Duke Research Blog

Following the people and events that make up the research community at Duke.

Category: Mathematics (Page 1 of 5)

Artificial Intelligence Knows How You Feel

Ever wondered how Siri works? Afraid that super smart robots might take over the world soon?

On April 3rd researchers from Duke, NCSU and UNC came together for Triangle Machine Learning Day to provoke everyone’s curiosities about the complex field that is Artificial Intelligence. A.I. is an overarching term for smart technologies, ranging from self-driving cars to targeted advertising. We can arrive at artificial intelligence through what’s known as “machine learning.” Instead of explicitly programming a machine with the basic capabilities we want it to have, we can make it so that its code is flexible and adapts based on information it’s presented with. Its knowledge grows as a result of training it. In other words, we’re teaching a computer to learn.

Matthew Philips is working with Kitware to get computers to “see,” also known as “machine vision.” By providing thousands and thousands of images, a computer with the right coding can learn to actually make sense of what an image is beyond different colored pixels.

Machine vision has numerous applications. An effective way to search satellite imagery for arbitrary objects could be huge in the advancement of space technology – a satellite could potentially identify obscure objects or potential lifeforms that stick out in those images. This is something we as humans can’t do ourselves just because of the sheer amount of data there is to go through. Similarly, we could teach a machine to identify cancerous or malignant cells in an image, thus giving us a quick diagnosis if someone is at risk of developing a disease.

The problem is, how do you teach a computer to see? Machines don’t easily understand things like similarity, depth or orientation — things that we as humans do automatically without even thinking about. That’s exactly the type of problem Kitware has been tackling.

One hugely successful piece of Artificial Intelligence you may be familiar with is IBM’s Watson. Labeled as “A.I. for professionals,” Watson was featured on Sixty Minutes and even played Jeopardy on live television. Watson has visual recognition capabilities, can work as a translator, and can even understand things like tone, personality or emotional state. And obviously it can answer crazy hard questions. What’s even cooler is that it doesn’t matter how you ask the question – Watson will know what you mean. Watson is basically Siri on steroids, and the world got a taste of its power after watching it smoke its competitors on Jeopardy. However, Watson is not to be thought of as a physical supercomputer. It is a collection of technologies that can be used in many different ways, depending on how you train it. This is what makes Watson so astounding – through machine learning, its knowledge can adapt to the context it’s being used in.

Source: CBS News.

IBM has been able to develop such a powerful tool thanks to data. Stacy Joines from IBM noted, “Data has transformed every industry, profession, and domain.” From our smart phones to fitness devices, data is being collected about us as we speak (see: digital footprint). While it’s definitely pretty scary, the point is that a lot of data is out there. The more data you feed Watson, the smarter it is. IBM has utilized this abundance of data combined with machine learning to produce some of the most sophisticated AI out there.

Sure, it’s a little creepy how much data is being collected on us. Sure, there are tons of movies and theories out there about how intelligent robots in the future will outsmart humans and take over. But A.I. isn’t a thing to be scared of. It’s a beautiful creation that surpasses all capabilities even the most advanced purely programmable model has. It’s joining the health care system to save lives, advising businesses and could potentially find a new inhabitable planet. What we choose to do with A.I. is entirely up to us.

Post by Will Sheehan

Will Sheehan

Jonathan Mattingly: Mathematics and Maps to Define Democracy

Jonathan Mattingly is the chair of mathematics at Duke and an alumnus of the NC School of Science and Math

What began as an undergraduate project looking at how to create a “typical” map of congressional districts expanded to a national investigation for Duke mathematics chair Jonathan Mattingly. He was generous enough to speak to me about some of his recent work in mathematically investigating gerrymandering and the communication which followed between lawmakers and statisticians.

By strategically manipulating certain lines, it is possible to ensure a certain number of seats for one party even if that party does not win the majority vote. What “Team Gerrymandering” set out to do was to create an algorithm which would create the least biased map possible. The use of the term “fair” is complex in this instance, as politics and geography are very rarely simple enough to be split fairly.

An example of a mathematical model of precincts and districts.

In Wisconsin, the algorithm which “Team Gerrymandering” developed was used to prove that the voting districts were being disproportionately drawn in favor of the Republican votes, a trend which had was also been seen after the 2015 elections in North Carolina districts.

By strategically manipulating certain lines, it is possible to ensure a certain number of seats for one party even if that party does not win the majority vote. What “Team Gerrymandering” set out to do was to create an algorithm which would create the least biased map possible. The use of the term “fair” is complex in this instance, as politics and geography are very rarely simple enough to be split fairly.

The algorithm developed was then submitted as an brief amicus curae brief and used (it was used as a piece of appellate evidence) in the Wisconsin case Whitford vs. Bill. case. The mathematicians hoped to , in an attempt to prove that the districting of Wisconsin is an outlier in comparison to thousands of other mapping simulations run under their algorithm, which provide statistically sound data.

A problem such as this is a prime example of the bridge between the Humanities and STEM fields, which become increasingly separate as the level of expertise rises. as this truly bridges the humanities and STEM fields:, a solution has been found, but effectively communicating it was not as simple.

When asked about explaining and publishing this work in order to submit it as evidence, Mattingly admitted that it was, at times difficult, but it only further proved how important the effort is.

“It starts with a conversation. I’m willing to explain it, but you have to be willing to listen.”

A team full of lawyers looking to win a case is arguably a highly motivated audience, but this is not always the case. Mattingly, who is a 1988 graduate of the NC School of Science and Math which I attend, mentioned being at parties and hearing people state, “Oh, I’m no good at math, it’s just numbers and letters to me,” but he could never recount anyone saying “Oh, I don’t see the point in using language, or reading a dictionary.” These may seem like harmless comments, but a subconscious form of selective ignorance is still selective ignorance.

In light of the gerrymandering case, and “Team Gerrymandering’s” involvement in it, we are called to think again about the importance of fields we are not necessarily involved in, especially the STEM fields. What other patterns aren’t we noticing because we failed to look? Where else could we be improving if we were willing to listen? If we both don’t try, then we aren’t getting anywhere.”

The results of the Whitman vs Gill case are expected in June of 2018, and until then, the conversation must continue.

UPDATE: On Jan. 9, a federal court panel struck down North Carolina’s Congressional district maps on the grounds that they had been gerrymandered to favor Republicans. Mattingly commented.

Guest post by Paris Geolas, a senior at the North Carolina School of Science and Math

Anita Layton: A Model of STEM Versatility

Using mathematics to model the kidney and its biological systems is a field of study located at the intersection of two disciplines.

Anita Layton is a math professor at Duke. (Photo by Chris Hildreth, Duke Photography)

But for Duke’s Anita Layton, PhD, the Robert R. and Katherine B. Penn Professor of Mathematics and a professor of biomedical engineering, that just adds to the fun of it.

Growing up, with her father as the head of mathematics at her school, she was always told she was going to be a mathematician just like him. So she knew that was the last thing she wanted to do.

When Layton arrived as an undergraduate at Duke, she began a major in physics, but she seemed rather cursed when it came to getting correct results from her experiments. She settled for a BA in physics, but her academic journey was far from over. She had also taken a computer science course at Duke and fallen in love with it. If an experiment went wrong “things didn’t smell or blow up” and you could fix your mistake and move on, she said.

While pursuing her PhD in computer science at the University of Toronto, Layton was performing very math-oriented computer science, working with and analyzing numbers. However, it would be a while before biology entered the mix

While she was never good at dissections, she told me she was always good at understanding things that ‘flow’ and she came to the realization that blood is something that flows. She thought, “Hey, I can do that.

Anita Layton, Duke

Anita Layton, Ph.D.

Layton began creating programs that could solve the equations that model blood flow quickly, using her background in computer science. She then started learning about physiology, focusing on the renal system, and making models

It was a journey that took her to many different places, with pit stops and U-turns throughout many different fields. Had Layton stuck with just physics or computer science or math, she never would have ventured out and found this field that she is an expert in now.

It’s her interest in many different fields that has set Layton apart from many other people in the STEM field. In learning a wide variety of things, she has gotten better at computer science, mathematics, biology, physics, and more

When asked about what advice she would give her younger self, or any young person going into college, it would be to do just that: “Learn more things that you’re not good at.” She encouraged just taking a chemistry or biology class once in a while, or a philosophy course that makes you think in ways that you don’t normally. It’s often in those classes that you unearth things that can truly set your life in a completely different direction, Layton said, and she’s living proof of that.

Cecilia Poston, NCSSM

Cecilia Poston

Guest Post by Cecilia Poston, a senior at North Carolina School of Science and Math

Morphogenesis: All Guts and Morning Glories

What is morphogenesis? Morphogenesis examines the development of the living organisms’ forms.

It also is an area of research for Lakshminarayanan Mahadevan, Professor of Applied Mathematics, Organismic and Evolutionary Biology and Physics at Harvard University. On his presentation in the Public Lectures Unveiling Math (PLUM) series here at Duke, he credited the beginnings of morphogenesis to D’Arcy Wentworth Thompson, author of the book On Growth and Form.

Mathematically, morphogenesis focuses on how different rates of growth change the shapes of organisms as they develop. Cell number, cell size, cell shape, and cell position comprise the primary cellular factors of multicellular morphogenesis, which studies larger structures than individual cells and is Mahadevan’s focus.

Effects on tissues appear through changes in sizes, connectivities, and shapes, altering the phenotype, or the outward physical appearance. All these variables change in space and time. Professor Mahadevan presented on morphogenesis studies that have been conducted on plant shoots, guts, and brains.

Research on plant shoots often concentrates on the question, “Why do plant shoots grow in such a wide variety of directions and what determines their shapes?” The picture below shows the different postures appearances of plant shoots from completely straight to leaning to hanging.

Can morphogenesis make sense of these differences? Through mathematical modeling, two stimuli for shoots’ shapes was determined: gravity and itself. Additionally, elasticity as a function of the shoots’ weight plays a role in the mathematical models of plant shoots’ shapes which appear in Mahadevan’s paper co-written with a fellow professor, Raghunath Chelakkot. Mahadevan also explored the formation of flower and leaf shapes with these morphogenesis studies. 

Over twenty feet of guts are coiled up inside you. In order to fit these intestines inside the mammals, they must coil and loop. But what variables determine how these guts loop around? To discover the answer to this question, Mahadevan and other researchers examined chick embryos which increase their gut lengths by a factor greater than twenty over a twelve-day span. They were able to create a physical model using a rubber tube sewn to a sheet that followed the same patterns as the chicks’ guts. Through their observation of not only chicks but also quail and mice, Mahadevan determined that the morphogenesis of the guts has no dependence on genetics or any other microscopic factors.

Mahadevan’s study of how the brain folds occurs through MRI images of human fetal development. Initially, barely any folding exists on fetal brains but eventually the geometry of the surrounding along with local stress forms folds on the brain. By creating a template with gel and treating it to mimic the relationship between the brain’s gray matter and white matter, Mahadevan along with other researchers discovered that they could reproduce the brain’s folds. Because they were able to recreate the folds through only global geometry and local stress, they concluded that morphogenesis evolution does not depend on microscopic factors such as genetics. Further, by examining if folding regions correlate with the activity regions of the brain, questions about the effect of physical form on abilities and the inner functions of the brain.



Cheating Time to Watch Liquids do the Slow Dance

Colorful spheres simulating liquid molecules shift around inside a cube shape

The team’s new algorithm is able to simulate molecular configurations of supercooled liquids below the glass transition. The properties of these configurations are helping to solve a 70-year paradox about the entropy of glasses. Credit: Misaki Ozawa and Andrea Ninarello, Université de Montpellier.

If you could put on a pair of swimming goggles, shrink yourself down like a character from The Magic School Bus and take a deep dive inside a liquid, you would see a crowd of molecules all partying like it’s 1999.

All this frenetic wiggling makes it easy for molecules to rearrange themselves and for the liquid as a whole to change shape. But for supercooled liquids — liquids like honey that are cooled below their freezing point without crystallizing – the lower temperature slows down the dancing like Etta James’ “At Last.” Lower the temperature enough, and the slow-down can be so dramatic that it takes centuries or even millennia for the molecules to rearrange and the liquid to move.

Scientists can’t study processes that last longer than their careers. But Duke chemists and their Simons Foundation collaborators have found a way to cheat time, simulating the slow dance of deeply supercooled liquids. Along the way, they have found new physical properties of “aged” supercooled liquids and glasses.

A droplet rises above a surface of water

Credit: Ruben Alexander via Flickr.

To understand just how slow deeply supercooled liquids move, consider the world’s longest-running experiment, the University of Queensland’s Pitch Drop Experiment. A single drop of pitch forms every eight to thirteen years — and this pitch is moving faster than deeply supercooled liquids.

“Experimentally there is a limit to what you can observe, because even if you managed to do it over your entire career, that is still a maximum of 50 years,” said Patrick Charbonneau, an associate professor of chemistry and physics at Duke. “For many people that was considered a hard glass ceiling, beyond which you couldn’t study the behavior of supercooled liquids.”

Charbonneau, who is an expert on numerical simulations, said that using computers to simulate the behavior of supercooled liquids has even steeper time limitations. He estimates that, given the current rate of computer advancement, it would take 50 to 100 years before computers would be powerful enough for simulations to exceed experimental capabilities – and even then the simulations would take months.

To break this glass ceiling, the Charbonneau group collaborated with Ludovic Berthier and his team, who were developing an algorithm to bypass these time constraints. Rather than taking months or years to simulate how each molecule in a supercooled liquids jiggles around until the molecules rearrange, the algorithm picks individual molecules to swap places with each other, creating new molecular configurations.

This allows the team to explore new configurations that could take millennia to form naturally. These “deeply supercooled liquids and ultra-aged glasses” liquids are at a lower energy, and more stable, than any observed before.

“We were cheating time in the sense that we didn’t have to follow the dynamics of the system,” Charbonneau said. “We were able to simulate deeply supercooled liquids well beyond is possible in experiments, and it opened up a lot of possibilities.”

Two columns of blue and red spheres represent simulations of vapor-deposited glasses.

Glasses that are grown one layer at a time have a much different structure than bulk glasses. The team used their new algorithm to study how molecules in these glasses rearrange, and found that at low temperatures (right), only the molecules at the surface are mobile. The results may be used to design better types of glass for drug delivery or protective coatings. Credit: Elijah Flenner.

Last summer, the team used this technique to discover a new phase transition in low-temperature glasses. They recently published two additional studies, one of which sheds light on the “Kauzmann paradox,” a 70-year question about the entropy of supercooled liquids below the glass transition. The second explores the formation of vapor-deposited glasses, which have applications in drug delivery and protective coatings.

“Nature has only one way to equilibrate, by just following the molecular dynamics,” said Sho Yaida, a postdoctoral fellow in Charbonneau’s lab. “But the great thing about numerical simulations is you can tweak the algorithm to accelerate your experiment.”

Configurational entropy measurements in extremely supercooled liquids that break the glass ceiling.” Ludovic Berthier, Patrick Charbonneau, Daniele Coslovich, Andrea Ninarello, Misaki Ozawa and Sho Yaida. PNAS, Oct. 24, 2017. DOI: 10.1073/pnas.1706860114

The origin of ultrastability in vapor-deposited glasses.” Ludovic Berthier, Patrick Charbonneau, Elijah Flenner and Francesco Zamponi. PRL, Nov. 1, 2017. DOI: 10.1103/PhysRevLett.119.188002

Post by Kara Manke

Durham Traffic Data Reveal Clues to Safer Streets

Ghost bikes are a haunting site. The white-painted bicycles, often decorated with flowers or photographs, mark the locations where cyclists have been hit and killed on the street.

A white-painted bike next to a street.

A Ghost Bike located in Chapel Hill, NC.

Four of these memorials currently line the streets of Durham, and the statistics on non-fatal crashes in the community are equally sobering. According to data gathered by the North Carolina Department of Transportation, Durham county averaged 23 bicycle and 116 pedestrian crashes per year between 2011 and 2015.

But a team of Duke researchers say these grim crash data may also reveal clues for how to make Durham’s streets safer for bikers, walkers, and drivers.

This summer, a team of Duke students partnered with Durham’s Department of Transportation to analyze and map pedestrian, bicycle and motor vehicle crash data as part of the 10-week Data+ summer research program.

In the Ghost Bikes project, the team created an interactive website that allows users to explore how different factors such as the time-of-day, weather conditions, and sociodemographics affect crash risk. Insights from the data also allowed the team to develop policy recommendations for improving the safety of Durham’s streets.

“Ideally this could help make things safer, help people stay out of hospitals and save lives,” said Lauren Fox, a Duke cultural anthropology major who graduated this spring, and a member of the DATA+ Ghost Bikes team.

A map of Durham county with dots showing the locations of bicycle crashes

A heat map from the team’s interactive website shows areas with the highest density of bicycle crashes, overlaid with the locations of individual bicycle crashes.

The final analysis showed some surprising trends.

“For pedestrians the most common crash isn’t actually happening at intersections, it is happening at what is called mid-block crossings, which happen when someone is crossing in the middle of the road,” Fox said.

To mitigate the risks, the team’s Executive Summary includes recommendations to install crosswalks, median islands and bike lanes to roads with a high density of crashes.

They also found that males, who make up about two-thirds of bicycle commuters over the age of 16, are involved in 75% of bicycle crashes.

“We found that male cyclists over age 16 actually are hit at a statistically higher rate,” said Elizabeth Ratliff, a junior majoring in statistical science. “But we don’t know why. We don’t know if this is because males are riskier bikers, if it is because they are physically bigger objects to hit, or if it just happens to be a statistical coincidence of a very unlikely nature.”

To build their website, the team integrated more than 20 sets of crash data from a wide variety of different sources, including city, county, regional and state reports, and in an array of formats, from maps to Excel spreadsheets.

“They had to fit together many different data sources that don’t necessarily speak to each other,” said faculty advisor Harris Solomon, an associate professor of cultural anthropology and global health at Duke.  The Ghost Bikes project arose out of Solomon’s research on traffic accidents in India, supported by the National Science Foundation Cultural Anthropology Program.

In Solomon’s Spring 2017 anthropology and global health seminar, students explored the role of the ghost bikes as memorials in the Durham community. The Data+ team approached the same issues from a more quantitative angle, Solomon said.

“The bikes are a very concrete reminder that the data are about lives and deaths,” Solomon said. “By visiting the bikes, the team was able to think about the very human aspects of data work.”

“I was surprised to see how many stakeholders there are in biking,” Fox said. For example, she added, the simple act of adding a bike lane requires balancing the needs of bicyclists, nearby residents concerned with home values or parking spots, and buses or ambulances who require access to the road.

“I hadn’t seen policy work that closely in my classes, so it was interesting to see that there aren’t really simple solutions,” Fox said.

[youtube https://www.youtube.com/watch?v=YHIRqhdb7YQ&w=629&h=354]


Data+ is sponsored by Bass Connections, the Information Initiative at Duke, the Social Science Research Institute, the departments of Mathematics and Statistical Science and MEDx.

Other Duke sponsors include DTECH, Duke Health, Sanford School of Public Policy, Nicholas School of the Environment, Development and Alumni Affairs, Energy Initiative, Franklin Humanities Institute, Duke Institute for Brain Sciences, Office for Information Technology and the Office of the Provost, as well as the departments of Electrical & Computer Engineering, Computer Science, Biomedical Engineering, Biostatistics & Bioinformatics and Biology.

Government funding comes from the National Science Foundation. Outside funding comes from Accenture, Academic Analytics, Counter Tools and an anonymous donation.

Community partnerships, data and interesting problems come from the Durham Police Department, Durham Neighborhood Compass, Cary Institute of Ecosystem Studies, Duke Marine Lab, Center for Child and Family Policy, Northeast Ohio Medical University, TD Bank, Epsilon, Duke School of Nursing, University of Southern California, Durham Bicycle and Pedestrian Advisory Commission, Duke Surgery, MyHealth Teams, North Carolina Museum of Art and Scholars@Duke.

Writing by Kara Manke; video by Lauren Mueller and Summer Dunsmore

Students Share Research Journeys at Bass Connections Showcase

From the highlands of north central Peru to high schools in North Carolina, student researchers in Duke’s Bass Connections program are gathering data in all sorts of unique places.

As the school year winds down, they packed into Duke’s Scharf Hall last week to hear one another’s stories.

Students and faculty gathered in Scharf Hall to learn about each other’s research at this year’s Bass Connections showcase. Photo by Jared Lazarus/Duke Photography.

The Bass Connections program brings together interdisciplinary teams of undergraduates, graduate students and professors to tackle big questions in research. This year’s showcase, which featured poster presentations and five “lightning talks,” was the first to include teams spanning all five of the program’s diverse themes: Brain and Society; Information, Society and Culture; Global Health; Education and Human Development; and Energy.

“The students wanted an opportunity to learn from one another about what they had been working on across all the different themes over the course of the year,” said Lori Bennear, associate professor of environmental economics and policy at the Nicholas School, during the opening remarks.

Students seized the chance, eagerly perusing peers’ posters and gathering for standing-room-only viewings of other team’s talks.

The different investigations took students from rural areas of Peru, where teams interviewed local residents to better understand the transmission of deadly diseases like malaria and leishmaniasis, to the North Carolina Museum of Art, where mathematicians and engineers worked side-by-side with artists to restore paintings.

Machine learning algorithms created by the Energy Data Analytics Lab can pick out buildings from a satellite image and estimate their energy consumption. Image courtesy Hoël Wiesner.

Students in the Energy Data Analytics Lab didn’t have to look much farther than their smart phones for the data they needed to better understand energy use.

“Here you can see a satellite image, very similar to one you can find on Google maps,” said Eric Peshkin, a junior mathematics major, as he showed an aerial photo of an urban area featuring buildings and a highway. “The question is how can this be useful to us as researchers?”

With the help of new machine-learning algorithms, images like these could soon give researchers oodles of valuable information about energy consumption, Peshkin said.

“For example, what if we could pick out buildings and estimate their energy usage on a per-building level?” said Hoël Wiesner, a second year master’s student at the Nicholas School. “There is not really a good data set for this out there because utilities that do have this information tend to keep it private for commercial reasons.”

The lab has had success developing algorithms that can estimate the size and location of solar panels from aerial photos. Peshkin and Wiesner described how they are now creating new algorithms that can first identify the size and locations of buildings in satellite imagery, and then estimate their energy usage. These tools could provide a quick and easy way to evaluate the total energy needs in any neighborhood, town or city in the U.S. or around the world.

“It’s not just that we can take one city, say Norfolk, Virginia, and estimate the buildings there. If you give us Reno, Tuscaloosa, Las Vegas, Pheonix — my hometown — you can absolutely get the per-building energy estimations,” Peshkin said. “And what that means is that policy makers will be more informed, NGOs will have the ability to best service their community, and more efficient, more accurate energy policy can be implemented.”

Some students’ research took them to the sidelines of local sports fields. Joost Op’t Eynde, a master’s student in biomedical engineering, described how he and his colleagues on a Brain and Society team are working with high school and youth football leagues to sort out what exactly happens to the brain during a high-impact sports game.

While a particularly nasty hit to the head might cause clear symptoms that can be diagnosed as a concussion, the accumulation of lesser impacts over the course of a game or season may also affect the brain. Eynde and his team are developing a set of tools to monitor both these impacts and their effects.

A standing-room only crowd listened to a team present on their work “Tackling Concussions.” Photo by Jared Lazarus/Duke Photography.

“We talk about inputs and outputs — what happens, and what are the results,” Eynde said. “For the inputs, we want to actually see when somebody gets hit, how they get hit, what kinds of things they experience, and what is going on in the head. And the output is we want to look at a way to assess objectively.”

The tools include surveys to estimate how often a player is impacted, an in-ear accelerometer called the DASHR that measures the intensity of jostles to the head, and tests of players’ performance on eye-tracking tasks.

“Right now we are looking on the scale of a season, maybe two seasons,” Eynde said. “What we would like to do in the future is actually follow some of these students throughout their career and get the full data for four years or however long they are involved in the program, and find out more of the long-term effects of what they experience.”

Kara J. Manke, PhD

Post by Kara Manke

Visualizing the Fourth Dimension

Living in a 3-dimensional world, we can easily visualize objects in 2 and 3 dimensions. But as a mathematician, playing with only 3 dimensions is limiting, Dr. Henry Segerman laments.  An Assistant Professor in Mathematics at Oklahoma State University, Segerman spoke to Duke students and faculty on visualizing 4-dimensional space as part of the PLUM lecture series on April 18.

What exactly is the 4th dimension?

Let’s break down spatial dimensions into what we know. We can describe a point in 2-dimensional space with two numbers x and y, visualizing an object in the xy plane, and a point in 3D space with 3 numbers in the xyz coordinate system.

Plotting three dimensions in the xyz coordinate system.

While the green right-angle markers are not actually 90 degrees, we are able to infer the 3-dimensional geometry as shown on a 2-dimensional screen.

Likewise, we can describe a point in 4-dimensional space with four numbers – x, y, z, and w – where the purple w-axis is at a right angle to the other regions; in other words, we can visualize 4 dimensions by squishing it down to three.

Plotting four dimensions in the xyzw coordinate system.

One commonly explored 4D object we can attempt to visualize is known as a hypercube. A hypercube is analogous to a cube in 3 dimensions, just as a cube is to a square.

How do we make a hypercube?

To create a 1D line, we take a point, make a copy, move the copied point parallely to some distance away, and then connect the two points with a line.

Similarly, a square can be formed by making a copy of a line and connecting them to add the second dimension.

So, to create a hypercube, we move identical 3D cubes parallel to each other, and then connect them with four lines, as depicted in the image below.

To create an n–dimensional cube, we take 2 copies of the (n−1)–dimensional cube and connecting corresponding corners.

Even with a 3D-printed model, trying to visualize the hypercube can get confusing. 

How can we make a better picture of a hypercube? “You sort of cheat,” Dr. Segerman explained. One way to cheat is by casting shadows.

Parallel projection shadows, depicted in the figure below, are caused by rays of light falling at a  right angle to the plane of the table. We can see that some of the edges of the shadow are parallel, which is also true of the physical object. However, some of the edges that collide in the 2D cast don’t actually collide in the 3D object, making the projection more complicated to map back to the 3D object.

Parallel projection of a cube on a transparent sheet of plastic above the table.

One way to cast shadows with no collisions is through stereographic projection as depicted below.

The stereographic projection is a mapping (function) that projects a sphere onto a plane. The projection is defined on the entire sphere, except the point at the top of the sphere.

For the object below, the curves on the sphere cast shadows, mapping them to a straight line grid on the plane. With stereographic projection, each side of the 3D object maps to a different point on the plane so that we can view all sides of the original object.

Stereographic projection of a grid pattern onto the plane. 3D print the model at Duke’s Co-Lab!

Just as shadows of 3D objects are images formed on a 2D surface, our retina has only a 2D surface area to detect light entering the eye, so we actually see a 2D projection of our 3D world. Our minds are computationally able to reconstruct the 3D world around us by using previous experience and information from the 2D images such as light, shade, and parallax.

Projection of a 3D object on a 2D surface.

Projection of a 4D object on a 3D world

How can we visualize the 4-dimensional hypercube?

To use stereographic projection, we radially project the edges of a 3D cube (left of the image below) to the surface of a sphere to form a “beach ball cube” (right).

The faces of the cube radially projected onto the sphere.

Placing a point light source at the north pole of the bloated cube, we can obtain the projection onto a 2D plane as shown below.

Stereographic projection of the “beach ball cube” pattern to the plane. View the 3D model here.

Applied to one dimension higher, we can theoretically blow a 4-dimensional shape up into a ball, and then place a light at the top of the object, and project the image down into 3 dimensions.

Left: 3D print of the stereographic projection of a “beach ball hypercube” to 3-dimensional space. Right: computer render of the same, including the 2-dimensional square faces.

Forming n–dimensional cubes from (n−1)–dimensional renderings.

Thus, the constructed 3D model of the “beach ball cube” shadow is the projection of the hypercube into 3-dimensional space. Here the 4-dimensional edges of the hypercube become distorted cubes instead of strips.

Just as the edges of the top object in the figure can be connected together by folding the squares through the 3rd dimension to form a cube, the edges of the bottom object can be connected through the 4th dimension

Why are we trying to understand things in 4 dimensions?

As far as we know, the space around us consists of only 3 dimensions. Mathematically, however, there is no reason to limit our understanding of higher-dimensional geometry and space to only 3, since there is nothing special about the number 3 that makes it the only possible number of dimensions space can have.

From a physics perspective, Einstein’s theory of Special Relativity suggests a connection between space and time, so the space-time continuum consists of 3 spatial dimensions and 1 temporal dimension. For example, consider a blooming flower. The flower’s position it not changing: it is not moving up or sideways. Yet, we can observe the transformation, which is proof that an additional dimension exists. Equating time with the 4th dimension is one example, but the 4th dimension can also be positional like the first 3. While it is possible to visualize space-time by examining snapshots of the flower with time as a constant, it is also useful to understand how space and time interrelate geometrically.

Explore more in the 4th dimension with Hypernom or Dr. Segerman’s book “Visualizing Mathematics with 3D Printing“!

Post by Anika Radiya-Dixit.



Hidden No More: Women in STEM reflect on their Journeys

Back when she was a newly-minted Ph.D., Ayana Arce struggled to picture her future life as an experimental physicist. An African American woman in a field where the number of black women U.S. doctorates is still staggeringly small, Arce could not identify many role models who looked like her.

“I didn’t know what my life would look like as a black postdoc or faculty member,” Arce said.

But in the end, Arce – an associate professor of physics at Duke who went on to join the international team of physicists who discovered the Higgs Boson in 2012 — drew inspiration from her family.

“I looked to the women such as my mother who had had academic careers, and tried to think about how I could shape my life to look something like that, and I realized that it could be something I could make work,” Arce said.

Adrienne Stiff-Roberts, Fay Cobb Payton, Kyla McMullen, Robin Coger and Valerie Ashby on stage at the Hidden Figures No More panel discussion.

Adrienne Stiff-Roberts, Fay Cobb Payton, Kyla McMullen, Robin Coger and Valerie Ashby on stage at the Hidden Figures No More panel discussion. Credit: Chris Hildreth, Duke Photography.

Arce joined five other African American women faculty on the stage of Duke’s Griffith Film Theater March 23 for a warm and candid discussion on the joys and continuing challenges of their careers in science, technology, engineering and math (STEM) fields.

The panel, titled “Hidden Figures No More: Highlighting Phenomenal Women in STEM,” was inspired by Hidden Figures, a film which celebrates three pioneering African American women mathematicians who overcame racial segregation and prejudice to play pivotal roles in NASA’s first manned space flight.

The panel discussion was spearheaded by Johnna Frierson, Director of the Office of Diversity and Inclusion at the Pratt School of Engineering, and co-sponsored by the Duke Women’s Center. It was followed by a free screening of the film.

Though our society has made great strides since the days depicted in the film, women and minorities still remain under-represented in most STEM fields. Those who do pursue careers in STEM must overcome numerous hurdles, including unconscious bias and a lack of colleagues and role models who share their gender and race.

“In my field, at some of the smaller meetings, I am often the only black woman present at the conference, many times I’m the only black person at all,” said Adrienne Stiff-Roberts, an Associate Professor of Electrical and Computer Engineering at Duke. “In that atmosphere often it can be very challenging to engage with others in the way that you are supposed to, and you can feel like an outsider.”

Valerie Ashby and Ayana Arce onstage at the Hidden Figures No More panel discussion

Valerie Ashby and Ayana Arce shared their experiences. Credit: Chris Hildreth, Duke Photography

Stiff-Roberts and the other panelists have all excelled in the face of these challenges, making their marks in fields that include physics, chemistry, computer science, mechanical engineering and electrical engineering. On Thursday they shared their thoughts and experiences with a diverse audience of students, faculty, community members and more than a few kids.

Many of the panelists credited teams of mentors and sponsors for bolstering them when times got tough, and encouraged young scientists to form their own support squads.

Valerie Ashby, Dean at Duke’s Trinity College of Arts and Sciences, advised students to look for supporters who have a vision for what they can become, and are eager to help them get there. “Don’t assume that your help might come from people who you might expect your help to come from,” Ashby said.

The importance of cheerleading from friends, and particularly parents, can never be overestimated, the panelists said.

“Having someone who will celebrate every single positive with you is a beautiful thing,” said Ashby, in response to a mother seeking advice for how to support a daughter majoring in biomedical engineering. “If your daughter is like many of us, we’ll do 99 great things but if we do one wrong thing we will focus on the one wrong thing and think we can’t do anything.”

Women in STEM can also be important and powerful allies to each other, noted Kyla McMullen, an Assistant Professor of Computer and Information Science at the University of Florida.

“I have seen situations where a woman suggests something and then the male next her says the same thing and gets the credit,” McMullen said. “That still happens, but one thing that I see help is when women make an effort to reiterate the points made by other women so people can see who credit should be attributed to.”

With all the advice out there for young people who are striving to succeed in STEM – particularly women and underrepresented minorities – the panelists advocated that everyone to stay true to themselves, above all.

“I want to encourage everyone in the room – whether you are a budding scientist or woman scholar – you can be yourself,” Ashby said. “You should make up in your mind that you are going to be yourself, no matter what.”

Kara J. Manke, PhD

Post by Kara Manke

The Man Who Knew Infinity, and his biggest fan

Ken Ono, a distinguished professor of mathematics at Emory University, was visibly thrilled to be at Duke last Thursday, January 26. Grinning from ear to ear, he announced that he was here to talk about three of his favorite things: math, movies, and “one of the most inspirational figures in my life”: Srinivasa Ramanujan.

Professor Ken Ono of Emory University poses with a bust of Newton and one of Ramanujan’s legendary notebook pages. Source: IFC Films.

Ramanujan, I learned, is one of the giants of mathematics; an incontestable genius, his scrawls in letters and notebooks have spawned whole fields of study, even up to 100 years after his death. His life story continues to inspire mathematicians around the globe—as well as, most recently, a movie which Ono helped produce: The Man Who Knew Infinity, featuring Hollywood stars Dev Patel and Jeremy Irons.

I didn’t realize until much too late that this lecture was essentially one massive spoiler for the movie. Nevertheless, I got to appreciate the brains and the heart behind the operation in hearing Ono express his passion for the man who, at age 16, inspired him to see learning in a new light. Ramanujan’s story follows.

Ramanujan was born in Kambakunam, India in 1887, the son of a cloth merchant and a singer at a local temple. He was visibly gifted from a young age, not only an outstanding student, but also a budding intellectual: by age 13, he had discovered most of modern trigonometry by himself.

Ramanujan’s brilliance earned him scholarships to attend college, only for him to flunk out not once, but twice: he was so engrossed in mathematics that he paid little heed to his actual schoolwork and let his grades suffer. His family and friends, aware of his genius, supported him anyway.

Thus, he spent the daytime in a low-level accounting job that earned him barely enough income to live, and spent the night scribbling groundbreaking mathematics in his notebooks.

A photo portrait of Srinivasa Ramanujan, a brilliant Indian mathematician born in the late 19th century. Source: IFC Films.

Unable to share his discoveries and explain their importance to those around him, Ramanujan finally grew so frustrated that, in desperation, he wrote to dozens of prominent English mathematics professors asking for help. The first of these to respond was G. H. Hardy (for any Biology nerds, this is the Hardy of the Hardy-Weinberg equilibrium), who examined the mathematics Ramanujan included in his letters and was so astounded by what he found that, at first, he thought it was a hoax perpetrated by his friend.

Needless to say, it wasn’t a hoax.

Ramanujan left India to join Hardy in England and publish his discoveries. The meat of the movie, according to Ono, is “the transformation of the relationship between these two characters:” one, a devout Hindu with no formal experience in higher education; the other, a haughty English professor who happened to be an atheist.

The two push past their differences and manage to jointly publish 30 papers based on Ramanujan’s work. Overcoming impossible odds—poverty, World War I, and racism in particular—Ramanujan’s discoveries finally found the light of day.

Sadly, Ramanujan’s story was cut short: a lifelong vegetarian, he fell ill of malnutrition while working in England, returning to India for the last year of his life in the hopes that the warmer climate would improve his health. He died in 1920, at 32 years old.

He continued writing to Hardy from his deathbed, his last letter including revolutionary ideas, which, like much of his work, were so far ahead of his time that mathematicians only began to wrap their minds around them decades after his death.

“Ramanujan was a great anticipator of mathematics, writing formulas that seemed foreign or random at the time but later inspired deep and revolutionary discoveries in math,” Ono said.

Ono’s infatuation with Ramanujan began when he was 16 years old, himself the son of a mathematics professor at Johns Hopkins University. Upon receiving a letter from Ramanujan’s widow, Ono’s father—by Ono’s account, a very stoic, stern man—was brought to tears. Shocked, Ono began to research the origin of the letter, discovering Ramanujan’s story and reaching a turning point in his own life when he realized that there were aspects to learning that were far more important than grades.

That seems to have worked out quite well for Ono, considering his success and expertise in his own area of study—not to mention that he now has “Hollywood producer” under his belt.

Professor Ken Ono chats with actor Dev Patel on the set of The Man Who Knew Infinity. Photo credit: Sam Pressman.


Post by Maya Iskandarani

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