The Magic of Math

This professor uses optical illusion to not only challenge your visual understanding, but to also show the magic that is mathematics.

Walking into Kokichi Sugihara’s office feels closer to entering a magic shop than a mathematician’s space. The bookcases lining his workspace house dozens of optical illusions, ranging from 2-D creations that twist one’s perspective around when turned, or 3-D objects that look like a cylinder at first glance… but from another angle appear as a different shape entirely.

Man in office

The distinguished professor emeritus at Meiji University’s Institute for Advanced Study of Mathematical Sciences laughs when asked if they are organized in any particular way, admitting it has gotten a bit messy. “I invented a system of equations that represent all possible 3-D objects,” Sugihara says.

Using that model, I can predict how the human brain behaves when we show 3-D structures. Then I construct it physically, and show it to humans to verify.

Among the illusions is “Triply Ambiguous Objects,” a 2-D image depicting a series of steps. Place a 3-D item on top of these steps though – such as a tiny flag, as Sugihara demonstrates – and see its reflection via a mirror… and suddenly you see a very differently shaped 2-D image.

Add a second mirror, and you see three distinct versions of it at once. Even having the professor break down how the illusion works in person doesn’t completely erase the feeling that one’s mind has been broken.

The Prize

“Triply Ambiguous Objects” won first prize in the 2018 Best Illusion of the Year Contest, awarded by the Neural Correlate Society. 

That marked the third time Sugihara took home the top honor in the past 10 years, alongside several second-place finishes. His work has inspired jewelry pieces, and a selection of his impossible objects are currently on display at the National Palace Museum in Taipei until February 2020. “This summer I’ll visit Las Vegas and give a talk for a professional magician’s conference,” he says with a laugh.

Diagram

Sugihara says it was the attitude of Meiji University that helped make this acclaim and international recognition possible, thanks to an open-minded approach to study and research that other institutions in the country don’t offer. Since joining the school in 2009, he’s been allowed to burrow into the field of optical illusions. That tolerance carries over to the rest of Meiji, making it one of the premier academic destinations in Japan for people to pursue their passions.

Man on stage

Sugihara first encountered optical illusions as an elementary school student, via simple perception-warping line drawings such as the Zollner illusion and the Muller-Lyer illusion. “I was so surprised by them, because of their simplicity,” he says.

“We know these two lines are the same length, but if we add something, it looks like they have different lengths. Even though I know the truth. That unreliability of our eyes and brains surprised me very much.” This, coupled with a life-long interest in the art of M.C. Escher, influenced his work moving forward.

Diagram

Since the early 1980s, he has taught and researched at several universities in Japan, but says his focus there was always on scientific fields such as computer vision. During these stints and in his own time, he developed a personal interest in turning impossible objects — optical illusions that traditionally are two-dimensional but which is interpreted by viewers as three-dimensional — into 3-D constructions.

He couldn’t find the time to really explore them as much as he’d like, “but Meiji University invited me and offered me a position in which I could concentrate on the research of optical illusions,” Sugihara says.

Founded in the late 1800s and boasting one of the best reputations in Japan, Meiji University has four campuses in the Tokyo area. Sugihara says the school offers many strong points, such as an alumni association which makes finding jobs after graduation easier and convenient facilities (including his own location at the Nakano campus, a breezy 4-minute train ride from Shinjuku).

Not long after being accepted into Meiji, Sugihara captured his first Best Illusion of the Year Contest top prize in 2010 with “Magnet-like Slopes,” which uses perspective to make it appear that marbles are moving up an incline as if they were pulled and not pushed. Over the years he’s gone larger with this creation, literally, going as far as to create life-sized versions at ski resorts in Niigata prefecture and even one on the Nakano school grounds.

Besides offering space both physically and through their open-minded approach to research, the approach Meiji’s students bring to the university stand out to Sugihara as especially beneficial. He points to his second win at the Best Illusion of the Year Contest in 2013, which he worked on alongside a colleague and a graduate student. “The students here have an active love of things, it goes beyond just thinking,” he says.

“They can sometimes find something unexpected, and I think that’s how the second illusion of the year was generated.”

Thanks to Meiji University’s progressive academic climate, Sugihara’s exploration of math via illusions has generated valuable findings. He has found that the human brain loves 90 degree angles and that many of his creations use this as a way to play with our perspective. He’s also found that, even when we understand the true shape of an object, a 3-D illusion can still occur. His ambiguous cylinder illusion, with its arrow that always points in one direction – even when spun – highlight this well.

Sugihara has big ambitions for the future, including a dream to develop a theme park built around optical illusions. For now, he’s continuing to show that the impossible is possible, thanks in part to an educational environment allowing one to use unconventional means to solve problems.

“I myself have always felt like mathematics is a very strong tool to create new things,” he says, with the optical illusions surrounding him being his strongest example. “Because the [optical illusions] are mysterious, powerful and beautiful, it’s easy for students to see just how important and useful math is.”

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