current physics-education research, but you need to get them out of bed and through the classroom door for all your hard work to pay off!

This is something I have struggled with during the pandemic with all my teaching done remotely. Being jokey and improvisational with my students has been incredibly difficult when only a tiny fraction of the class turns their camera and/or mic on – it is almost impossible to ‘read the room’ in Zoom. Even with these challenges, it seems more important than ever to find ways to inject some levity given the state of the world around us: I’m a firm believer in the notion that laughter truly is the best medicine. So, after grumbling about the countless ways in which technology foils my best-laid plans for an engaging virtual class, I decided to take advantage of the humour in the farcical: my students and I created a bingo game based on the myriad ways in which our sessions go sideways! (A spherical cow stress ball, pictured below, is the prize for completing a row or column.)

Break Down Barriers

In addition to incentivizing lecture attendance, injecting humour into our sessions also helps to break down barriers between my students and me. As evidenced by our bingo card, my humour tends towards the self-deprecating kind, which subtly communicates to my class that I really don’t take myself too seriously. I want my classroom to be a safe haven, where students can ask questions to each other and to me without worrying about being judged. No one, including me, has all the answers or is right all the time; mistakes are the best opportunities for learning.

So, rather than projecting an air of unassailable authority, one who, for example, would never drop a negative sign, I celebrate these moments with my students with Smarties. As we work through an example together, I want them to be trying to follow along with each step as we go, rather than just passively writing it all down to figure out later (i.e. the night before the next test). In our sessions, everyone is on high alert because the first person to catch a mistake, such as that inevitable loss of a pesky negative sign, is rewarded with a box of Smarties. Usually, in our face-to-face sessions, I have a couple of boxes in my laptop bag to hand out on the spot. With remote teaching I had to find a work-around and now I keep a running tally during the semester on my Smarties List, with payout at the end of term. This year, since I was actually keeping track, I added a personalized Smarties certificate to this special delivery, with one pictured below on proud display on the family fridge door! Students seemed to really appreciate the personalized touch so I’m keeping this new tradition going even after we are back in face-to-face sessions. (FYI, if you want to adopt this approach, definitely have extra boxes on hand if you need to teach after taking cold medication. Just saying.)

A COVID-inspired modification to my use of SmartiesTM in class to encourage careful scrutiny during our in-class examples as well as to normalize mistakes.

Improve Student Attitudes

I am not a stand-up comic in academic regalia. But you don’t have to be hysterical to make good use of humour in the classroom and it can have a profound effect on students’ attitudes. A recent study looked at the effect of a positive attitude towards math and the brain’s ability to learn and remember (3). The study concluded that even when they controlled for IQ, working memory, math anxiety, general anxiety and general attitude toward academics, children with poor attitudes toward math rarely performed well in the subject. And, let’s face it, in our big first-year service courses there are a lot of incoming students who do not have a positive attitude towards physics. In addition to teaching second-year Electricity & Magnetism to physics majors, a preacher to the converted, I frequently find myself in front of hundreds of first-year biology students, which is more like being an emissary to a hostile nation. In my very first lecture every year, I ask them to tell me how they are feeling about taking this course using their clickers to respond anonymously, and the breakdown invariably looks like this:

Typical responses from first-semester biology students who are required to take my introductory physics course.

There are times when there are identically zero responses in the “excited” category. In a room full of 400 students. *sigh* Making a concerted effort to help shift our students’ attitudes towards the subject is key to opening the door for them to do well, as that positive attitude results in enhanced memory and more efficient engagement of the brain’s problem-solving capacities according to the work of the team at Stanford (3).

A Little Goes a Long Way

Whenever I can I’m injecting little scenarios to make students smile. We work through a lot of example problems in class and I try to give them a little twist. For example, in free fall calculations, we work through a problem in which “Gord is spitting (sunflower seeds) from a bridge. Just to see how far down it really is.” This precise turn of phrase won’t mean much to non-Canadian readers, but this is a line from a Tragically Hip song, a hugely popular band that appeals across generations. The delivery is important when I read the question aloud and, although not everyone will get the reference, there are plenty of smiles around the room. We have also been known to analyze the position versus time graph of someone canoeing across a lake to an island where they suddenly encounter a mumma bear and her cub, as well as determining how far across an ice floe a sweet little penguin will slide before coming to a stop. This sometimes leads to a discussion on how one would experimentally determine the coefficient of kinetic friction between a penguin’s belly and an icy surface in terms of the factors one would have to control for, but I digress.

I am always looking for opportunities for student engagement and these often take the form of multiple-choice conceptual questions to use in the peer instruction mode of delivery, originally popularized by Eric Mazur at Harvard (4). I ask a question, students vote individually and we see the results. They then chat about their choices and vote again, after which we have a discussion about the correct answer (and, often, why the incorrect answers are wrong). Throwing a ridiculous choice or fun scenario into the mix often lightens the mood, like the following question that I pose as we start talking about conservation of momentum right around Halloween:

Two physicists decide to blow up a pumpkin. When it explodes, it splits into 3 pieces of equal mass, as shown from above in the illustration here. What is the direction of the velocity of the third piece?

(Yes, the image here is a still from a video that my colleague Chris and I made of blowing up a pumpkin with a bottle of liquid nitrogen!)

I like to couple our peer-instruction discussions with demonstrations, trying to include student helpers whenever possible. I always structure these as Predict-Observe-Explain, where students vote on what they think will happen, the demonstration/experiment is conducted, and then we discuss what happened and why. A popular one we do when talking about resistors in series and in parallel is the smoothie race – a head-to-head student battle with two straws in series vs. two straws in parallel to see which “battery” generates the greater current. This is a fun way to visualize currents/batteries/resistors as well as another good opportunity to discuss experimental design. But just as importantly, there is lots of cheering and laughter; a spoonful of sugar to help the physics go down!

The great smoothie race: an in-class demonstration of the difference between resistors in series and in parallel.

Every teacher develops their own style with practice and guidance, and I had the good fortune to be mentored by exceptional educators as I began to shape mine at the University of Guelph almost 20 years ago. I learned from some of the best in the business that humour is a powerful tool in the lecture hall for incentivizing class attendance, creating a welcoming environment, and improving student attitudes. To my mind, there is always room for a little tomfoolery in teaching physics!

References:

  1. F Finkenberg and T Trefzger 2019 Flipped classroom in secondary school physics education J. Phys.: Conf. Ser. 1286 012015
  2. David C.D. van Alten, Chris Phielix, Jeroen Janssen, Liesbeth Kester 2019 Effects of flipping the classroom on learning outcomes and satisfaction: A meta-analysis Educational Research Review 28 100281 https://doi.org/10.1016/j.edurev.2019.05.003
  3. Chen L, Bae SR, Battista C, et al. Positive Attitude Toward Math Supports Early Academic Success: Behavioral Evidence and Neurocognitive Mechanisms. Psychological Science. 2018;29(3):390-402. doi:10.1177/0956797617735528
  4. Peer Instruction: A User’s Manual | mazur (harvard.edu)

This is Why … Campfire Smoke Follows You Around

They said someday you’ll find

All who love are blind

When your heart’s on fire

You must realize

Smoke gets in your eyes

Jerome Kern and Otto Harbach, 1933

(“Watching the Campfire” by Mr Moss is licensed under CC BY 2.0)

Despite legions of campers swearing to the contrary, campfire smoke does NOT actually follow you around like your friend’s labrador retriever who really, really, really wants that wiener you just roasted. Of course, if there was no wind, if there were no complex air currents generated by the mixing of hot and cold air, and if you were the only person near the fire, then perhaps you could claim otherwise. But, in any realistic campfire scenario, the airflow is dictated by a large number of variables and where you happen to be sitting is likely the least important factor of them all.

For argument’s sake, let’s assume that we had the perfect evening. There is no wind and the air is moving in the simplified path shown in the sketch below. Denser cool air is drawn in towards the fire at ground level as the warmer, less dense air moves upward in a process called convection. In this scenario, the smoke travels directly upward with the heated air, unless there is a local disruption to these convection currents.

Now, if we add in just one person near the campfire, this could create a local obstruction at ground level:

The camper blocks cool air from flowing in toward the base of the fire at that point in the circle. Like a large boulder in a quickly moving stream, this person is an obstruction to the flow. The disrupted airflow could lead to the smoke swirling around in the region between the fire and the camper, like eddy currents or vortices downstream from the boulder. If she moves to another spot around the circle, the same disrupted airflow and eddies go with her, hence the feeling that the smoke is following her.

This is the same principle by which obstructions are diagnosed in blood vessels with Doppler ultrasound: regular blood flow becomes turbulent with vortices and eddies behind obstructions. The resulting colour Doppler image clearly shows these regions of anomalous flow.



Doppler image suggests 50‐60% blockage (yellow arrow) that was confirmed by angiography (black arrow).
Image from Journal of Neuroimaging, Volume: 28, Issue: 6, Pages: 683-687, First published: 19 June 2018, DOI: (10.1111/jon.12532)

But for the effect of these eddies and vortices to be so pronounced with smoke, there would have to be no other people around the campfire and no wind. You would likely need to be relatively close to the fire as well. So, in reality, campfire smoke doesn’t really follow you around, it just seems like it does. This is a classic example of confirmation basis, a phenomenon observed as long ago as the time of Sir Francis Bacon (1561 – 1626), who said:

“And such is the way of all superstitions,
whether in astrology, dreams, omens, divine judgments, or the like; wherein men, having a delight in
such vanities, mark the events where they are fulfilled,
but where they fail, although this happened much
oftener, neglect and pass them by”

Novum organum by Francis Bacon, 1620

It does make sense. After all, it is super annoying when it swirls in your face and makes your eyes water, so we remember these instances vividly. We don’t, however, tend to notice or remember all the times that the smoke was swirling in the face of your friend with the moochy labrador retriever.

To prevent smoke from swirling in front of you at the campfire, several survivalist references suggest that you build your fire with a large object nearby, such as a rock, a pile of sand, or firewood. This object will create a bigger disruption in airflow and, in theory, the smoke will swirl in front of the wood pile rather than your face. Again, however, this effect will only be observable with very specific circumstances. Let’s just call me skeptical on this one. Rather than going to great lengths to ensure there is a smoke-attracting decoy near the fire in case you have the perfect evening, why not look at it as an opportunity to do some fun experiments on confirmation bias instead?

References

Comparison of Carotid Doppler Ultrasound to Other Angiographic Modalities in the Measurement of Carotid Artery Stenosis 2018 Matthew Boyko, Hayrapet Kalashyan, et al Journal of Neuroimaging 28(6) Pages 683-687 https://doi-org.subzero.lib.uoguelph.ca/10.1111/jon.12532

Confirmation Bias: A Ubiquitous Phenomenon in Many Guises, Raymond S. Nickerson 1998 Review of General Psychology Vol. 2, No. 2, 175-220 PDF available at: nickersonConfirmationBias.pdf (ucsd.edu)

In Burtt, E. A. (Ed.), The English philosophers from Bacon to Mill (pp. 24-123). New York: Random House, published in 1939

Basic Wilderness Survival Skills, Bradford Angier and Lamar Underwood, The Lyons Press, Guilford, Conneticut, 2002; page 65

The Family Guide to Survival (Skills That Can Save Your Life and the Lives of Your Family), Alan Corson, Balboa Press, Bloomington, IN, 2013; page 137

Backpacker – The Magazine of Wilderness Travel, Volume 24, Issue 147, Number 1 (February 1996, page 85)

This is Why … The Snowflake is “Six-Cornered”

Image: https://www.shutterstock.com/g/Bobkov+Evgeniy

Back in 1611, Johannes Kepler, court mathematician to the Holy Roman Emperor Rudolf II, found himself flat broke in the lead-up to the holiday season. (Apparently Rudolf didn’t pay his court all that regularly.) As he wandered across the Charles Bridge in Prague, he noticed a delicate snowflake attached to the lapel of his coat and marvelled at its geometric beauty. Why does the snowflake always appear ‘six cornered’?, he wondered. Never five or eight or twelve. The theory he then developed became the foundation of the field of crystallography, proposing that the geometric shapes of crystals is determined by the most efficient way for molecules to arrange themselves.

Stop and think about that for a second …. Kepler came up with a theory about how molecules arrange themselves almost 200 years before the notion of atoms forming molecules was even formulated! He wrote his musings down in the 24-page booklet “De nive sexangula”, which he presented to his friend Johannes Matthäus Wackher von Wackenfels as a New Year’s gift. (Anyone feeling strapped for cash right now? Now that’s a priceless homemade gift!)

“Cannonballs, Whitepoint Gardens, Charleston, SC” by Martin LaBar is licensed under CC BY-NC 2.0

The crux of Kepler’s argument was that the six-sided hexagonal shape arises because of how the ‘smallest natural units of a liquid like water’ (i.e. molecules before we called them that) arrange themselves to fill the space most efficiently. This idea of how molecules arrange themselves in close-packed configurations was inspired by earlier work from an English mathematician Thomas Harriot, who had advised Raleigh on how to stack cannonballs on the ship’s deck in the most efficient manner: hexagonal, like the snowflakes.

Harriot shared his cannonball-stacking analysis with Kepler, who immediately saw the connection to the symmetric crystals decorating his coat, suggesting that the six-sided structure comes about naturally since “in no other arrangement could more pellets be stuffed into the same container”. Now here is another truly mind-blowing part to this story: Kepler’s conjecture from 1611 was not formally proved until almost 400 years later by mathematician Thomas Hales and collaborators, with the formal proof eventually published by Forum of Mathematics, Pi, in 2017!

Hexagons are found just about everywhere you care to look in nature, not just in wintertime on your lapel, like honeycombs made by bees, a collection (“raft”) of bubbles on a frothy pond surface, or the compound eyes of insects. If you want to cover a flat surface as efficiently as possible with objects that all have the same shape and size, you have three options: equilateral triangles, squares, and hexagons. Hexagons often come out as the winner because they require the least amount of side length to contain a given area.

For example, let’s think about an equilateral triangle, a square, and a hexagon, each with a total area of 100 cm2. For a square, that means that each side is 10 cm long, for a total perimeter/wall length of 10 + 10 + 10 + 10 = 40 cm. For an equilateral triangle, the area is calculated as the side length squared multiplied by the square root of 3, all divided by 4. This tells us that an equilateral triangle of area 100 cm2 has a side length of 15.2 cm. The total perimeter here is then 15.2 + 15.2 + 15.2 = 45.6 cm. Lastly, the naturally-preferred hexagon. The area for such a shape is calculated as the side length squared multiplied by the factor 3/2 times the square root of 3. (This is the same thing as six-times the area of an equilateral triangle, since a hexagon is the same thing as six close-packed equilateral triangles.) For a total area of 100 cm2, the hexagon has a side length of only 6.2 cm. Adding these up for the perimeter gives us a total wall length of 37.2 cm, the smallest of these three shapes.

  • close up view of the structure of a beehive with the hexagonal shaped regions packed tightly together
  • Close up view of the compound eye of a longhorned beetle with the close-packing arrangement of hexagons
  • soap bubbles packed together on a surface, forming hexagonal shapes to fill the space

It takes energy to build a honeycomb structure. None other than Charles Darwin noted that the observed shape made sense from the evolutionary perspective, declaring that the hexagonal honeycomb is “absolutely perfect in economizing labor and wax.” Once again, Mother Nature, you are brilliant!

References

The honeycomb conjecture: Proving mathematically that honeybee constructors are on the right track – Peterson – 1999 – Science News – Wiley Online Library

In retrospect: On the Six-Cornered Snowflake, Philip Ball Nature volume 480, page 455 (2011)

On the Six-Cornered Snowflake, Kepler’s Discovery

THE SIX-CORNERED SNOWFLAKE (De nive sexangula) By Johannes Kepler, Translated by Jacques Bromberg, March 11, 2010 Paul Dry Books ISBN – 10:1589880536

A proof of the Kepler conjecture, Thomas C. Hales et al, Forum of Mathematics, Pi , Volume 5 , 2017 , e2

How Physics Gives Structure to Nature by Philip Ball, April 2016, for Nautilus