Seeing Math: Transforming Mathematical Equations into Art

03 January 2013 Written by  Jody Hansen
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Technology is intricately intertwined into our daily lives, and yet most of us barely grasp the concepts that underlie the electronic devices that we have come to rely on. Underneath the plastic casing of our cell phones, computers and TVs are layers of complicated math and physics that direct the inner workings of our gadgets.

James Nagel, Ph.D., a research scientist at Terahertz Device Corporation based in Salt Lake City, was not content with taking gadgets for granted. As an undergraduate at Brigham Young University, he wondered how deceptively mundane materials, such as ceramics, plastics, and metals, convert electrical signals into images and sounds. His curiosity eventually led him to electromagnetism, which governs the forces between electrically charged particles. For example, when current flows through a wire, it generates a magnetic field, creating something new that combines properties of both magnetic and electric fields.

During the course of his investigations as a graduate student at the University of Utah, Nagel began to literally illustrate mathematical equations that govern the cryptic theories of electromagnetism. Creating the images helped Nagel and others to understand math through visualization. “It’s very easy to write down an equation but it’s very difficult to picture what that means at the end of the day,” he says. However, it doesn’t take a rocket scientist to observe that the brightly colored images express an aesthetically pleasing mathematical solution.

Take the case of the image Two Right Circular Rods of Opposite Charge, which resembles two blue eyes bulging through a red and yellow mask. It depicts a dipole, electrical fields distributed between positively and negatively charged poles, which in this image look like dark pupils within the “eyes”. The cool-colored electric fields that make up the steely blue eyes are weaker than those represented by the hot colors immediately surrounding them. Arrows, demonstrating directions of electric fields, erupt from the dark blue negatively charged “pupil” and terminate in the positively charged one.

“Human beings are very visual by nature and if you can visualize something, you have a lot more intuitive grasp then you would just reading it on paper,” says Nagel.

Originally devised as a teaching tool, today he uses his mathematical artwork to visualize answers to technological puzzles for his job as a research scientist. Nagel’s most recent abstract conglomeration of blue, red, and green hues will improve the efficiency of infra-red light sources, used in devices such as night vision goggles. The image is a visual representation of a calculation that allows the maximal amount of light to be used for its purpose, in this case seeing at night, rather than much of it being wasted as light scattered throughout the inside of the device. The mathematical information will guide engineers in manipulating surfaces to maximize light extraction.

“What I’m trying to do is really capture a very complicated mathematical story into one little image,” says Nagel. Nagel’s colorful orbs, waves, and ripples are arresting in their own right, but perhaps what is most striking is that it has practical applications for our increasingly technological world.

The Hertzian Electric Dipole






 Surface Plasmon Polariton





 Beam Forming by a Six-Element Dipole Array






Images (James Nagel):

The Hertzian Electric Dipole

Surface Plasmon Polariton

Beam Forming by a Six-Element Dipole Array




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