My favorite optical illusions are motion illusions: static images that appear to spin, shimmer, and shimmy, like Piet Mondrian’s *Broadway Boogie*–*Woogie* or the psychedelic pinwheels of Japanese psychologist Akiyoshi Kitaoka. So I was pretty pleased to come across a new class of them, known as *fuyuu*, or floating, illusions. I visited their co-creator, mathematician Hitoshi Arai, at the University of Tokyo last month. Here’s one that makes great computer wallpaper. It should look like a languidly flowing river; if not, try moving your head toward or away from the screen.

Arai and his wife, Shinobu Arai, also a mathematician at Tokyo, have an extensive online gallery with commentary in Japanese, as well as an abridged English version and a book (also in Japanese, but you can just look at the pretty pictures).

The Arais approach optical illusions as mathematicians rather than as neuroscientists. Earlier studies relied on Fourier analysis, which decomposes an image (or any set of data) into a family of sine waves representing the constituent frequencies. Low frequencies capture broad humps, while high frequencies correspond to fine details. Yet Fourier analysis takes the image as an indivisible whole; the sinusoids are completely delocalized, stretching across the entire frame. So mathematicians have developed a more general technique, wavelet analysis, that breaks an image into pieces that are localized both in frequency and in space. The Arais use a version of wavelet analysis in which the building blocks have a strictly finite extent, thus representing patterns that visual-processing neurons might realistically respond to. The goal of studying illusions, after all, is to lay bare how our brains form an impression of the world.

The Arais’ technique does not merely dissect illusions, but can generate them, taking an image that looks boringly normal and making subtle changes to color and contrast to fool our brains. Run in reverse, the algorithm can turn an illusion into an ordinary image. The algorithm does not merely turn an illusion on or off, but can dial it in strength. Consider this variant of Fraser’s spiral illusion, which you’d swear is a spiral, but is really a series of concentric circles. The algorithm can either de-illusion it (center) or intensify it (right).

What is more, you can apply the algorithm to *any* image, not just a geometric pattern. If you run a photograph of a natural scene through the algorithm, obscure details pop out. Hitoshi Arai showed me a mountain scene in Japan. After enhancement, the image reveals power lines and pylons that you can barely make out in the original.

The Arais’ work demonstrates the central truth of illusions. They are not a deviation from reality. They are how we view reality. We see things by seeing things.

**Update (21 June 2017)**: A modified version of this post appears at Nautilus magazine.

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