Category Archives: Mathematics

Mathematics/Music/Art/Architecture/Education/Culture: Bridges 2017 conference in Waterloo, Canada

Bridges 2017 will be held in Waterloo, Canada from July 27 – 31, 2017. Here’s the invitation which was released last year,

To give you a sense of the range offered, here’s more from Bridges 2017 events page,

Every Bridges conference includes a number of events other than paper presentations. Please click on one of the events below to learn more about it.

UWAG Exhibition

The University of Waterloo Art Gallery (UWAG) has partnered with Bridges to create an exhibition of five local artists who explore mathematical themes in their work. The exhibition runs concurrently with the conference.


Theatre Night

An evening dramatic performance that explores themes of art, mathematics and teaching, performed by Peter Taylor and Judy Wearing from Queen’s University.


Formal Music Night

An evening concert of mathematical choral music, performed by a specially-formed ensemble of choristers and professional soloists.


Family Day

An afternoon of community activities, games, workshops, interactive demonstrations, presentations, performances, and art exhibitions for children and adults, free and open to all.


Poetry Reading

A session of invited readings of poetry exploring mathematical themes, in a wide range of styles. Attendees will also be invited to share their own poetry in an open mic session. A printed anthology will be available at the conference.


Informal Music Night

A longstanding tradition at Bridges—a casual variety show in which all conference participants are invited to share their talents, musical or otherwise, with a brief performance.

I have some more details about the exhibition at the University of Waterloo Art Gallery (UWAG) from a July 19, 2017 ArtSci Salon notice received via email,

P A S S A G E  +  O B S T A C L E

JULY 27–30


PASSAGE + OBSTACLE features a selection of work by multidisciplinary
area artists Patrick Cull, Paul Dignan, Laura De Decker, Soheila
Esfahani, and Andrew James Smith. Sharing a rigorous approach to
materials and subject matter, their artworks parallel Bridges’ stated
goal to explore “mathematical connections in art, music, architecture,
education and culture”. The exhibition sets out to complement and
expand on the theme by contrasting subtle and overt links between the
use of geometry, pattern, and optical effects across mediums ranging
from painting and installation to digital media. Using the bridge as a
metaphor, the artworks can be appreciated as a means of getting from A
to B by overcoming obstructions, whether perceptual or otherwise.


University of Waterloo Art Gallery
East Campus Hall 1239
519.888.4567 ext. 33575 [9] [10]

Ivan Jurakic, Director / Curator
519.888.4567 ext. 36741

263 Phillip Street, Waterloo
East Campus Hall (ECH) is located north of University Avenue West
across from Engineering 6

Visitor Parking is available in Lot E6 or Q for a flat rate of $5 [11]

University of Waterloo Art Gallery
200 University Avenue West
Waterloo, ON, Canada N2L 3G1

You can find out more about Bridges 2017 including how to register here (the column on the left provides links to registration, program, and more information.


Brain stuff: quantum entanglement and a multi-dimensional universe

I have two brain news bits, one about neural networks and quantum entanglement and another about how the brain operates on more than three dimensions.

Quantum entanglement and neural networks

A June 13, 2017 news item on describes how machine learning can be used to solve problems in physics (Note: Links have been removed),

Machine learning, the field that’s driving a revolution in artificial intelligence, has cemented its role in modern technology. Its tools and techniques have led to rapid improvements in everything from self-driving cars and speech recognition to the digital mastery of an ancient board game.

Now, physicists are beginning to use machine learning tools to tackle a different kind of problem, one at the heart of quantum physics. In a paper published recently in Physical Review X, researchers from JQI [Joint Quantum Institute] and the Condensed Matter Theory Center (CMTC) at the University of Maryland showed that certain neural networks—abstract webs that pass information from node to node like neurons in the brain—can succinctly describe wide swathes of quantum systems.

An artist’s rendering of a neural network with two layers. At the top is a real quantum system, like atoms in an optical lattice. Below is a network of hidden neurons that capture their interactions (Credit: E. Edwards/JQI)

A June 12, 2017 JQI news release by Chris Cesare, which originated the news item, describes how neural networks can represent quantum entanglement,

Dongling Deng, a JQI Postdoctoral Fellow who is a member of CMTC and the paper’s first author, says that researchers who use computers to study quantum systems might benefit from the simple descriptions that neural networks provide. “If we want to numerically tackle some quantum problem,” Deng says, “we first need to find an efficient representation.”

On paper and, more importantly, on computers, physicists have many ways of representing quantum systems. Typically these representations comprise lists of numbers describing the likelihood that a system will be found in different quantum states. But it becomes difficult to extract properties or predictions from a digital description as the number of quantum particles grows, and the prevailing wisdom has been that entanglement—an exotic quantum connection between particles—plays a key role in thwarting simple representations.

The neural networks used by Deng and his collaborators—CMTC Director and JQI Fellow Sankar Das Sarma and Fudan University physicist and former JQI Postdoctoral Fellow Xiaopeng Li—can efficiently represent quantum systems that harbor lots of entanglement, a surprising improvement over prior methods.

What’s more, the new results go beyond mere representation. “This research is unique in that it does not just provide an efficient representation of highly entangled quantum states,” Das Sarma says. “It is a new way of solving intractable, interacting quantum many-body problems that uses machine learning tools to find exact solutions.”

Neural networks and their accompanying learning techniques powered AlphaGo, the computer program that beat some of the world’s best Go players last year (link is external) (and the top player this year (link is external)). The news excited Deng, an avid fan of the board game. Last year, around the same time as AlphaGo’s triumphs, a paper appeared that introduced the idea of using neural networks to represent quantum states (link is external), although it gave no indication of exactly how wide the tool’s reach might be. “We immediately recognized that this should be a very important paper,” Deng says, “so we put all our energy and time into studying the problem more.”

The result was a more complete account of the capabilities of certain neural networks to represent quantum states. In particular, the team studied neural networks that use two distinct groups of neurons. The first group, called the visible neurons, represents real quantum particles, like atoms in an optical lattice or ions in a chain. To account for interactions between particles, the researchers employed a second group of neurons—the hidden neurons—which link up with visible neurons. These links capture the physical interactions between real particles, and as long as the number of connections stays relatively small, the neural network description remains simple.

Specifying a number for each connection and mathematically forgetting the hidden neurons can produce a compact representation of many interesting quantum states, including states with topological characteristics and some with surprising amounts of entanglement.

Beyond its potential as a tool in numerical simulations, the new framework allowed Deng and collaborators to prove some mathematical facts about the families of quantum states represented by neural networks. For instance, neural networks with only short-range interactions—those in which each hidden neuron is only connected to a small cluster of visible neurons—have a strict limit on their total entanglement. This technical result, known as an area law, is a research pursuit of many condensed matter physicists.

These neural networks can’t capture everything, though. “They are a very restricted regime,” Deng says, adding that they don’t offer an efficient universal representation. If they did, they could be used to simulate a quantum computer with an ordinary computer, something physicists and computer scientists think is very unlikely. Still, the collection of states that they do represent efficiently, and the overlap of that collection with other representation methods, is an open problem that Deng says is ripe for further exploration.

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

Quantum Entanglement in Neural Network States by Dong-Ling Deng, Xiaopeng Li, and S. Das Sarma. Phys. Rev. X 7, 021021 – Published 11 May 2017

This paper is open access.

Blue Brain and the multidimensional universe

Blue Brain is a Swiss government brain research initiative which officially came to life in 2006 although the initial agreement between the École Politechnique Fédérale de Lausanne (EPFL) and IBM was signed in 2005 (according to the project’s Timeline page). Moving on, the project’s latest research reveals something astounding (from a June 12, 2017 Frontiers Publishing press release on EurekAlert),

For most people, it is a stretch of the imagination to understand the world in four dimensions but a new study has discovered structures in the brain with up to eleven dimensions – ground-breaking work that is beginning to reveal the brain’s deepest architectural secrets.

Using algebraic topology in a way that it has never been used before in neuroscience, a team from the Blue Brain Project has uncovered a universe of multi-dimensional geometrical structures and spaces within the networks of the brain.

The research, published today in Frontiers in Computational Neuroscience, shows that these structures arise when a group of neurons forms a clique: each neuron connects to every other neuron in the group in a very specific way that generates a precise geometric object. The more neurons there are in a clique, the higher the dimension of the geometric object.

“We found a world that we had never imagined,” says neuroscientist Henry Markram, director of Blue Brain Project and professor at the EPFL in Lausanne, Switzerland, “there are tens of millions of these objects even in a small speck of the brain, up through seven dimensions. In some networks, we even found structures with up to eleven dimensions.”

Markram suggests this may explain why it has been so hard to understand the brain. “The mathematics usually applied to study networks cannot detect the high-dimensional structures and spaces that we now see clearly.”

If 4D worlds stretch our imagination, worlds with 5, 6 or more dimensions are too complex for most of us to comprehend. This is where algebraic topology comes in: a branch of mathematics that can describe systems with any number of dimensions. The mathematicians who brought algebraic topology to the study of brain networks in the Blue Brain Project were Kathryn Hess from EPFL and Ran Levi from Aberdeen University.

“Algebraic topology is like a telescope and microscope at the same time. It can zoom into networks to find hidden structures – the trees in the forest – and see the empty spaces – the clearings – all at the same time,” explains Hess.

In 2015, Blue Brain published the first digital copy of a piece of the neocortex – the most evolved part of the brain and the seat of our sensations, actions, and consciousness. In this latest research, using algebraic topology, multiple tests were performed on the virtual brain tissue to show that the multi-dimensional brain structures discovered could never be produced by chance. Experiments were then performed on real brain tissue in the Blue Brain’s wet lab in Lausanne confirming that the earlier discoveries in the virtual tissue are biologically relevant and also suggesting that the brain constantly rewires during development to build a network with as many high-dimensional structures as possible.

When the researchers presented the virtual brain tissue with a stimulus, cliques of progressively higher dimensions assembled momentarily to enclose high-dimensional holes, that the researchers refer to as cavities. “The appearance of high-dimensional cavities when the brain is processing information means that the neurons in the network react to stimuli in an extremely organized manner,” says Levi. “It is as if the brain reacts to a stimulus by building then razing a tower of multi-dimensional blocks, starting with rods (1D), then planks (2D), then cubes (3D), and then more complex geometries with 4D, 5D, etc. The progression of activity through the brain resembles a multi-dimensional sandcastle that materializes out of the sand and then disintegrates.”

The big question these researchers are asking now is whether the intricacy of tasks we can perform depends on the complexity of the multi-dimensional “sandcastles” the brain can build. Neuroscience has also been struggling to find where the brain stores its memories. “They may be ‘hiding’ in high-dimensional cavities,” Markram speculates.


About Blue Brain

The aim of the Blue Brain Project, a Swiss brain initiative founded and directed by Professor Henry Markram, is to build accurate, biologically detailed digital reconstructions and simulations of the rodent brain, and ultimately, the human brain. The supercomputer-based reconstructions and simulations built by Blue Brain offer a radically new approach for understanding the multilevel structure and function of the brain.

About Frontiers

Frontiers is a leading community-driven open-access publisher. By taking publishing entirely online, we drive innovation with new technologies to make peer review more efficient and transparent. We provide impact metrics for articles and researchers, and merge open access publishing with a research network platform – Loop – to catalyse research dissemination, and popularize research to the public, including children. Our goal is to increase the reach and impact of research articles and their authors. Frontiers has received the ALPSP Gold Award for Innovation in Publishing in 2014.

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

Cliques of Neurons Bound into Cavities Provide a Missing Link between Structure and Function by Michael W. Reimann, Max Nolte, Martina Scolamiero, Katharine Turner, Rodrigo Perin, Giuseppe Chindemi, Paweł Dłotko, Ran Levi, Kathryn Hess, and Henry Markram. Front. Comput. Neurosci., 12 June 2017 |

This paper is open access.

Time traveling at the University of British Columbia

Anyone who dreams of timetraveling is going to have to wait a bit longer as this form of timetraveling is theoretical. From an April 27, 2017 news item on ScienceDaily,

After some serious number crunching, a UBC [University of British Columbia] researcher has come up with a mathematical model for a viable time machine.

Ben Tippett, a mathematics and physics instructor at UBC’s Okanagan campus, recently published a study about the feasibility of time travel. Tippett, whose field of expertise is Einstein’s theory of general relativity, studies black holes and science fiction when he’s not teaching. Using math and physics, he has created a formula that describes a method for time travel.

An April 27, 2017 UBC at Okanagan news release (also on EurekAlert), which originated the news item, elaborates on the work.

“People think of time travel as something fictional,” says Tippett. “And we tend to think it’s not possible because we don’t actually do it. But, mathematically, it is possible.”

Ever since H.G. Wells published his book Time Machine in 1885, people have been curious about time travel—and scientists have worked to solve or disprove the theory. In 1915 Albert Einstein announced his theory of general relativity, stating that gravitational fields are caused by distortions in the fabric of space and time. More than 100 years later, the LIGO Scientific Collaboration—an international team of physics institutes and research groups—announced the detection of gravitational waves generated by colliding black holes billions of light years away, confirming Einstein’s theory.

The division of space into three dimensions, with time in a separate dimension by itself, is incorrect, says Tippett. The four dimensions should be imagined simultaneously, where different directions are connected, as a space-time continuum. Using Einstein’s theory, Tippett explains that the curvature of space-time accounts for the curved orbits of the planets.

In “flat” or uncurved space-time, planets and stars would move in straight lines. In the vicinity of a massive star, space-time geometry becomes curved and the straight trajectories of nearby planets will follow the curvature and bend around the star.

“The time direction of the space-time surface also shows curvature. There is evidence showing the closer to a black hole we get, time moves slower,” says Tippett. “My model of a time machine uses the curved space-time—to bend time into a circle for the passengers, not in a straight line. That circle takes us back in time.”

While it is possible to describe this type of time travel using a mathematical equation, Tippett doubts that anyone will ever build a machine to make it work.

“H.G. Wells popularized the term ‘time machine’ and he left people with the thought that an explorer would need a ‘machine or special box’ to actually accomplish time travel,” Tippett says. “While is it mathematically feasible, it is not yet possible to build a space-time machine because we need materials—which we call exotic matter—to bend space-time in these impossible ways, but they have yet to be discovered.”

For his research, Tippett created a mathematical model of a Traversable Acausal Retrograde Domain in Space-time (TARDIS). He describes it as a bubble of space-time geometry which carries its contents backward and forward through space and time as it tours a large circular path. The bubble moves through space-time at speeds greater than the speed of light at times, allowing it to move backward in time.

“Studying space-time is both fascinating and problematic. And it’s also a fun way to use math and physics,” says Tippett. “Experts in my field have been exploring the possibility of mathematical time machines since 1949. And my research presents a new method for doing it.”

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

Traversable acausal retrograde domains in spacetime by Benjamin K Tippett and David Tsang. Classical and Quantum Gravity, Volume 34, Number 9 DOI: Published 31 March 2017

© 2017 IOP Publishing Ltd

This paper is behind a paywall.

Did artists lead the way in mathematics?

There is no way to definitively answer the question of whether artists have led the way in mathematics but the question does provide interesting fodder for an essay (h/t April 28, 2017 news item on by Henry Adams, professor of Art History at Case Western Reserve University , in his April 28, 2017 essay for,

Mathematics and art are generally viewed as very different disciplines – one devoted to abstract thought, the other to feeling. But sometimes the parallels between the two are uncanny.

From Islamic tiling to the chaotic patterns of Jackson Pollock, we can see remarkable similarities between art and the mathematical research that follows it. The two modes of thinking are not exactly the same, but, in interesting ways, often one seems to foreshadow the other.

Does art sometimes spur mathematical discovery? There’s no simple answer to this question, but in some instances it seems very likely.

Patterns in the Alhambra

Consider Islamic ornament, such as that found in the Alhambra in Granada, Spain.

In the 14th and 15th centuries, the Alhambra served as the palace and harem of the Berber monarchs. For many visitors, it’s a setting as close to paradise as anything on earth: a series of open courtyards with fountains, surrounded by arcades that provide shelter and shade. The ceilings are molded in elaborate geometric patterns that resemble stalactites. The crowning glory is the ornament in colorful tile on the surrounding walls, which dazzles the eye in a hypnotic way that’s strangely blissful. In a fashion akin to music, the patterns lift the onlooker into an almost out-of-body state, a sort of heavenly rapture.

It’s a triumph of art – and of mathematical reasoning. The ornament explores a branch of mathematics known as tiling, which seeks to fill a space completely with regular geometric patterns. Math shows that a flat surface can be regularly covered by symmetric shapes with three, four and six sides, but not with shapes of five sides.

It’s also possible to combine different shapes, using triangular, square and hexagonal tiles to fill a space completely. The Alhambra revels in elaborate combinations of this sort, which are hard to see as stable rather than in motion. They seem to spin before our eyes. They trigger our brain into action and, as we look, we arrange and rearrange their patterns in different configurations.

An emotional experience? Very much so. But what’s fascinating about such Islamic tilings is that the work of anonymous artists and craftsmen also displays a near-perfect mastery of mathematical logic. Mathematicians have identified 17 types of symmetry: bilateral symmetry, rotational symmetry and so forth. At least 16 appear in the tilework of the Alhambra, almost as if they were textbook diagrams.

The patterns are not merely beautiful, but mathematically rigorous as well. They explore the fundamental characteristics of symmetry in a surprisingly complete way. Mathematicians, however, did not come up with their analysis of the principles of symmetry until several centuries after the tiles of the Alhambra had been set in place.

Tiles at the Alhambra. Credit: Wikimedia Commons, CC BY-SA

Quasicrystalline tiles

Stunning as they are, the decorations of the Alhambra may have been surpassed by a masterpiece in Persia. There, in 1453, anonymous craftsmen at the Darbi-I Imam shrine in Isfahan discovered quasicrystalline patterns. These patterns have complex and mysterious mathematical properties that were not analyzed by mathematicians until the discovery of Penrose tilings in the 1970s.

Such patterns fill a space completely with regular shapes, but in a configuration which never repeats itself – indeed, is infinitely nonrepeated – although the mathematical constant known as the Golden Section occurs over and over again.

Daniel Schectman won the 2001 Nobel Prize [Schechtman was awarded the Nobel Prize for Chemistry in 2011 as per his Wikipedia entry] or the discovery of quasicrystals, which obey this law of organization. This breakthrough forced scientists to reconsider their conception of the very nature of matter.

In 2005, Harvard physicist Peter James Lu showed that it’s possible to generate such quasicrystalline patterns relatively easily using girih tiles. Girih tiles combine several pure geometric shapes into five patterns: a regular decagon, an irregular hexagon, a bow tie, a rhombus and a regular pentagon.

Whatever the method, it’s clear that the quasicrystalline patterns at Darbi-I Imam were created by craftsmen without advanced training in mathematics. It took several more centuries for mathematicians to analyze and articulate what they were doing. In other words, intuition preceded full understanding.

It’s a fascinating essay and, if you have the time and the interest, it’s definitely a worthwhile read (Henry’s April 28, 2017 essay ).

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.