“Trion” may sound like the name of one of the theoretical particles blamed for mucking up operations aboard the Starship Enterprise.
But believe it or not, trions are real — and they may soon play a key role in electronic devices. Duke researchers have for the first time pinned down some of the behaviors of these one-of-a-kind particles, a first step towards putting them to work in electronics.
Three-in-one particles called trions — carrying charge, energy and spin — zoom through special polymer-wrapped carbon nanotubes at room temperature. Credit: Yusong Bai.
Trions are what scientists call “quasiparticles,” bundles of energy, electric charge and spin that zoom around inside semiconductors.
“Trions display unique properties that you won’t be able to find in conventional particles like electrons, holes (positive charges) and excitons (electron-hole pairs that are formed when light interacts with certain materials),” said Yusong Bai, a postdoctoral scholar in the chemistry department at Duke. “Because of their unique properties, trions could be used in new electronics such as photovoltaics, photodetectors, or in spintronics.”
Usually these properties – energy, charge and spin – are carried by separate particles. For example, excitons carry the light energy that powers solar cells, and electrons or holes carry the electric charge that drives electronic devices. But trions are essentially three-in-one particles, combining these elements together into a single entity – hence the “tri” in trion.
A trion is born when a particle called a polaron (top) marries an exciton (middle). Credit: Yusong Bai.
“A trion is this hybrid that involves a charge marrying an exciton to become a uniquely distinct particle,” said Michael Therien, the William R. Kenan, Jr. Professor of Chemistry at Duke. “And the reason why people are excited about trions is because they are a new way to manipulate spin, charge, and the energy of absorbed light, all simultaneously.”
Until recently, scientists hadn’t given trions much attention because they could only be found in semiconductors at extremely low temperatures – around 2 Kelvin, or -271 Celcius. A few years ago, researchers observed trions in carbon nanotubes at room temperature, opening up the potential to use them in real electronic devices.
Bai used a laser probing technique to study how trions behave in carefully engineered and highly uniform carbon nanotubes. He examined basic properties including how they are formed, how fast they move and how long they live.
He was surprised to find that under certain conditions, these unusual particles were actually quite easy to create and control.
“We found these particles are very stable in materials like carbon nanotubes, which can be used in a new generation of electronics,” Bai said. “This study is the first step in understanding how we might take advantage of their unique properties.”
The team published their results Jan. 8 in the Proceedings of the National Academy of Sciences.
Mixing black glitter with oobleck allowed researchers to track the movement of individual cornstarch particles after a sudden impact. A computer program locked onto pieces of glitter and illustrated their motion. Credit: Melody Lim.
What do gelatin and glitter have to do with serious science? For some experiments, a lot! Duke alumna Melody Lim used jiggly Jell-O and a just a pinch of glitter to solve a scientific mystery about the curious goo many like to call oobleck.
To the uninitiated, oobleck is almost magic. The simple mixture of cornstarch and water feels solid if you squeeze it, but moments later runs through your fingers like water. You can dance across a bathtub full of oobleck, but stand still for too long and you will be sucked into a goopy mess. Not surprisingly, the stuff is a Youtubefavorite.
Oobleck is an example of what scientists call a non-Newtonian fluid, a liquid whose viscosity – how easily it changes shape and flows – depends upon the force that is applied. But exactly how it is that this material switches from solid to liquid and back again has remained a mystery to scientists.
This blogger mixed up a batch of jello to see the photoelastic effect for herself. When viewed with polarized light – from an iPhone screen and a circular polarizer – the jello changes color when squeezed.
“Water is simple to understand, and so is cornstarch,” said Lim, ’16, who is currently a graduate student at the University of Chicago. “However, a combination of the two produces this ‘liquid’ that ripples and flows, solidifies beneath your feet if you run on it, then turns back into a liquid if you stop running and stand still. I wanted to know why.”
The question beguiling scientists was whether sudden impact causes the cornstarch particles to “jam” into a solid like cement, or whether the suspension remains liquid but simply moves too slowly for its liquid-like properties to be apparent — similar to what happens if you skip a rock off the surface of a lake.
“There are these two opposing pictures,” said Robert Behringer, James B. Duke Professor of Physics at Duke. “Either you squish the material and turn it into cement temporarily, or you simply transmit the stress from the impactor straight to the boundary.”
Lim did two sets of experiments to find out which way oobleck works. In one experiment, she mixed black glitter into a transparent channel filled with oobleck, and then used a high-speed camera to watch how the material responded to the impact. The glitter let her track the motion of individual particles after the disc hit.
The photoelastic effect in gelatin.
Her video shows that the particles near the impact site jam and become solid, forming what the researchers call a “mass shock” wave that travels slowly through the suspension.
In a second set of experiments, Lim placed the oobleck in a container lined with gelatin, the main ingredient in Jell-O – besides sugar and food dye, of course. Gelatin is what is called a photoelastic material, which means that applying pressure bends light that travels through it, like a prism.
“Next time you eat Jell-O, get out your sunglasses and get somebody else’s sunglasses and look between them,” Behringer said. “Because if you give it a shake you should see all these stress patterns bouncing around.”
After the metal disc hit the oobleck, the gelatin let Lim see how fast the resulting pressure wave traveled through the material and reached the boundary.
The researchers poured oobleck into a clear container lined with gelatin, a material that bends light when a pressure is applied to it. They saw that the force of a sudden impact is rapidly transmitted through the oobleck and to the boundary with the gelatin. Credit: Melody Lim.
They found that when the impact is sudden, the pressure wave traveled to the gelatin boundary faster than the “mass shock” wave. This means that the reason oobleck appears solid after a sudden impact is because the force of the collision is quickly transmitted to a solid boundary.
“If you are running across the water, that actually puts you into an impact velocity range where the pressure wave is significantly faster than the mass shock,” Behringer said. “Whereas if you try to walk across it, the impact speeds are slow, and the system actually doesn’t have the ability to transport the momentum quickly through the material and so you just sink in.”
“If you’d told me when I started that I would line a narrow container with Jell-o, add cornstarch, water, and black glitter, drop a piece of metal on it, then publish a paper on the results, I would have laughed at you,” Lim said.
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.
This year’s listing of the world’s most-cited researchers is out from Clarivate Analytics, and Duke has 34 names on the list of 3,400 researchers from 21 fields of science and social science.
Having your publication cited in a paper written by other scientists is a sign that your work is significant and advances the field. The highly-cited list includes the top 1 percent of scientists cited by others in the years 2005 to 2015.
“Citations by other scientists are an acknowledgement that the work our faculty has published is significant to their fields,” said Vice Provost for Research Lawrence Carin. “In research, we often talk about ‘standing on the shoulders of giants,’ as a way to explain how one person’s work builds on another’s. For Duke to have so many of our people in the top 1 percent indicates that they are leading their fields and their work is indeed something upon which others can build.”
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.
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.”
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.”
Force chains erupt as an “intruder” is yanked from beneath a pile of circular discs, which are designed to simulate a granular material. The entire process takes less than one second. Credit: Yue Zhang, Duke University.
You can easily walk across the sand on a beach. But step into a ball pit, and chances are you’ll fall right through.
Sand and ball pits are both granular materials, or materials that are made of collections of much smaller particles or grains. Depending on their density and how much force they experience, granular materials sometimes behave like liquids — something you fall right through — and sometimes “jam” into solids, making them something you can stand on.
“In some cases, these little particles have figured out how to actually form solid-like structures,” said Robert P. Behringer, James B. Duke Professor of Physics. “So why don’t they always just go squirting sideways and relax all the stress?”
Physicists do not yet understand exactly when and how jamming occurs, but Behringer’s team at Duke is on the case. The group squishes, stretches, hits, and pulls at granular materials to get a better picture of how and why they behave like they do. The team recently presented a whopping 10 papers at the 2017 Powders and Grains Conference, which occurred from July 3-7, 2017 in Montpellier, France.
Many of these studies use one of the lab’s favorite techniques, which is to create granular materials from small transparent discs that are about half an inch to an inch in diameter. These discs are made of a material which, thanks to the special way it interacts with light, changes color when squished. This effect allows the team to watch how the stress within the material changes as various forces are applied.
As the wheels turn, shear strain between the discs creates a dense web of inter-particle forces. Credit: Yiqiu Zhao, Duke University.
In one experiment, graduate student Yue Zhang used a high-speed camera to catch the stress patterns as a ball on a string is yanked out from a pile of these discs. In the video, the ball first appears to be stuck under the pile, and then suddenly gives way after enough force is applied — not unlike what you might experience pulling a tent stake out of the ground, or opening the lid on a pesky pickle jar.
“The amusing thing is that you start trying to pull, you add more force, you add more force, and then at some point you pull so hard that you hit yourself in the head,” Behringer said.
The team was surprised to find that the stress patterns created by the ball, which Behringer says look “like hair all standing on end,” are almost identical to the stress of impact, only in reverse.
“What you see is even though you are just gradually gradually pulling harder and harder, the final dynamics are in some sense the same dynamics that you get on impact,” Behringer said.
In another experiment, the team examined what happens in granular materials under shear strain, which is similar to the force your fingers exert on one another when you rub them together.
Graduate student Yiqiu Zhao placed hundreds of these discs onto a circular platform made of a series of flat, concentric rings, each of which is controlled by a separate motor. As the rings turn at different speeds, the particles rub against one another, creating a shear stress.
Beneath the small transparent discs lie a series of concentric wheels, each attached to its own motor. By turning these platforms at different speeds, Yiqiu Zhao can observe how shear strain affects the discs.
“We have about twenty stepper motors here, so that we can rotate all the rings to apply a shear not only from the outside boundary, but also from everywhere inside the bulk of the material,” Zhao said. This ensures that each particle in the circle experiences a similar amount of shear.
“One of the key intents of this new experiment was to find a way that we could shear until the cows come home,” Behringer said. “And if it takes a hundred times more shear than I could get with older experiments, well we’ll get it.”
As the rings turn, videos of the material show forces snaking out from the inner circle like lightning bolts. They found that by applying enough shear, it is possible to make the material like a solid at much lower densities than had been seen before.
“You can actually turn a granular fluid into a granular solid by shearing it,” Behringer said. “So it is like you don’t put your ice in the refrigerator, you put it in one of these trays and you shear the tray and it turns into ice.”
The Duke team used their 3D virus cam to spy on this small lentivirus as it danced through a salt water solution.
Before germs like viruses can make you sick, they first have to make a landing on one of your cells — Mars Rover style — and then punch their way inside.
A team of physical chemists at Duke is building a microscope so powerful that it can spot these minuscule germs in the act of infection.
The team has created a new 3D “virus cam” that can spy on tiny viral germs as they wriggle around in real time. In a video caught by the microscope, you can watch as a lentivirus bounces and jitters through an area a little wider that a human hair.
Next, they hope to develop this technique into a multi-functional “magic camera” that will let them see not only the dancing viruses, but also the much larger cell membranes they are trying breech.
“Really what we are trying to investigate is the very first contacts of the virus with the cell surface — how it calls receptors, and how it sheds its envelope,” said group leader Kevin Welsher, assistant professor of chemistry at Duke. “We want to watch that process in real time, and to do that, we need to be able to lock on to the virus right from the first moment.”
To test out the microscope, the team attached a fluorescent bead to a motion controller and tracked its movements as it spelled out a familiar name.
This isn’t the first microscope that can track real-time, 3D motions of individual particles. In fact, as a postdoctoral researcher at Princeton, Welsher built an earlier model and used it to track a bright fluorescent bead as it gets stuck in the membrane of a cell.
But the new virus cam, built by Duke postdoc Shangguo Hou, can track particles that are faster-moving and dimmer compared to earlier microscopes. “We were trying to overcome a speed limit, and we were trying to do so with the fewest number of photons collected possible,” Welsher said.
The ability to spot dimmer particles is particularly important when tracking viruses, Welsher said. These small bundles of proteins and DNA don’t naturally give off any light, so to see them under a microscope, researchers first have to stick something fluorescent on them. But many bright fluorescent particles, such as quantum dots, are pretty big compared to the size of most viruses. Attaching one is kind of like sticking a baseball onto a basketball – there is a good chance it might affect how the virus moves and interacts with cells.
The new microscope can detect the fainter light given off by much smaller fluorescent proteins – which, if the virus is a basketball, are approximately the size of a pea. Fluorescent proteins can also be inserted to the viral genome, which allows them to be incorporated into the virus as it is being assembled.
“That was the big move for us,” Welsher said, “We didn’t need to use a quantum dot, we didn’t need to use an artificial fluorescent bead. As long as the fluorescent protein was somewhere in the virus, we could spot it.” To create their viral video, Welsher’s team enlisted Duke’s Viral Vector Core to insert a yellow fluorescent protein into their lentivirus.
Now that the virus-tracking microscope is up-and-running, the team is busy building a laser scanning microscope that will also be able to map cell surfaces nearby. “So if we know where the particle is, we can also image around it and reconstruct where the particle is going,” Welsher said. “We hope to adapt this to capturing viral infection in real time.”
A simplified version of Sara Haravifard’s recipe for new superconductors, by the National High Magnetic Field Laboratory
Duke physics professor Sara Haravifard is mixing, cooking, squishing and freezing “frustrated” magnetic crystals in search of the origins of superconductivity.
Superconductivity refers to the ability of electrons to travel endlessly through certain materials, called superconductors, without adding any energy — think of a car that can drive forever with no gas or electricity. And just the way gas-less, charge-less cars would make travel vastly cheaper, superconductivity has the potential to revolutionize electronics and energy industry.
But superconductors are extremely rare, and are usually only superconductive at extremely cold temperatures — too cold for any but a few highly specialized applications. A few “high-temperature” superconductors have been discovered, but scientists are still flummoxed at why and how these superconductors exist.
Haravifard hopes that her magnet experiments will reveal the origins of high-temperature superconductivity so that researchers can design and build new materials with this amazing property. In the process, her team may also discover materials that are useful in quantum computing, or even entirely new states of matter.
Professor Maiken H. Mikkelsen uses optics to tailor the properties of materials, making them stronger and lighter than anything found in nature. This distinguished researcher also teaches my ECE 340: Optics and Photonics course, giving me a wonderful opportunity to ask about her research and experience at the Photonics Asia conference held in China in October 2016.
Below is an edited transcript of our interview.
Q: What sparked your interest in optics and photonics?
I was really excited about doing hands-on research where you could actually probe nanoscale and quantum phenomena from optical experiments. I started out looking into condensed matter and quantum information science and currently observe delicately designed nanostructures. Optics is, to some extent, a tool to modify the properties of materials.
Q: What does your lab do and how do students contribute?
During the last few years, my students and I have been structuring materials on the nanoscale to modify the local electromagnetic environment, which makes these materials behave in new ways. Students play a key role in all aspects of the research, from nanofabrication to performing optical experiments and presenting the results to the scientific community at conferences all over the world. The lab uses tiny metal structures to concentrate the incoming electromagnetic field of light to very small volumes — a research area known as plasmonics. Placing other materials in the near field of this modified environment causes the electrons to behave completely differently.
Platform based on metal nanostructures that allows the lab to dramatically enhance the radiative properties of emitters and other materials.
By controlling how these electrons behave and modifying the geometry of the material, we can gain a deeper understanding of the light-matter interactions. Combining these techniques with our optical experiments shows modifications to material properties that are much stronger than has been seen before. It’s been very exciting!
Q: And this research is what you presented at the Photonics Asia conference?
Yes. With this knowledge, we can enhance the properties of materials significantly, which in the future could lead to ultra-fast and much better LEDs, more efficient photodetectors, or more efficient solar cells and sensors. In Beijing, China, I gave an overview of this research at the leading meeting for the photonics and optics industries in Asia, as well as several other conferences and universities. It was very fulfilling to see how the research I do in a dark lab actually gets noticed around the world. It is always deeply inspiring to learn about recent research breakthroughs from other research groups.
Q: What is the main purpose of trying to find these improved materials?
I am motivated by furthering our fundamental understanding, such as how do light and matter interact when we get to really small scales and how this interaction can be leveraged to achieve useful properties. I believe you often achieve the biggest technological breakthroughs when you’re not trying to solve one particular problem, but creating new materials that could lay the groundwork for a wide range of new technologies. For example, semiconductor materials, with a set of properties that are found naturally, are the cornerstone of most modern technologies. But if you imagine that you now have an entirely new set of building blocks with tailored properties instead, we could revolutionize a lot of different technologies down the road.
The Mikkelsen Research Group. Back row, left to right: Qixin Shen, Andrew Traverso, Maiken Mikkelsen, Guoce Yang, Jon Stewart, Andrew Boyce. Front row, left to right: Wade Wilson, Daniela Cruz, Jiani Huang, Tamra Nebabu.
By improving or completely changing the fabrication technique of these light-matter interactions, new properties begin to emerge. Generally, there’s always a big desire to have something that’s lighter, smaller, more efficient and more flexible. One of the applications we’re targeting with this research is ultrafast LEDs. While future devices might not use this exact approach, the underlying physics will be crucial.
About a year ago, Facebook contacted me and was interested in utilizing our research for omnidirectional detectors that could be ultrafast and detect signals from a large range of incidence angles. This has led to a fruitful collaboration and is one example of how fundamental research can have applications in a wide range of areas — some that you may not even have imagined when you started!
Q: What would be your advice to young researchers still trying to decide a career path for themselves or those interested in optics and photonics?
What really helped me was starting to do undergraduate research. I listened to talks by different faculty, asked them to do undergraduate research, and worked on a volunteer basis in their labs. I think that’s really a great way to see if you’re interested in research — use the amazing opportunities both at Duke and around the country. Doing research requires a lot of patience, but I think no two days are the same; there’s always a lot of creativity involved while troubleshooting new problems. After all, if it was easy or if we knew how to do it, it would have already been done. But it hasn’t, so we have to figure it out — I think that is a lot of fun. Doing internships in optics and photonics companies is also another option to learn more about research and development in the industry. Get as many experiences as possible and give things a chance!
Professor Mikkelsen is best known for the first demonstration of nondestructive readout of a single electron spin, ultrafast manipulation of a single spin using all-optical techniques, and extreme radiative decay engineering using nanoantennas.
Mikkelsen has received numerous accolades, including the Cottrell Scholar Award, the Maria Goeppert Mayer Award, and a “triple crown” of Young Investigator Awards from the Air Force, Army and Navy. Her work has been published in the journals Science, Nature Photonics, and NaturePhysics, to name a few. Professor Mikkelsen enjoys hiking, gardening, playing tennis, and traveling in her free time.
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.
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.