Category Archives: Mathematics

Mathematicians get illustrative

Frank A. Farris, an associate Professor of Mathematics at Santa Clara University (US), writes about the latest in mathematicians and data visualization in an April 4, 2017 essay on The Conversation (Note: Links have been removed),

Today, digital tools like 3-D printing, animation and virtual reality are more affordable than ever, allowing mathematicians to investigate and illustrate their work at the same time. Instead of drawing a complicated surface on a chalkboard, we can now hand students a physical model to feel or invite them to fly over it in virtual reality.

Last year, a workshop called “Illustrating Mathematics” at the Institute for Computational and Experimental Research in Mathematics (ICERM) brought together an eclectic group of mathematicians and digital art practitioners to celebrate what seems to be a golden age of mathematical visualization. Of course, visualization has been central to mathematics since Pythagoras, but this seems to be the first time it had a workshop of its own.

Visualization plays a growing role in mathematical research. According to John Sullivan at the Technical University of Berlin, mathematical thinking styles can be roughly categorized into three groups: “the philosopher,” who thinks purely in abstract concepts; “the analyst,” who thinks in formulas; and “the geometer,” who thinks in pictures.

Mathematical research is stimulated by collaboration between all three types of thinkers. Many practitioners believe teaching should be calibrated to connect with different thinking styles.

Borromean Rings, the logo of the International Mathematical Union. John Sullivan

Sullivan’s own work has benefited from images. He studies geometric knot theory, which involves finding “best” configurations. For example, consider his Borromean rings, which won the logo contest of the International Mathematical Union several years ago. The rings are linked together, but if one of them is cut, the others fall apart, which makes it a nice symbol of unity.

Apparently this new ability to think mathematics visually has influenced mathematicians in some unexpected ways,

Take mathematician Fabienne Serrière, who raised US$124,306 through Kickstarter in 2015 to buy an industrial knitting machine. Her dream was to make custom-knit scarves that demonstrate cellular automata, mathematical models of cells on a grid. To realize her algorithmic design instructions, Serrière hacked the code that controls the machine. She now works full-time on custom textiles from a Seattle studio.

In this sculpture by Edmund Harriss, the drill traces are programmed to go perpendicular to the growth rings of the tree. This makes the finished sculpture a depiction of a concept mathematicians know as ‘paths of steepest descent.’ Edmund Harriss, Author provided

Edmund Harriss of the University of Arkansas hacked an architectural drilling machine, which he now uses to make mathematical sculptures from wood. The control process involves some deep ideas from differential geometry. Since his ideas are basically about controlling a robot arm, they have wide application beyond art. According to his website, Harriss is “driven by a passion to communicate the beauty and utility of mathematical thinking.”

Mathematical algorithms power the products made by Nervous System, a studio in Massachusetts that was founded in 2007 by Jessica Rosenkrantz, a biologist and architect, and Jess Louis-Rosenberg, a mathematician. Many of their designs, for things like custom jewelry and lampshades, look like naturally occurring structures from biology or geology.

Farris’ essay is a fascinating look at mathematics and data visualization.

Fractal imagery (from nature or from art or from mathematics) soothes

Jackson Pollock’s work is often cited when fractal art is discussed. I think it’s largely because he likely produced the art without knowing about the concept.

No. 5, 1948 (Jackson Pollock, downloaded from Wikipedia essay about No. 5, 1948)

Richard Taylor, a professor of physics at the University of Oregon, provides more information about how fractals affect us and how this is relevant to his work with retinal implants in a March 30, 2017 essay for The Conversation (h/t Mar. 31, 2017 news item on, Note: Links have been removed),

Humans are visual creatures. Objects we call “beautiful” or “aesthetic” are a crucial part of our humanity. Even the oldest known examples of rock and cave art served aesthetic rather than utilitarian roles. Although aesthetics is often regarded as an ill-defined vague quality, research groups like mine are using sophisticated techniques to quantify it – and its impact on the observer.

We’re finding that aesthetic images can induce staggering changes to the body, including radical reductions in the observer’s stress levels. Job stress alone is estimated to cost American businesses many billions of dollars annually, so studying aesthetics holds a huge potential benefit to society.

Researchers are untangling just what makes particular works of art or natural scenes visually appealing and stress-relieving – and one crucial factor is the presence of the repetitive patterns called fractals.

When it comes to aesthetics, who better to study than famous artists? They are, after all, the visual experts. My research group took this approach with Jackson Pollock, who rose to the peak of modern art in the late 1940s by pouring paint directly from a can onto horizontal canvases laid across his studio floor. Although battles raged among Pollock scholars regarding the meaning of his splattered patterns, many agreed they had an organic, natural feel to them.

My scientific curiosity was stirred when I learned that many of nature’s objects are fractal, featuring patterns that repeat at increasingly fine magnifications. For example, think of a tree. First you see the big branches growing out of the trunk. Then you see smaller versions growing out of each big branch. As you keep zooming in, finer and finer branches appear, all the way down to the smallest twigs. Other examples of nature’s fractals include clouds, rivers, coastlines and mountains.

In 1999, my group used computer pattern analysis techniques to show that Pollock’s paintings are as fractal as patterns found in natural scenery. Since then, more than 10 different groups have performed various forms of fractal analysis on his paintings. Pollock’s ability to express nature’s fractal aesthetics helps explain the enduring popularity of his work.

The impact of nature’s aesthetics is surprisingly powerful. In the 1980s, architects found that patients recovered more quickly from surgery when given hospital rooms with windows looking out on nature. Other studies since then have demonstrated that just looking at pictures of natural scenes can change the way a person’s autonomic nervous system responds to stress.

Are fractals the secret to some soothing natural scenes? Ronan, CC BY-NC-ND

For me, this raises the same question I’d asked of Pollock: Are fractals responsible? Collaborating with psychologists and neuroscientists, we measured people’s responses to fractals found in nature (using photos of natural scenes), art (Pollock’s paintings) and mathematics (computer generated images) and discovered a universal effect we labeled “fractal fluency.”

Through exposure to nature’s fractal scenery, people’s visual systems have adapted to efficiently process fractals with ease. We found that this adaptation occurs at many stages of the visual system, from the way our eyes move to which regions of the brain get activated. This fluency puts us in a comfort zone and so we enjoy looking at fractals. Crucially, we used EEG to record the brain’s electrical activity and skin conductance techniques to show that this aesthetic experience is accompanied by stress reduction of 60 percent – a surprisingly large effect for a nonmedicinal treatment. This physiological change even accelerates post-surgical recovery rates.

Pollock’s motivation for continually increasing the complexity of his fractal patterns became apparent recently when I studied the fractal properties of Rorschach inkblots. These abstract blots are famous because people see imaginary forms (figures and animals) in them. I explained this process in terms of the fractal fluency effect, which enhances people’s pattern recognition processes. The low complexity fractal inkblots made this process trigger-happy, fooling observers into seeing images that aren’t there.

Pollock disliked the idea that viewers of his paintings were distracted by such imaginary figures, which he called “extra cargo.” He intuitively increased the complexity of his works to prevent this phenomenon.

Pollock’s abstract expressionist colleague, Willem De Kooning, also painted fractals. When he was diagnosed with dementia, some art scholars called for his retirement amid concerns that that it would reduce the nurture component of his work. Yet, although they predicted a deterioration in his paintings, his later works conveyed a peacefulness missing from his earlier pieces. Recently, the fractal complexity of his paintings was shown to drop steadily as he slipped into dementia. The study focused on seven artists with different neurological conditions and highlighted the potential of using art works as a new tool for studying these diseases. To me, the most inspiring message is that, when fighting these diseases, artists can still create beautiful artworks.

Recognizing how looking at fractals reduces stress means it’s possible to create retinal implants that mimic the mechanism. Nautilus image via

My main research focuses on developing retinal implants to restore vision to victims of retinal diseases. At first glance, this goal seems a long way from Pollock’s art. Yet, it was his work that gave me the first clue to fractal fluency and the role nature’s fractals can play in keeping people’s stress levels in check. To make sure my bio-inspired implants induce the same stress reduction when looking at nature’s fractals as normal eyes do, they closely mimic the retina’s design.

When I started my Pollock research, I never imagined it would inform artificial eye designs. This, though, is the power of interdisciplinary endeavors – thinking “out of the box” leads to unexpected but potentially revolutionary ideas.

Fabulous essay, eh?

I have previously featured Jackson Pollock in a June 30, 2011 posting titled: Jackson Pollock’s physics and and briefly mentioned him in a May 11, 2010 visual arts commentary titled: Rennie Collection’s latest: Richard Jackson, Georges Seurat & Jackson Pollock, guns, the act of painting, and women (scroll down about 45% of the way).

Dancing quantum entanglement (Ap. 20 – 22, 2017) and performing mathematics (Ap. 26 – 30, 2017) in Vancouver, Canada

I have listings for two art/science events in Vancouver (Canada).

Dance, poetry and quantum entanglement

From April 20, 2017 (tonight) – April 22, 2017, there will be 8 p.m. performances of Lesley Telford’s ‘Three Sets/Relating At A Distance; My tongue, your ear / If / Spooky Action at a Distance (phase 1)’ at the Scotiabank Dance Centre, 677 Davie St, Yes, that third title is a reference to Einstein’s famous phrase describing his response of the concept of quantum entanglement.

An April 19, 2017 article by Janet Smith for the Georgia Straight features the dancer’s description of the upcoming performances,

One of the clearest definitions of quantum entanglement—a phenomenon Albert Einstein dubbed “spooky action at a distance”—can be found in a vampire movie.

In Jim Jarmusch’s Only Lovers Left Alive Tom Hiddleston’s depressed rock-star bloodsucker explains it this way to Tilda Swinton’s Eve, his centuries-long partner: “When you separate an entwined particle and you move both parts away from the other, even at opposite ends of the universe, if you alter or affect one, the other will be identically altered or affected.”

In fact, it was by watching the dark love story that Vancouver dance artist Lesley Telford learned about quantum entanglement—in which particles are so closely connected that they cannot act independently of one another, no matter how much space lies between them. She became fascinated not just with the scientific possibilities of the concept but with the romantic ones. …

 “I thought, ‘What a great metaphor,’ ” the choreographer tells the Straight over sushi before heading into a Dance Centre studio. “It’s the idea of quantum entanglement and how that could relate to human entanglement.…It’s really a metaphor for human interactions.”

First, though, as is so often the case with Telford, she needed to form those ideas into words. So she approached poet Barbara Adler to talk about the phenomenon, and then to have her build poetry around it—text that the writer will perform live in Telford’s first full evening of work here.

“Barbara talked a lot about how you feel this resonance with people that have been in your life, and how it’s tied into romantic connections and love stories,” Telford explains. “As we dig into it, it’s become less about that and more of an underlying vibration in the work; it feels like we’ve gone beyond that starting point.…I feel like she has a way of making it so down-to-earth and it’s given us so much food to work with. Are we in control of the universe or is it in control of us?”

Spooky Action at a Distance, a work for seven dancers, ends up being a string of duets that weave—entangle—into other duets. …

There’s more information about the performance, which concerns itself with more than quantum entanglement in the Scotiabank Dance Centre’s event webpage,

Lesley Telford’s choreography brings together a technically rigorous vocabulary and a thought-provoking approach, refined by her years dancing with Nederlands Dans Theater and creating for companies at home and abroad, most recently Ballet BC. This triple bill features an excerpt of a new creation inspired by Einstein’s famous phrase “spooky action at a distance”, referring to particles that are so closely linked, they share the same existence: a collaboration with poet Barbara Adler, the piece seeks to extend the theory to human connections in our phenomenally interconnected world. The program also includes a new extended version of If, a trio based on Anne Carson’s poem, and the duet My tongue, your ear, with text by Wislawa Szymborska.

Here’s what appears to be an excerpt from a rehearsal for ‘Spooky Action …’,

I’m not super fond of the atonal music/sound they’re using. The voice you hear is Adler’s and here’s more about Barbara Adler from her Wikipedia entry (Note: Links have been removed),

Barbara Adler is a musician, poet, and storyteller based in Vancouver, British Columbia. She is a past Canadian Team Slam Champion, was a founding member of the Vancouver Youth Slam, and a past CBC Poetry Face Off winner.[1]

She was a founding member of the folk band The Fugitives with Brendan McLeod, C.R. Avery and Mark Berube[2][3] until she left the band in 2011 to pursue other artistic ventures. She was a member of the accordion shout-rock band Fang, later Proud Animal, and works under the pseudonym Ten Thousand Wolves.[4][5][6][7][8]

In 2004 she participated in the inaugural Canadian Festival of Spoken Word, winning the Spoken Wordlympics with her fellow team members Shane Koyczan, C.R. Avery, and Brendan McLeod.[9][10] In 2010 she started on The BC Memory Game, a traveling storytelling project based on the game of memory[11] and has also been involved with the B.C. Schizophrenia Society Reach Out Tour for several years.[12][13][14] She is of Czech-Jewish descent.[15][16]

Barbara Adler has her bachelor’s degree and MFA from Simon Fraser University, with a focus on songwriting, storytelling, and community engagement.[17][18] In 2015 she was a co-star in the film Amerika, directed by Jan Foukal,[19][20] which premiered at the Karlovy Vary International Film Festival.[21]

Finally, Telford is Artist in Residence at the Dance Centre and TRIUMF, Canada’s national laboratory for particle and nuclear physics and accelerator-based science.

To buy tickets ($32 or less with a discount), go here. Telford will be present on April 21, 2017 for a post-show talk.

Pi Theatre’s ‘Long Division’

This theatrical performance of concepts in mathematics runs from April 26 – 30, 2017 (check here for the times as they vary) at the Annex at 823 Seymour St.  From the Georgia Straight’s April 12, 2017 Arts notice,

Mathematics is an art form in itself, as proven by Pi Theatre’s number-charged Long Division. This is a “refreshed remount” of Peter Dickinson’s ambitious work, one that circles around seven seemingly unrelated characters (including a high-school math teacher, a soccer-loving imam, and a lesbian bar owner) bound together by a single traumatic incident. Directed by Richard Wolfe, with choreography by Lesley Telford and musical score by Owen Belton, it’s a multimedia, movement-driven piece that has a strong cast. …

Here’s more about the play from Pi Theatre’s Long Division page,

Long Division uses text, multimedia, and physical theatre to create a play about the mathematics of human connection.

Long Division focuses on seven characters linked – sometimes directly, sometimes more obliquely – by a sequence of tragic events. These characters offer lessons on number theory, geometry and logic, while revealing aspects of their inner lives, and collectively the nature of their relationships to one another.

Playwright: Peter Dickinson
Director: Richard Wolfe
Choreographer: Lesley Telford, Inverso Productions
Composer: Owen Belton
Assistant Director: Keltie Forsyth

Cast:  Anousha Alamian, Jay Clift, Nicco Lorenzo Garcia, Jennifer Lines, Melissa Oei, LInda Quibell & Kerry Sandomirsky

Costume Designer: Connie Hosie
Lighting Designer: Jergus Oprsal
Set Designer: Lauchlin Johnston
Projection Designer: Jamie Nesbitt
Production Manager: Jayson Mclean
Stage Manager: Jethelo E. Cabilete
Assistant Projection Designer: Cameron Fraser
Lighting Design Associate: Jeff Harrison

Dates/Times: April 26 – 29 at 8pm, April 29 and 30 at 2pm
Student performance on April 27 at 1pm

A Talk-Back will take place after the 2pm show on April 29th.

Shawn Conner engaged the playwright, Peter Dickinson in an April 20, 2017 Q&A (question and answer) for the Vancouver Sun,

Q: Had you been working on Long Division for a long time?

A: I’d been working on it for about five years. I wrote a previous play called The Objecthood of Chairs, which has a similar style in that I combine lecture performance with physical and dance theatre. There are movement scores in both pieces.

In that first play, I told the story of two men and their relationship through the history of chair design. It was a combination of mining my research about that and trying to craft a story that was human and where the audience could find a way in. When I was thinking about a subject for a new play, I took the profession of one of the characters in that first play, who was a math teacher, and said, “Let’s see what happens to his character, let’s see where he goes after the breakup of his relationship.”

At first, I wrote it (Long Division) in an attempt at completely real, kitchen-sink naturalism, and it was a complete disaster. So I went back into this lecture-style performance.

Q: Long Division is set in a bar. Is the setting left over from that attempt at realism?

A: I guess so. It’s kind of a meta-theatrical play in the sense that the characters address the audience, and they’re aware they’re in a theatrical setting. One of the characters is an actress, and she comments on the connection between mathematics and theatre.

Q: This is being called a “refreshed” remount. What’s changed since its first run 

A: It’s mostly been cuts, and some massaging of certain sections. And I think it’s a play that actually needs a little distance.

Like mathematics, the patterns only reveal themselves at a remove. I think I needed that distance to see where things were working and where they could be better. So it’s a gift for me to be given this opportunity, to make things pop a little more and to make the math, which isn’t meant to be difficult, more understandable and relatable.

You may have noticed that Lesley Telford from Spooky Action is also choreographer for this production. I gather she’s making a career of art/science pieces, at least for now.

In the category of ‘Vancouver being a small town’, Telford lists a review of one of her pieces,  ‘AUDC’s Season Finale at The Playhouse’, on her website. Intriguingly, the reviewer is Peter Dickinson who in addition to being the playwright with whom she has collaborated for Pi Theatre’s ‘Long Division’ is also the Director of SFU’s (Simon Fraser University’s) Institute for Performance Studies. I wonder how many more ways these two crisscross professionally? Personally and for what it’s worth, it might be a good idea for Telford (and Dickinson, if he hasn’t already done so) to make readers aware of their professional connections when there’s a review at stake.

Final comment: I’m not sure how quantum entanglement or mathematics with the pieces attributed to concepts from those fields but I’m sure anyone attempting to make the links will find themselves stimulated.

ETA April 21, 2017: I’m adding this event even though the tickets are completely subscribed. There will be a standby line the night of the event (from the Peter Wall Institute for Advanced Studies The Hidden Beauty of Mathematics event page,

02 May 2017

7:00 pm (doors open at 6:00 pm)

The Vogue Theatre

918 Granville St.

Vancouver, BC


Good luck!

3D picture language for mathematics

There’s a new, 3D picture language for mathematics called ‘quon’ according to a March 3, 2017 news item on,

Galileo called mathematics the “language with which God wrote the universe.” He described a picture-language, and now that language has a new dimension.

The Harvard trio of Arthur Jaffe, the Landon T. Clay Professor of Mathematics and Theoretical Science, postdoctoral fellow Zhengwei Liu, and researcher Alex Wozniakowski has developed a 3-D picture-language for mathematics with potential as a tool across a range of topics, from pure math to physics.

Though not the first pictorial language of mathematics, the new one, called quon, holds promise for being able to transmit not only complex concepts, but also vast amounts of detail in relatively simple images. …

A March 2, 2017 Harvard University news release by Peter Reuell, which originated the news item, provides more context for the research,

“It’s a big deal,” said Jacob Biamonte of the Quantum Complexity Science Initiative after reading the research. “The paper will set a new foundation for a vast topic.”

“This paper is the result of work we’ve been doing for the past year and a half, and we regard this as the start of something new and exciting,” Jaffe said. “It seems to be the tip of an iceberg. We invented our language to solve a problem in quantum information, but we have already found that this language led us to the discovery of new mathematical results in other areas of mathematics. We expect that it will also have interesting applications in physics.”

When it comes to the “language” of mathematics, humans start with the basics — by learning their numbers. As we get older, however, things become more complex.

“We learn to use algebra, and we use letters to represent variables or other values that might be altered,” Liu said. “Now, when we look at research work, we see fewer numbers and more letters and formulas. One of our aims is to replace ‘symbol proof’ by ‘picture proof.’”

The new language relies on images to convey the same information that is found in traditional algebraic equations — and in some cases, even more.

“An image can contain information that is very hard to describe algebraically,” Liu said. “It is very easy to transmit meaning through an image, and easy for people to understand what they see in an image, so we visualize these concepts and instead of words or letters can communicate via pictures.”

“So this pictorial language for mathematics can give you insights and a way of thinking that you don’t see in the usual, algebraic way of approaching mathematics,” Jaffe said. “For centuries there has been a great deal of interaction between mathematics and physics because people were thinking about the same things, but from different points of view. When we put the two subjects together, we found many new insights, and this new language can take that into another dimension.”

In their most recent work, the researchers moved their language into a more literal realm, creating 3-D images that, when manipulated, can trigger mathematical insights.

“Where before we had been working in two dimensions, we now see that it’s valuable to have a language that’s Lego-like, and in three dimensions,” Jaffe said. “By pushing these pictures around, or working with them like an object you can deform, the images can have different mathematical meanings, and in that way we can create equations.”

Among their pictorial feats, Jaffe said, are the complex equations used to describe quantum teleportation. The researchers have pictures for the Pauli matrices, which are fundamental components of quantum information protocols. This shows that the standard protocols are topological, and also leads to discovery of new protocols.

“It turns out one picture is worth 1,000 symbols,” Jaffe said.

“We could describe this algebraically, and it might require an entire page of equations,” Liu added. “But we can do that in one picture, so it can capture a lot of information.”

Having found a fit with quantum information, the researchers are now exploring how their language might also be useful in a number of other subjects in mathematics and physics.

“We don’t want to make claims at this point,” Jaffe said, “but we believe and are thinking about quite a few other areas where this picture-language could be important.”

Sadly, there are no artistic images illustrating quon but this is from the paper,

An n-quon is represented by n hemispheres. We call the flat disc on the boundary of each hemisphere a boundary disc. Each hemisphere contains a neutral diagram with four boundary points on its boundary disk. The dotted box designates the internal structure that specifies the quon vector. For example, the 3-quon is represented as

Courtesy: PNAS and Harvard University

I gather the term ‘quon’ is meant to suggest quantum particles.

Here’s a link and a citation for the paper,

Quon 3D language for quantum information by Zhengwei Liu, Alex Wozniakowski, and Arthur M. Jaffe. Proceedins of the National Academy of Sciences Published online before print February 6, 2017, doi: 10.1073/pnas.1621345114 PNAS March 7, 2017 vol. 114 no. 10

This paper appears to be open access.

Revisiting the scientific past for new breakthroughs

A March 2, 2017 article on features a thought-provoking (and, for some of us, confirming) take on scientific progress  (Note: Links have been removed),

The idea that science isn’t a process of constant progress might make some modern scientists feel a bit twitchy. Surely we know more now than we did 100 years ago? We’ve sequenced the genome, explored space and considerably lengthened the average human lifespan. We’ve invented aircraft, computers and nuclear energy. We’ve developed theories of relativity and quantum mechanics to explain how the universe works.

However, treating the history of science as a linear story of progression doesn’t reflect wholly how ideas emerge and are adapted, forgotten, rediscovered or ignored. While we are happy with the notion that the arts can return to old ideas, for example in neoclassicism, this idea is not commonly recognised in science. Is this constraint really present in principle? Or is it more a comment on received practice or, worse, on the general ignorance of the scientific community of its own intellectual history?

For one thing, not all lines of scientific enquiry are pursued to conclusion. For example, a few years ago, historian of science Hasok Chang undertook a careful examination of notebooks from scientists working in the 19th century. He unearthed notes from experiments in electrochemistry whose results received no explanation at the time. After repeating the experiments himself, Chang showed the results still don’t have a full explanation today. These research programmes had not been completed, simply put to one side and forgotten.

A March 1, 2017 essay by Giles Gasper (Durham University), Hannah Smithson (University of Oxford) and Tom Mcleish (Durham University) for The Conversation, which originated the article, expands on the theme (Note: Links have been removed),

… looping back into forgotten scientific history might also provide an alternative, regenerative way of thinking that doesn’t rely on what has come immediately before it.

Collaborating with an international team of colleagues, we have taken this hypothesis further by bringing scientists into close contact with scientific treatises from the early 13th century. The treatises were composed by the English polymath Robert Grosseteste – who later became Bishop of Lincoln – between 1195 and 1230. They cover a wide range of topics we would recognise as key to modern physics, including sound, light, colour, comets, the planets, the origin of the cosmos and more.

We have worked with paleographers (handwriting experts) and Latinists to decipher Grosseteste’s manuscripts, and with philosophers, theologians, historians and scientists to provide intellectual interpretation and context to his work. As a result, we’ve discovered that scientific and mathematical minds today still resonate with Grosseteste’s deeply physical and structured thinking.

Our first intuition and hope was that the scientists might bring a new analytic perspective to these very technical texts. And so it proved: the deep mathematical structure of a small treatise on colour, the De colore, was shown to describe what we would now call a three-dimensional abstract co-ordinate space for colour.

But more was true. During the examination of each treatise, at some point one of the group would say: “Did anyone ever try doing …?” or “What would happen if we followed through with this calculation, supposing he meant …”. Responding to this thinker from eight centuries ago has, to our delight and surprise, inspired new scientific work of a rather fresh cut. It isn’t connected in a linear way to current research programmes, but sheds light on them from new directions.

I encourage you to read the essay in its entirety.

The mathematics of Disney’s ‘Moana’

The hit Disney movie “Moana” features stunning visual effects, including the animation of water to such a degree that it becomes a distinct character in the film. Courtesy of Walt Disney Animation Studios

Few people think to marvel over the mathematics when watching an animated feature but without mathematicians, the artists would not be able to achieve their artistic goals as a Jan. 4, 2017 news item on makes clear (Note: A link has been removed),

UCLA [University of California at Los Angeles] mathematics professor Joseph Teran, a Walt Disney consultant on animated movies since 2007, is under no illusion that artists want lengthy mathematics lessons, but many of them realize that the success of animated movies often depends on advanced mathematics.

“In general, the animators and artists at the studios want as little to do with mathematics and physics as possible, but the demands for realism in animated movies are so high,” Teran said. “Things are going to look fake if you don’t at least start with the correct physics and mathematics for many materials, such as water and snow. If the physics and mathematics are not simulated accurately, it will be very glaring that something is wrong with the animation of the material.”

Teran and his research team have helped infuse realism into several Disney movies, including “Frozen,” where they used science to animate snow scenes. Most recently, they applied their knowledge of math, physics and computer science to enliven the new 3-D computer-animated hit, “Moana,” a tale about an adventurous teenage girl who is drawn to the ocean and is inspired to leave the safety of her island on a daring journey to save her people.

A Jan. 3, 2017 UCLA news release, which originated the news item, explains in further nontechnical detail,

Alexey Stomakhin, a former UCLA doctoral student of Teran’s and Andrea Bertozzi’s, played an important role in the making of “Moana.” After earning his Ph.D. in applied mathematics in 2013, he became a senior software engineer at Walt Disney Animation Studios. Working with Disney’s effects artists, technical directors and software developers, Stomakhin led the development of the code that was used to simulate the movement of water in “Moana,” enabling it to play a role as one of the characters in the film.

“The increased demand for realism and complexity in animated movies makes it preferable to get assistance from computers; this means we have to simulate the movement of the ocean surface and how the water splashes, for example, to make it look believable,” Stomakhin explained. “There is a lot of mathematics, physics and computer science under the hood. That’s what we do.”

“Moana” has been praised for its stunning visual effects in words the mathematicians love hearing. “Everything in the movie looks almost real, so the movement of the water has to look real too, and it does,” Teran said. “’Moana’ has the best water effects I’ve ever seen, by far.”

Stomakhin said his job is fun and “super-interesting, especially when we cheat physics and step beyond physics. It’s almost like building your own universe with your own laws of physics and trying to simulate that universe.

“Disney movies are about magic, so magical things happen which do not exist in the real world,” said the software engineer. “It’s our job to add some extra forces and other tricks to help create those effects. If you have an understanding of how the real physical laws work, you can push parameters beyond physical limits and change equations slightly; we can predict the consequences of that.”

To make animated movies these days, movie studios need to solve, or nearly solve, partial differential equations. Stomakhin, Teran and their colleagues build the code that solves the partial differential equations. More accurately, they write algorithms that closely approximate the partial differential equations because they cannot be solved perfectly. “We try to come up with new algorithms that have the highest-quality metrics in all possible categories, including preserving angular momentum perfectly and preserving energy perfectly. Many algorithms don’t have these properties,” Teran said.

Stomakhin was also involved in creating the ocean’s crashing waves that have to break at a certain place and time. That task required him to get creative with physics and use other tricks. “You don’t allow physics to completely guide it,” he said.  “You allow the wave to break only when it needs to break.”

Depicting boats on waves posed additional challenges for the scientists.

“It’s easy to simulate a boat traveling through a static lake, but a boat on waves is much more challenging to simulate,” Stomakhin said. “We simulated the fluid around the boat; the challenge was to blend that fluid with the rest of the ocean. It can’t look like the boat is splashing in a little swimming pool — the blend needs to be seamless.”

Stomakhin spent more than a year developing the code and understanding the physics that allowed him to achieve this effect.

“It’s nice to see the great visual effect, something you couldn’t have achieved if you hadn’t designed the algorithm to solve physics accurately,” said Teran, who has taught an undergraduate course on scientific computing for the visual-effects industry.

While Teran loves spectacular visual effects, he said the research has many other scientific applications as well. It could be used to simulate plasmas, simulate 3-D printing or for surgical simulation, for example. Teran is using a related algorithm to build virtual livers to substitute for the animal livers that surgeons train on. He is also using the algorithm to study traumatic leg injuries.

Teran describes the work with Disney as “bread-and-butter, high-performance computing for simulating materials, as mechanical engineers and physicists at national laboratories would. Simulating water for a movie is not so different, but there are, of course, small tweaks to make the water visually compelling. We don’t have a separate branch of research for computer graphics. We create new algorithms that work for simulating wide ranges of materials.”

Teran, Stomakhin and three other applied mathematicians — Chenfanfu Jiang, Craig Schroeder and Andrew Selle — also developed a state-of-the-art simulation method for fluids in graphics, called APIC, based on months of calculations. It allows for better realism and stunning visual results. Jiang is a UCLA postdoctoral scholar in Teran’s laboratory, who won a 2015 UCLA best dissertation prize.  Schroeder is a former UCLA postdoctoral scholar who worked with Teran and is now at UC Riverside. Selle, who worked at Walt Disney Animation Studios, is now at Google.

Their newest version of APIC has been accepted for publication by the peer-reviewed Journal of Computational Physics.

“Alexey is using ideas from high-performance computing to make movies,” Teran said, “and we are contributing to the scientific community by improving the algorithm.”

Unfortunately, the paper does not seem to have been published early online so I cannot offer a link.

Final comment, it would have been interesting to have had a comment from one of the film’s artists or animators included in the article but it may not have been possible due to time or space constraints.

Maths gallery at the UK’s Science Museum takes flight

Mathematics: The Winton Gallery at the Science Museum, Zaha Hadid Architects’ only permanent public museum exhibition design. London. Photograph: Nicholas Guttridge/NIck Guttridge

This exhibition looks great in the picture, I wonder what the experience is like. Alex Bellos is certainly enthusiastic in his Dec. 7, 2016 posting on the Guardian’s website,

Mathematics underlies all science, so for a science museum to be worthy of the name, maths needs to included somewhere. Yet maths, which deals mainly in abstract objects, is [a] challenge for museums, which necessarily contain physical ones. The Science Museum’s approach in its new gallery is to tell historical stories about the influence of mathematics in the real world, rather than actually focussing directly on the mathematical ideas involved. The result is a stunning gallery, with fascinating objects beautifully laid out, yet which eschews explaining any maths. (If you want to learn simple mathematical ideas, you can always head to the museum’s new interactive gallery, Wonderlab).

Much of the attention on Mathematics: The Winton Gallery – the main funders are David Harding, founder and CEO of investment firm Winton, and his wife Claudia – has been on Zaha Hadid’s design. The gallery is the first UK project by Zaha Hadid Architects to open since her unexpected death in March [2016], and the only permanent public museum exhibition she designed. Her first degree was in maths, before she turned to architecture.

Hanging from the ceiling is an aeroplane – the Handley Page ‘Gugnunc’, built in 1929 for a competition to build safe aircraft – and surrounding it is a swirly ceiling sculpture that represents the mathematical equations that describe airflow. In fact, the entire gallery follows the contours of the flow, providing the positions of the cabinets below.

The Science Museum’s previous maths gallery, which had not been updated in decades, contained about 600 objects, including cabinets crammed with geometrical objects and many examples of the same thing, such as medieval slide rules or Victorian curve-drawing machines. The new gallery has less than a quarter of that number of objects in the same space.

Every object now is in its own cabinet, and the extra space means you can walk around them from all angles, as well as making the gallery feel more manageable. Rather than being bombarded with stuff, you are given a single object to contemplate that tells part of a wider story.

In a section on “form and beauty”, there is a modern replica of a 1920s chair based on French architect’s Le Corbusier’s Modulor system of proportions, and two J W Turner sketches from his Royal Academy lectures on perspective.

The section “trade and travel” has a 3-metre long replica of the 1973 Globtik Tokyo oil tanker, then the largest ship in the world. In its massive cabinet it looks as terrifying as a Damien Hirst shark. The maths link? Because British mathematician William Froode a century before had worked out that bulbous bows were better than sharp bows at the fronts of boats and ships.

The new maths gallery is a wonderfully attractive space, full of interesting and thought-provoking objects, and a very welcome addition [geddit?] to London’s museums. Go!

A Dec. 8 (?), 2016 [London, UK] Science Museum press release is the first example I’ve seen of the funders being highlighted quite so prominently, i.e., before the press release proper,

Mathematics: The Winton Gallery designed by Zaha Hadid Architects opens at the Science Museum

  • A stunning new permanent gallery that reveals the importance of mathematics in all our lives through remarkable historical artefacts, stories and design
  • Free to visit and open daily from 8 December 2016
  • The only permanent public museum exhibition designed by Zaha Hadid anywhere in the world

Principal Funder: David and Claudia Harding
Principal Sponsor: Samsung
Major Sponsor: MathWorks

On 8 December 2016 the Science Museum will open an inspirational new mathematics gallery, designed by Zaha Hadid Architects.

Mathematics: The Winton Gallery brings together remarkable stories, historical artefacts and design to highlight the central role of mathematical practice in all our lives, and explores how mathematicians, their tools and ideas have helped build the modern world over the past four centuries.

More than 100 treasures from the Science Museum’s world-class science, technology, engineering and mathematics collections have been selected to tell powerful stories about how mathematics has shaped, and been shaped by, some of our most fundamental human concerns – from trade and travel to war, peace, life, death, form and beauty.

Curator Dr David Rooney said, ‘At its heart this gallery reveals a rich cultural story of human endeavour that has helped transform the world over the last four hundred years. Mathematical practice underpins so many aspects of our lives and work, and we hope that bringing together these remarkable stories, people and exhibits will inspire visitors to think about the role of mathematics in a new light.’

Positioned at the centre of the gallery is the Handley Page ‘Gugnunc’ aeroplane, built in 1929 for a competition to construct safe aircraft. Ground-breaking aerodynamic research influenced the wing design of this experimental aeroplane, helping to shift public opinion about the safety of flying and to secure the future of the aviation industry. This aeroplane encapsulates the gallery’s overarching theme, illustrating how mathematical practice has helped solve real-world problems and in this instance paved the way for the safe passenger flights that we rely on today.

Mathematics also defines Zaha Hadid Architects’ enlightening design for the gallery. Inspired by the Handley Page aircraft, the design is driven by equations of airflow used in the aviation industry. The layout and lines of the gallery represent the air that would have flowed around this historic aircraft in flight, from the positioning of the showcases and benches to the three-dimensional curved surfaces of the central pod structure.

Mathematics: The Winton Gallery is the first permanent public museum exhibition designed by Zaha Hadid Architects anywhere in the world. The gallery is also the first of Zaha Hadid Architects’ projects to open in the UK since Dame Zaha Hadid’s sudden death in March 2016. The late Dame Zaha first became interested in geometry while studying mathematics at university. Mathematics and geometry have a strong connection with architecture and she continued to examine these relationships throughout each of her projects; with mathematics always central to her work. As Dame Zaha said, ‘When I was growing up in Iraq, math was an everyday part of life. We would play with math problems just as we would play with pens and paper to draw – math was like sketching.’

Ian Blatchford, Director of the Science Museum Group, said, ‘We were hugely impressed by the ideas and vision of the late Dame Zaha Hadid and Patrik Schumacher when they first presented their design for the new mathematics gallery over two years ago. It was a terrible shock for us all when Dame Zaha died suddenly in March this year, but I am sure that this gallery will be a lasting tribute to this world-changing architect and provide inspiration for our millions of visitors for many years to come.’

From a beautiful 17th century Islamic astrolabe that uses ancient mathematical techniques to map the night sky, to an early example of the famous Enigma machine, designed to resist even the most advanced mathematical techniques for code breaking during the Second World War, each historic object within the gallery has an important story to tell. Archive photography and film helps to capture these stories, and introduces the wide range of people who made, used or were impacted by each mathematical device or idea.

Some instruments and objects within the gallery clearly reference their mathematical origin. Others may surprise visitors and appear rooted in other disciplines, from classical architecture to furniture design. Visitors will see a box of glass eyes used by Francis Galton in his 1884 Anthropometric Laboratory to help measure the physical characteristics of the British public and develop statistics to support a wider social and political movement he termed ‘eugenics’. On the other side of the gallery is the pioneering Wisard pattern-recognition machine built in 1981 to attempt to re-create the ‘neural networks’ of the brain. This early Artificial Intelligence machine worked, until 1995, on a variety of projects, from banknote recognition to voice analysis, and from foetal growth monitoring in hospitals to covert surveillance for the Home Office.

A richly illustrated book has been published by Scala to accompany the new gallery. Mathematics: How it Shaped Our World, written by David Rooney, expands on the themes and stories that are celebrated in the gallery itself and includes a series of newly commissioned essays written by world-leading experts in the history and modern practice of mathematics.

David Harding, Principal Funder of the gallery and Founder and CEO of Winton said, ‘Mathematics, whilst difficult for many, is incredibly useful. To those with an aptitude for it, it is also beautiful. I’m delighted that this gallery will be both useful and beautiful.’

Mathematics: The Winton Gallery is free to visit and open daily from 8 December 2016. The gallery has been made possible through an unprecedented donation from long-standing supporters of science, David and Claudia Harding. It has also received generous support from Samsung as Principal Sponsor, MathWorks as Major Sponsor, with additional support from Adrian and Jacqui Beecroft, Iain and Jane Bratchie, the Keniston-Cooper Charitable Trust, Dr Martin Schoernig, Steve Mobbs and Pauline Thomas.

After the press release, there is the most extensive list of ‘Abouts’ I’ve seen yet (Note: This includes links to the Science Museum and other agencies),

About the Science Museum
The Science Museum’s world-class collection forms an enduring record of scientific, technological and medical achievements from across the globe. Welcoming over 3 million visitors a year, the Museum aims to make sense of the science that shapes our lives, inspiring visitors with iconic objects, award-winning exhibitions and incredible stories of scientific achievement. More information can be found at

About Curator David Rooney
Mathematics: The Winton Gallery has been curated by Dr David Rooney, who was responsible for the award-winning 2012 Science Museum exhibition Codebreaker: Alan Turing’s Life and Legacy as well as developing galleries on time and navigation at the National Maritime Museum, Greenwich. David writes and speaks widely on the history of technology and engineering. His critically acclaimed first book, Ruth Belville: The Greenwich Time Lady, was described by Jonathan Meades as ‘an engrossing and eccentric slice of London history’, and by the Daily Telegraph as ‘a gem of a book’. He has recently authored Mathematics: How It Shaped Our World, to accompany the new mathematics gallery, and is currently writing a political history of traffic.

About David and Claudia Harding
David and Claudia Harding are associated with Winton, one of the world’s leading quantitative investment management firms which David founded in 1997. Winton uses mathematical and scientific methods to devise, evaluate and execute investment ideas on behalf of clients all over the world. A British-based company, Winton and David and Claudia Harding have donated to numerous scientific and mathematical causes in the UK and internationally, including Cambridge University, the Crick Institute, the Max Planck Institute, and the Science Museum. The main themes of their philanthropy have been supporting basic scientific research and the communication of scientific ideas. David and Claudia reside in London.

About Samsung’s Citizenship Programmes
Samsung is committed to help close the digital divide and skills gap in the UK. Samsung Digital Classrooms in schools, charities/non-profit organisations and cultural partners provide access to the latest technology. Samsung is also providing the training and maintenance support necessary to help make the transition and integration of the new technology as smooth as possible. Samsung also offers qualifications and training in technology for young people and teachers through its Digital Academies. These initiatives will inspire young people, staff and teachers to learn and teach in new exciting ways and to help encourage young people into careers using technology. Find out more

About MathWorks
MathWorks is the leading developer of mathematical computing software. MATLAB, the language of technical computing, is a programming environment for algorithm development, data analysis, visualisation, and numeric computation. Simulink is a graphical environment for simulation and Model-Based Design for multidomain dynamic and embedded systems. Engineers and scientists worldwide rely on these product families to accelerate the pace of discovery, innovation, and development in automotive, aerospace, electronics, financial services, biotech-pharmaceutical, and other industries. MATLAB and Simulink are also fundamental teaching and research tools in the world’s universities and learning institutions. Founded in 1984, MathWorks employs more than 3000 people in 15 countries, with headquarters in Natick, Massachusetts, USA. For additional information, visit

About Zaha Hadid Architects
Zaha Hadid founded Zaha Hadid Architects (ZHA) in 1979. Each of ZHA’s projects builds on over thirty years of exploration and research in the interrelated fields of urbanism, architecture and design. Hadid’s pioneering vision redefined architecture for the 21st century and captured imaginations across the globe. Her legacy is embedded within the DNA of the design studio she created as ZHA’s projects combine the unwavering belief in the power of invention with concepts of connectivity and fluidity.

ZHA is currently working on a diversity of projects worldwide including the new Beijing Airport Terminal Building in Daxing, China, the Sleuk Rith Institute in Phnom Penh, Cambodia and 520 West 28th Street in New York City, USA. The practice’s portfolio includes cultural, academic, sporting, residential, and transportation projects across six continents.

About Discover South Kensington
Discover South Kensington brings together the Science Museum and other leading cultural and educational organisations to promote innovation and learning. South Kensington is the home of science, arts and inspiration. Discovery is at the core of what happens here and there is so much to explore every day.

About Zaha Hadid: Early Paintings and Drawings at the Serpentine Sackler Gallery
This week an exhibition of paintings and drawings by Zaha Hadid will open at the Serpentine Galleries that will reveal her as an artist with drawing at the very heart of her work. It will include calligraphic drawings and rarely seen private notebooks, showing her complex thoughts about architecture’s forms and relationship to the world we live in. Zaha Hadid: Early Paintings and Drawings at the Serpentine Sackler Gallery is free to visit and runs from 8th December 2016 – 12th February 2017.

I found the mentions of Zaha Hadid fascinating and so I looked her up on Wikipedia, where I found this (Note: Links have been removed),

Dame Zaha Mohammad Hadid, DBE (Arabic: زها حديد‎‎ Zahā Ḥadīd; 31 October 1950 – 31 March 2016) was an Iraqi-born British architect. She was the first woman to receive the Pritzker Architecture Prize, in 2004.[1] She received the UK’s most prestigious architectural award, the Stirling Prize, in 2010 and 2011. In 2012, she was made a Dame by Elizabeth II for services to architecture, and in 2015 she became the first woman to be awarded the Royal Gold Medal from the Royal Institute of British Architects.[2]

She was dubbed by The Guardian as the ‘Queen of the curve’.[3] She liberated architectural geometry[4] with the creation of highly expressive, sweeping fluid forms of multiple perspective points and fragmented geometry that evoke the chaos and flux of modern life.[5] A pioneer of parametricism, and an icon of neo-futurism, with a formidable personality, her acclaimed work and ground-breaking forms include the aquatic centre for the London 2012 Olympics, the Broad Art Museum in the US, and the Guangzhou Opera House in China.[6] At the time of her death in 2016, Zaha Hadid Architects in London was the fastest growing British architectural firm.[7] Many of her designs are to be released posthumously, ranging in variation from the 2017 Brit Awards statuette to a 2022 FIFA World Cup stadium.[8][9]

Dubbed ‘Queen of the curve’, Hadid has a reputation as the world’s top female architect,[3][62][63][64][65] although her reputation is not without criticism. She is considered an architect of unconventional thinking, whose buildings are organic, dynamic and sculptural.[66][67] Stanton and others also compliment her on her unique organic designs: “One of the main characteristics of her work is that however clearly recognizable, it can never be pigeonholed into a stylistic signature. Digital knowledge, technology-driven mutations, shapes inspired by the organic and biological world, as well as geometrical interpretation of the landscape are constant elements of her practice. Yet, the multiplicity and variety of the combination among these facets prevent the risk of self-referential solutions and repetitions.”[68] Allison Lee Palmer considers Hadid a leader of Deconstructivism in architecture, writing that, “Almost all of Hadid’s buildings appear to melt, bend, and curve into a new architectural language that defies description. Her completed buildings span the globe and include the Jockey Club Innovation Tower on the north side of the Hong Kong Polytechnic University in Hong Kong, completed in 2013, that provides Hong Kong an entry into the world stage of cutting-edge architecture by revealing a design that dissolved traditional architecture, the so called modernist “glass box,” into a shattering of windows and melting of walls to form organic structures with halls and stairways that flow through the building, pooling open into rooms and foyers.”[69]

Hadid’s architectural language has been described by some as “famously extravagant” with many of her projects sponsored by “dictator states”. [emphasis mine] [70] Rowan Moore described Hadid’s Heydar Aliyev Center as “not so different from the colossal cultural palaces long beloved of Soviet and similar regimes”. Architect Sean Griffiths characterised Hadid’s work as “an empty vessel that sucks in whatever ideology might be in proximity to it”.[71] Art historian Maike Aden criticises in particular the foreclosure of Zaha Hadid’s architecture of the MAXXI in Rome towards the public and the urban life that undermines even the most impressive program to open the museum.[72]

If you think about it, most of the world’s great monuments were built by dictators or omnipotent rulers of one country or another. Getting the money and commitment can present an ethical/moral issue for any artist or architect who has a ‘grand design’.

Epic Scottish poetry and social network science

It’s been a while since I’ve run a social network story here and this research into a 250-year controversy piqued my interest anew. From an Oct. 20, 2016 Coventry University (UK) press release (also on EurekAlert) Note: A link has been removed,

The social networks behind one of the most famous literary controversies of all time have been uncovered using modern networks science.

Since James Macpherson published what he claimed were translations of ancient Scottish Gaelic poetry by a third-century bard named Ossian, scholars have questioned the authenticity of the works and whether they were misappropriated from Irish mythology or, as heralded at the time, authored by a Scottish equivalent to Homer.

Now, in a joint study by Coventry University, the National University of Ireland, Galway and the University of Oxford, published today in the journal Advances in Complex Systems, researchers have revealed the structures of the social networks underlying the Ossian’s works and their similarities to Irish mythology.

The researchers mapped the characters at the heart of the works and the relationships between them to compare the social networks found in the Scottish epics with classical Greek literature and Irish mythology.

The study revealed that the networks in the Scottish poems bore no resemblance to epics by Homer, but strongly resembled those in mythological stories from Ireland.

The Ossianic poems are considered to be some of the most important literary works ever to have emerged from Britain or Ireland, given their influence over the Romantic period in literature and the arts. Figures from Brahms to Wordsworth reacted enthusiastically; Napoleon took a copy on his military campaigns and US President Thomas Jefferson believed that Ossian was the greatest poet to have ever existed.

The poems launched the romantic portrayal of the Scottish Highlands which persists, in many forms, to the present day and inspired Romantic nationalism all across Europe.

Professor Ralph Kenna, a statistical physicist based at Coventry University, said:

By working together, it shows how science can open up new avenues of research in the humanities. The opposite also applies, as social structures discovered in Ossian inspire new questions in mathematics.”

Dr Justin Tonra, a digital humanities expert from the National University of Ireland, Galway said:

From a humanities point of view, while it cannot fully resolve the debate about Ossian, this scientific analysis does reveal an insightful statistical picture: close similarity to the Irish texts which Macpherson explicitly rejected, and distance from the Greek sources which he sought to emulate.”

A statistical physicist, eh? I find that specialty quite an unexpected addition to the team stretching my ideas about social networks in new directions.

Getting back to the research, the scientists have supplied this image to illustrate their work,

Caption: In the social network underlying the Ossianic epic, the 325 nodes represent characters appearing in the narratives and the 748 links represent interactions between them. Credit: Coventry University

Caption: In the social network underlying the Ossianic epic, the 325 nodes represent characters appearing in the narratives and the 748 links represent interactions between them. Credit: Coventry University

Here’s a link to and a citation for the paper,

A networks-science investigation into the epic poems of Ossian by Joseph Yose, Ralph Kenna, Pádraig MacCarron, Thierry Platini, Justin Tonra.  Complex Syst. DOI: Published: 21 October 2016

This paper is behind a paywall.