In materials science class, we examined wallpaper patterns for symmetries. Atoms and molecules can pack according to a variety of crystal structures. Mathematics obviously loves patterns too. There are fractal tilings and tessellations. Who doesn’t love Escher? There are probably practical applications to tiling, but more importantly they are great fun that tickles the brain. Recently I took my first stab at pattern making depicting (what else?) water polo.
Tag Archives: math
Turing Patterns: What do a leopard’s spots, vegetation in arid zones, and the formation of fingers have in common?
Please excuse my inconsistent posting of late, I have been deep down the rabbit hole of science. Last week, I attended the Society of Industrial and Applied Math (SIAM) dynamical systems conference. What fun!
I learned about Turing Patterns, named for mathematician Alan Turing. Complex patterns can arise from the balance between the diffusion of chemicals and the reaction of those chemicals. For this reason, Turing’s model is also called the Reaction-Diffusion model. In general, these kinds of patterns can arise when there’s some kind of competition.
This sounds abstract, but suspected examples in nature abound. Have you ever wondered how the leopard got his spots or what’s behind the patterns on seashells? We often don’t know the chemical mechanisms that produce the patterns, but we can mathematically reproduce them with generic models.
Mary Silber and her grad student Karna Gowda presented research on Turing patterns in the vegetation of arid regions. When there isn’t enough precipitation to support uniform vegetation, what vegetation will you observe? If there’s too little water, their model yields a vegetation-free desert. Between “not enough” and “plenty” the model generates patterns, from spots to labyrinths to gaps. Their work expands at least two decades worth of study of Turing patterns in vegetation.
Silber and Gowda considered an area in the Horn of Africa (the bit that juts east below the Middle East). Here, stable patterns in the vegetation have been documented since the 1950s. They wanted to know how the patterns have changed with time. Have the wavelengths between vegetation bands changed? Are there signs of distress due to climate change? By comparing pictures taken by the RAF in the 1950s to recent satellite images, they found that the pattern were remarkably stable. The bands slowly travelled uphill, but they had the same wavelength and the same pattern. They only observed damage in areas with lots of new roads.
Turing patterns have even been studied experimentally in zebrafish. Zebrafish stripes might appear stationary, but they will slowly change in response to perturbations. So scientists did just. Below is a figure from the paper. The left shows the pattern on the zebrafish, the right shows the predictions of the model.
The model has been used to explain the distribution of feather buds in chicks and hair follicles in mice. Turing’s equations have even been used to explain how fingers form.
If you want to learn more, the links above are a great start. And if you want to play with the patterns yourself, check out this super fun interactive. These waves aren’t stationary like the Turing patterns I described here, but they arise from similar mathematics. The interactive can make your computer work, fyi.
Here in Albuquerque, mathematical art adorns the schools. We are the Fractal Capital of the World. Fractals are a kind of math that considers the multi-scale aspects of nature. In school, we learn about rectangles, circles, and triangles, but which of these shapes best represents the coastline of Great Britain?
And even if learning fractal math isn’t your path, you probably appreciate what others have done with it. This documentary describes how lava in Star Wars was simulated using fractal approaches. Many natural objects have fractal aspects, and CGI versions of these objects utilize this approach.
I do research in nonlinear dynamics, which is a cousin to chaos theory and fractal math. Fractal math first emerged as a visual wonder with Benoit Mandelbrot; as a scientist and artist, fractals inspire me in multiple ways. I hope my forays into fractals might inspire, too!
M.C. Escher: revisiting a familiar name
M.C. Escher and Salvador Dalí are two of the greatest reality-bending artists. So, fittingly, the Salvador Dalí Museum in St. Petersburg, Florida recently hosted a special Escher exhibit. I’ve visited the Dalí collection many times, but I’d never seen Escher in person.
I had many Escher calendars as a kid. When I took crystallography, we studied the symmetries in Escher’s tessellations. I’ve always been interested in design and mathematics, and Escher is the purest intersection of the two. I was ecstatic to see the exhibit.
Before the exhibit, I was most familiar with Escher’s lithographs. Without too much elaboration, lithography is a high-fidelity technique which allows the artist to produce an image that is not directed by the mechanics of printing (I’m sure the method does direct some artistic choices, but as a non-expert, that’s my rough take on it).
The exhibit contained many of Escher’s woodcuts, which were new to me. Woodcuts are make by carving a plate of wood, coating the plate with ink, and pressing the plate to a page. The page will be white where the wood has been cut away and the page will be colored where the wood remained. Woodcuts have a distinctive style–they cannot render colors in between white and ink color. Multiple colors can be achieved with additional pressings, but the technique is inherently color-limited.Additionally, the resolution of the print is limited to the fidelity of the wood. These two aspects give woodcuts a distinctive artistic feeling. If you can’t tell, I’m currently a little in love with woodcuts.
Escher died in 1972, but thanks to Disney, his works remain out of the Creative Commons. However, I am allowed to use low-resolution works for discussion purposes. You’ll have to buy books if you want anything with much detail, though. Below are a few of my favorite Escher images that are available through Wikipedia, as I have linked them under the image.
Some of Escher’s early works were illustrations. There was a beautiful cathedral, half underwater. There were evil-looking creatures in forests. It was such a romantic side to an artist most think of as a master of geometry. Below is an example of one of his illustrations. Even though it’s of a conventional subject, the Tower of Babel, the perspective is beautiful. I love the lines; this work just wouldn’t be whole using a method besides woodcut.
Below was Escher’s first impossible reality. And look, it’s a woodcut! Hooray!
Below is one of Escher’s more famous images. It is a lithograph printing. See how various tones of gray are possible with this technique, as well as high-fidelity. It lends this images a very different tone than the one above. The lizard design is called a tessellation. Tessellations are plane-filling patterns. They occur in nature and area subject of mathematical study. Escher was inspired by the tiling work at the Alhambra in Spain, another example of tessellation.
Below is another Escher woodcut, done with several plates to achieve multiple colors. Even when Escher wasn’t exploring impossible realities of geometry puzzles, he chose interesting perspectives.
Art and Math: Poemotion (Takahiro Kurashima)
Poemotion and Poemotion 2 books of astonishingly beautiful patterns. They are beautiful because they are so simple and yet I struggle to describe them here. The book comes with a lined overlay, and when the images of the book combine with the overlay, they dance and amaze.
These dancing patterns arise from something called a Moiré pattern, a creature of math and physics. These kinds of patterns naturally arise when two patterns are overlaid.
You’ve probably seen Moiré patterns when people wear busy patterns on tv:
We usually associate Moiré patterns with annoying visual artifacts, but science has found several ways to exploit them. Moiré patterns can be used to measure strain in materials. They can also be exploited to take microscope images at high magnification. The little lines on US dollars are designed to create Moiré lines when scanned, as a mechanism for defeating counterfeiters.
Kurashima’s Poemotion (just in black) and Poemotion 2 (in color), contain dozens of Moiré patterns. Every time I look at them, I feel such simple joy. The patterns are so deeply familiar and yet I had never consciously noticed them before. These books made me look at the world differently.
Fun science: An easy fractal to make at home
Viscous fingering is a fractal pattern that occurs when a less viscous (or thick) fluid spreads through a more viscous (or thick) fluid. Such systems are present in oil extraction, when we pump one fluid underground to push another one out. Fractals are common in nature even though they’re new to our mathematics, and they are beautiful.
The pictures in this post were created with basic watercolor paints using one simple principle: water containing paint is more viscous than regular water. It’s easy to try at home!
For the top picture, I laid down red paint. Before the paint dried, I added salt, then let the square dry. Water from the still-damp paper rushed to the salt (because of entropy, systems tend towards uniform distributions of things if they can help it– in this case, the lowest energy state is to have a uniform distribution of salt). But because paint molecules are larger than water molecules, they don’t move as well. The water that accumulates around the salt has less paint than the water in the rest of the paper, and thus we have a less viscous fluid spreading into a more viscous one. Try it at home! If the paint is too wet or too dry when you add the salt, the results won’t be as dramatic, so play around a bit. Larger salt crystals can be especially fun.
For the three pictures below, I simply placed a drop of water into a damp square of paint. The patterns vary depending upon the size of my drop, the wetness of the paint, and the paint color (the chemistry of which influences the viscosity of the paint).
Below are a couple of examples from the University of Alberta of viscous fingering with pentane into oil and water into oil. This particular research aims to improve the flow rate of oil during extraction. And it looks pretty similar to some humble watercolors.
Left: pentane displacing mineral oil. Right: Water displacing mineral oil (University of Alberta).
Interesting facts: 50-75
Today I post interesting facts 51-75. These last weeks have been incredibly intense, and it’s been tough meeting my 100 fact challenge celebrating reaching 100 posts. I will follow with the last 25 later this week or next Monday. The blog has had to take a backseat to work and to my book writing efforts; I will hit 70,000 words tomorrow and I am closing in on the finish. And without further ado, more facts!
51. Earth’s magnetic poles switch every few hundred thousand years, as a result of natural movements in iron in the crust. I wondered how this might affect migratory species using magnetic senses, but there isn’t enough evidence from the last switch 41,000 years ago to tell.
52. The creator of Kellogg’s cornflakes was at war with sexuality. The cornflakes were a part of this, as an unstimulating food. He was a strong advocate against masturbation– advocating circumcision and application of acid to the genitals.
53. Left-handed people are at a higher risk for numerous ailments, including schizophrenia, ADHD, and depression. I am what they call mixed-handed– I do some tasks with my right hand (writing), and some with the other (sports).
54. Eta Carinae is sometimes one of the brightest stars in the sky, and sometimes not. It is a system including a luminous blue variable, which grows a coat of obscuring gas, and then periodically blasts it off. In 1843, it was the second brightest object in the sky. It currently cannot be seen with the naked eye.
55. George Washington did not have wooden teeth. His dentures were made of gold, hippopotamus ivory, lead, and human and animal teeth (including horse and donkey teeth).
56. Goldfish actually have memories of about three months. As anyone who ever owned a goldfish should know.
57. Alfred Tennyson was troubled and interested by the science of his time. Themes about evolution and references to the contemporary phrase “ontogeny recapitulates phylogeny” (a since debunked scientific concept which claims an organism develops in vitro according to its phylum order) can be found in his poetry, specially In Memoriam.
58. Water-induced wrinkles are not caused by the skin absorbing water and swelling. They are caused by the autonomic nervous system, which triggers localized vasoconstriction in response to wet skin, yielding a wrinkled appearance. This may have evolved because it gives ancestral primates a better grip in slippery, wet environments.
59. Eating nuts, popcorn, or seeds does not increase the risk of diverticulitis.These foods may actually have a protective effect.
60. The Coriolis effect does not determine the direction that water rotates in a bathtub drain or a flushing toilet. The Coriolis effect induced by the Earth’s daily rotation is too small to affect the direction of water in a typical bathtub drain. The effect becomes significant and noticeable only at large scales, such as in weather systems or oceanic currents. Other forces dominate the dynamics of water in drains.
61. Abner Doubleday did not invent baseball.
62. Nikola Tesla was a badass scientist. Thomas Edison isn’t as great as you thought. Tesla pioneered AC current distribution and the lightbulb. Edison stole ideas from Tesla and attempted to undermine him to increase his own profits.
63. Paul Erdös wrote over 1500 math papers. If you’ve heard of six degrees of Kevin Bacon, this was originally known as the Erdös number, the number of degrees of separation from publishing a paper with Erdös. He was very eccentric. For years, he lived out of his suitcase, traveling across the world and collaborating on papers. He didn’t know how to open juice containers and used amphetamines to give him energy. His epitaph was “I’ve finally stopped getting dumber.”
64. Catherine the Great was the longest-ruling female monarch in Russian history. She was actually prussian, and married Peter the Great’s grandson. She probably conspired in his assassination, and took the throne. Her son changed Peter the Great’s succession laws to exclude women from rule.
65. Marie Curie was the first woman to win a nobel prize, and the only person to win in multiple sciences. She discovered polonium and radium and x-rays. She used x-rays to help diagnose injuries in WW1. She eventually died due to radiation-related illness.
66. Ramanujan was an indian-born mathematical genius. With little formal instruction, he devised many theorems that are still being incorporated into mathematical theory. He died at 32.
67. Michael Faraday was a pioneering scientist in electromagnetism, although he also received little formal education. He discovered benzene, and discovered the relationship between light and magnetism. He knew little math beyond trigonometry. The unit of capacitance, Farad, is named after him, as well as numerous constants and devices.
68. the symbol pi, π, originally referred to the perimeter of a circle. only in 1706 was it used to mean the ratio of perimeter to diameter.
69. James Tiptree Jr., a prominent science fiction writer, was actually a woman. She wrote under the pseudonym for two decades until she killed her husband and then herself.
70. In the early years of the Soviet Union, a type of genetics besides Mendelian genetics became accepted as correct, known as Lysenkoism. In Lysenkoism, the way you raised a crop determined its outcome, not the type of seed. Widespread starvation occurred in the Soviet five-year plans, partially due to Lysenkoism.
71. There are over 20,000 species of orchids, or four times the number of mammalian species. Many of them are epiphytes, meaning they grow above the ground in tree-borne habitats.
72. East germans could only buy Trabant cars. Used Trabants were more expensive than new ones, because the waiting line was shorter.
73. A girl in Sweden survived her body temperature dropping to 55 F (13 C) in 2010.
74. Hypothermia is highly correlated to age. Older people suffer hypothermia at a much higher rate.
75. The Cape Hatteras lighthouse is the tallest base to tip lighthouse in the United States. Due to shore encroachment, it was moved in 1999. Its light can be seen 20 miles out to sea.
Happy 50th Anniversary, Chaos
This month, the American Physics Society magazine, Physics Today, published an article about the 50th anniversary of the Lorenz model. At the link, you can read the entire article. In it, experts describe the history of chaos, Lorenz’s discovery of it, and some of the state of the field today, but with a great deal less technical jargon.
50 years ago, Edward Lorenz first captured the mathematical phenomena we now know as chaos, known popularly as the “butterfly effect“. Below is a picture from the Lorenz model exhibiting chaos. The idea of chaos boils down to highly structured behavior that cannot be predicted. No matter how precisely we measure, after some time we cannot know the state of the system. We can say that the system will stay in a certain region of weather; in the picture below, there are definitely places the trajectory does not visit. We observe this with weather models– the forecast is good for a couple of days, so-so for a couple of days after that, and completely inaccurate for any time farther in the future. Analogously, we can say that it will not be -100 C tomorrow. Appropriately, Lorenz’s discovery of chaos came about as he tried to develop a model for the weather. Chaos is all around us and can be observed in a number of systems.
At this link, you can play with a fun Lorenz model java applet. The trick with the applet is choosing the right parameters. Try setting the “spread” to 0.1, the “variation” to 20, the “number of series” to 2, and the “refresh period” to 100. Then push the button “reset the parameters” and “restart”. This will start 2 trajectories in the Lorenz model that differ by only 0.1. You will quickly see the two paths diverge and become completely unrelated. If you reduce the “spread” to 0.01, the same thing will happen, though it will take longer. As long as the spread is more than 0, the two paths will eventually diverge.
This is why we cannot predict the state of a chaotic system, because our ability to measure the state of the system is inevitably flawed. If we could measure the state of the weather to 99.99999% accuracy, that 0.00001% inaccuracy would eventually lead to divergence. And you can imagine that getting 99.99999% accuracy is much harder and more expensive than 99.9% accuracy.
Did you know that Pluto’s orbit is chaotic? Or a double pendulum? Or the logistic model for population dynamics? So check out the Lorenz model, and happy chaos-ing.
Fun Science: Small World Networks
The small-world phenomenon refers to the fact that even in a very large population, it takes relatively few connections to go from any element 1 to another random element. Amongst people, we know this concept as the “six degrees of separation” game. Any population of objects with connections can be conceptualized this way. Examples include crickets communicating by audible chirps, websites with links, electrical elements with wiring, board members with common members, or authors on mutual scientific papers. All of the examples I list have been examined in various scientific studies.
In a small-world network, elements are first connected in a regular lattice; for example, each element is connected to one or two nearby neighbors on each side. The leftmost picture below shows a regular lattice of elements. A connection between element i and element j is then removed. Then we add a connection between element i and any other element x, like the middle picture below. If x is across the network from i, then the number of steps between i and x has been reduced from some large number to 1. All of the elements connected to i are now 2 steps from element x. This reduces the diameter of the network, which is the maximum number of steps between any two elements, although the number of connections remains constant. In the six-degrees of separation game, the diameter would be 6. As we replace more of the lattice connections with random ones, the network becomes more and more random. We quantify a small-world network by its randomness, as in the picture below.
The small-world network has been explored as a means of sending information efficiently through a population. As the diameter reduces, the time it takes information to spread through the entire network reduces. Neurons in the brain have been explored as small-world networks; certain regions of the brain are highly interconnected with a few long distance connections to other regions of the brain. Protein networks and gene transcription networks have also been described with the small-world model. Further information with scholarly references is available on the scholarpedia page (which is generally a great resource for complex systems problems).
Here you can read a good scientific paper by Steven Strogatz, one of the premier scientists in the area. This is a paper published in Nature, one of the highest scientific publications. There are some equations, but the figures are also excellent if you are uncomfortable with the math. The paper models the power grid, boards of directors, and coauthorship using network ideas. I mention this paper specifically because I find Strogatz a very relatable and clear writer. Also consider reading his recent nontechnical book about math, The Joy of X, for more math fun.
Check out my other science posts on graph theory, chaos, fractals, the mandelbrot set, and synchrony. And drop a note with any questions!
Fun Science: Network Theory and Graphs
If you have a set of items and you can connect or sequence them in many ways, you probably have a graph or network. Clearly if you have these objects, some connection arrangements might be preferable to others. Heart cells are connected in patterns that contract the heart in the proper pattern. If you must deliver items to ten different locations, different paths may be more efficient (the traveling salesman problem).
Euler’s 1735 Koenigsberg bridge problem is considered the first graph theory problem. At the time, the city of Koenigsberg had seven bridges (shown above). Euler wished to find a path which crossed each bridge exactly once. He showed mathematically that no path satisfied those constraints.
The famous game “six degrees of Kevin Bacon” is a network theory problem. This game says that with six steps, any actor can be linked to Kevin Bacon through films pairs of actors appeared in. This idea was originally introduced at the Erdös number. Paul Erdös was a brilliant and highly published mathematician (over 1500 papers!) who worked in graph theory and combinatorics. The Erdös number was how many papers it took by coauthoring to connect you to Erdös. He was also wonderfully eccentric. Once, visiting a friend, he woke in the night to get some juice. In the morning, his friend found red liquid all over the floor. Erdös, puzzled by the juice carton, had simply stabbed a hole in the side to drink from. His biography is a fascinating glimpse into a nearly alien mind.
In my own research, I look at how oscillators synchronize in small networks, such as rings. Even in a simple ring, many new types of synchrony occur, compared to all-to-all connections. It is easy to believe that the structure of the brain, and how various regions and subregions connect, might greatly influence human thinking. On a more science fiction note, I suspect that artificial intelligence will not exist in machines without complex networks of elements.
This was just a very quick overview of a huge field. In the future, I plan to write on topics like small-world networks, scale-free networks, and synchrony on networks. Check out my other science posts on synchrony, fractals, the Mandelbrot set, and chaos.