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

Young Scientists, Making the Rounds

“Can you make a photosynthetic human?!” an 8th grader enthusiastically asks me while staring at a tiny fern in a jar.

He’s not the only one who asked me that either — another student asked if Superman was a plant, since he gets his power from the sun.

These aren’t the normal questions I get about my research as a Biology PhD candidate studying how plants get nutrients, but they were perfect for the day’s activity –A science round robin with Durham eighth-graders.

Biology grad student Leslie Slota showing Durham 8th graders some fun science.

After seeing a post under #scicomm on Twitter describing a public engagement activity for scientists, I put together a group of Duke graduate scientists to visit local middle schools and share our science with kids. We had students from biomedical engineering, physics, developmental biology, statistics, and many others — a pretty diverse range of sciences.

With help from David Stein at the Duke-Durham Neighborhood Partnership, we made connections with science teachers at the Durham School of the Arts and Lakewood Montessori school, and the event was in motion!

The outreach activity we developed works like speed dating, where people pair up, talk for 3-5 mins, and then rotate. We started out calling it “Science Speed Dating,” but for a middle school audience, we thought “Science Round-Robin” was more appropriate. Typically, a round-robin is a tournament where every team plays each of the other teams. So, every middle schooler got to meet each of us graduate students and talk to us about what we do.

The topics ranged from growing back limbs and mapping the brain, to using math to choose medicines and manipulating the different states of matter.

The kids were really excited for our visit, and kept asking their teachers for the inside scoop on what we did.

After much anticipation, and a little training and practice with Jory Weintraub from the Science & Society Initiative, two groups of 7-12 graduate students armed themselves with photos, animals, plants, and activities related to our work and went to visit these science classes full of eager students.

First-year MGM grad student Tulika Singh (top right) brought cardboard props to show students how antibodies match up with cell receptors.

“The kids really enjoyed it!” said Alex LeMay, middle- and high-school science teacher at the Durham School of the Arts. “They also mentioned that the grad students were really good at explaining ideas in a simple way, while still not talking down to them.”

That’s the ultimate trick with science communication: simplifying what we do, but not talking to people like they’re stupid.

I’m sure you’ve heard the old saying, “dumb it down.” But it really doesn’t work that way. These kids were bright, and often we found them asking questions we’re actively researching in our work. We don’t need to talk down to them, we just need to talk to them without all of the exclusive trappings of science. That was one thing the grad students picked up on too.

“It’s really useful to take a step back from the minutia of our projects and look at the big picture,” said Shannon McNulty, a PhD candidate in Molecular Genetics and Microbiology.

The kids also loved the enthusiasm we showed for our work! That made a big difference in whether they were interested in learning more and asking questions. Take note, fellow scientists: share your enthusiasm for what you do, it’s contagious!

Another thing that worked really well was connecting with the students in a personal way. According to Ms. LeMay, “if the person seemed to like them, they wanted to learn more.” Several of the grad students would ask each student their names and what they were passionate about, or even talk about their own passions outside of their research, and these simple questions allowed the students to connect as people.

There was one girl who shared with me that she didn’t know what she wanted to do when she grew up, and I told her that’s exactly where I was when I was in 8th grade too. We then bonded over our mutual love of baking, and through that interaction she saw herself reflected in me a little bit; making a career in science seem like a possibility, which is especially important for a young girl with a growing interest in science.

Making the rounds in these science classrooms, we learned just as much from the students we spoke to as they did from us. Our lesson being: science outreach is a really rewarding way to spend our time, and who knows, maybe we’ll even spark someone who loves Superman to figure out how to make the first photosynthesizing super-person!

Guest post by Ariana Eily , PhD Candidate in Biology, shown sharing her floating ferns at left.


Acoustic Metamaterials: Designing Plastic to Bend Sound

I recently toured Dr. Steven Cummer’s lab in Duke Engineering to learn about metamaterials, synthetic materials used to manipulate sound and light waves.

Acoustic metamaterials recently bent an incoming sound into the shape of an A, which the researchers called an acoustic hologram.

Acoustic metamaterials recently bent an incoming sound into the shape of an A, which the researchers called an acoustic hologram.

Cummer’s graduate student Abel Xie first showed me the Sound Propagator. It was made of small pieces that looked similar to legos stacked in a wall. These acoustic metamaterials were made of plastic and contained many winding pathways that delay and propagate, or change the direction, of sound waves. The pieces were configured in certain ways so they could design a sound field, a sort of acoustic hologram.

These metamaterials can be configured to direct a 4 kHz sound wave into the shape of a letter ‘A’. The researchers measured the outgoing sound wave using a 2D sweeping microphone that passed back and forth over the A-shaped sound like a lawnmower, moving to the right, then up, then left, etc. The arrangement of metamaterials that reconfigures sound waves is called a lens, because it can focus sound waves to one or more points like a light-bending lens.

Xie then showed me a version of the acoustic metamaterials 10 times smaller that propagated ultrasonic (40 KHz) sound waves. He told me that since 40 kHz was well out of the human range of hearing, it could be a viable option for the wireless non-contact charging of devices like phones. The smaller wave propagator could direct inaudible sound waves to your device, and then another piece of technology called a transfuser would convert acoustic energy into electrical energy.

This structure, with a microphone in the middle, can perform the "cocktail party" trick that humans can -- figuring out where in the room a sound is coming from.

This structure with a microphone in the middle can perform the “cocktail party” trick that humans can — picking out once voice among many.

Now that the waves have been directed, how do we read them? Xie directed me to what looked like a plastic cheesecake in the middle of the table. It was deep and beige and was split into many ‘slices.’ Each slice was further divided into a unique honeycomb of varying depth. The slices were separated from each by glass panes. This directed the soundwaves across the unique honeycomb of each slice towards the lone microphone in the middle. A microphone would be able to recognize where the sound was coming from based on how the wave had changed while it passed over the different honeycomb pattern of each slice.

Xie described the microphone’s ability to distinguish where a sound is coming from and comprehend that specific sound as the “cocktail party effect,” or the human ability to pick out one person speaking in a noisy room. This dense plastic sound sensor is able to distinguish up to three different people speaking and determine where they are in relation to the microphone. He explained how this technology could be miniaturized and implemented in devices like the Amazon Echo to make them more efficient.

Dr. Cummer and Abel Xie’s research is changing the way we think about microphones and sound, and may one day improve all kinds of technology ranging from digital assistants to wirelessly charging your phone.

Frank diLustro

Frank diLustro is a senior at the North Carolina School for Science and Math.


3D-Printable Material Sets Terminator’s Eyes Aglow

Pumpkins just not cutting it for you this year?

If you want a unique, hand-made Halloween decoration – and happen to have access to a 3D printer – Duke graduate student Patrick Flowers has just the project for you: this 3D-printed Terminator head, complete with shining, blood-red eyes.

Flowers, a PhD candidate in Benjamin Wiley’s lab, is not spending his time studying early eighties action flicks or the Governator’s best break-out roles. Instead, he and his labmates are working hard to brew up highly-conductive, copper-based materials that can be 3D printed into multilayer circuits – just like the one powering this Terminator’s glowing LED eyes.

Their latest copper concoction, which they have named “Electrifi,” is about 100 times more conductive than other materials on the market. The team has a taken out a provisional patent on Electrifi and also started a company, named Multi3D, where 3D-printing aficionados can purchase the material to include in their very own devices.

Micro CT scan of the 3D Terminator head

This X-ray view of Terminator’s head, collected with Duke SMIF’s Micro CT scanner, shows the embedded 3D circuit powering his LED eyes.

Creating a conductive, 3D-printable material is a lot trickier than just throwing some copper into a printer and going to town, Flowers said.

“Copper is really conductive originally, but if you try to extrude it out of a hot nozzle like you have to do in order to do this 3D printing, then it quickly loses all its properties,” Flowers said. And conductive materials that can stand the heat, like silver, are too expensive to use on any sort of scale, he added.

To bring the benefits of 3D printing to the world of electric circuits, Flowers and his labmates are experimenting with mixing copper with other materials to help it stay conductive through this extrusion process.

“This lab has a long history of working with copper – copper nanowires, copper particles, copper nanoparticles – so we’ve got a lot of little tricks that we use to maintain the conductivity,” Flowers said.

The team is currently testing the limits of their new material and plans to publish their findings soon. In the meantime, Flowers is busy exploring the other capabilities of Electrifi — outside of plastic android noggins.

“The circuit inside this guy is really simple, but it does show the capabilities of the material: it is embedded, it shows that I can go down, over, up, out, and go to a couple of eyes,” Flowers said. “Now I want to expand on that and show that you can make these really complicated embedded structures that have multiple layers and multiple components, other than just LEDs.”


Kara J. Manke, PhD

Post by Kara Manke

Girls Get An Eye-Opening Introduction to Photonics


Demonstration of the Relationship between Solar Power and Hydrogen Fuel. Image courtesy of DukeEngineering.

Last week I attended the “Exploring Light Technologies” open house hosted by the Fitzpatrick Institute for Photonics, held to honor International “Introduce a Girl to Photonics” Week. It was amazing!

I was particularly enraptured by a MEDx Wireless Technology presentation and demonstration titled “Using Light to Monitor Health and View Health Information.” There were three “stations” with a presenter at each station.

At the first station, the presenter, Julie, discussed how wearable technologies are used in optical heart rate monitoring. For example, a finger pulse oximeter uses light to measure blood oxygen levels and heart rates, and fitness trackers typically contain LED lights in the band. These lights shine into the skin and the devices use algorithms to read the amount of light scattered by the flow of blood, thus measuring heart rate.

At the second station, the presenter, Jackie, spoke about head-mounted displays and their uses. The Google Glass helped inspire the creation of the Microsoft Hololens, a new holographic piece of technology resembling a hybrid of laboratory goggles and a helmet. According to Jackie, the Microsoft Hololens “uses light to generate 3D objects we can see in our environment.”


Using the Microsoft Hololens. Image courtesy of DukeEngineering.

After viewing a video on how the holographic technology worked, I put on the Microsoft Hololens at the demonstration station. The team had set up 3D images of a cat, a dog and a chimpanzee. “Focus the white point of light on the object and make an L-shape with your fingers,” directed Eric, the overseer. “Snap to make the objects move.” With the heavy Hololens pressing down on my nose, I did as he directed. Moving my head moved the point of light. Using either hand to snap made the dog bark, the cat meow and lick its paws, and the chimpanzee eat. Even more interesting was the fact that I could move around the animals and see every angle, even when the objects were in motion. Throughout the day, I saw visitors of all ages with big smiles on their faces, patting and “snapping” at the air.

Applications of the Microsoft Hololens are promising. In the medical field, they can be used to display patient health information or electronic health records in one’s line of sight. In health education, students can view displays of interactive 3D anatomical animations. Architects can use the Hololens to explore buildings. “Imagine learning about Rome in the classroom. Suddenly, you can actually be in Rome, see the architecture, and explore the streets,” Jackie said. “[The Microsoft Hololens] deepens the educational experience.”


Tour of the Facilities. Image courtesy of DukeEngineering.

Throughout the day, I oo-ed and aw-ed at the three floors-worth of research presentations lining the walls. Interesting questions were posed on easy-to-comprehend posters, even for a non-engineer such as myself. The event organizers truly did make sure that all visitors would find at least one presentation to pique their interest. There were photonic displays and demonstrations with topics ranging from art to medicine to photography to energy conservation…you get my point.

Truly an eye-opening experience!

Post by Meg Shiehmeg_shieh_100hed