The Perfect Valentine? A Math Formula


Meet Süss, a math widget after your own heart. (You can also visit the widget on its website here, which you might want to do if you’re reading this on a smartphone.)

Like many geometric figures, a heart can be captured in all its curvaceous glory by a single algebraic equation. The equation for a sphere looks simple enough: x²+y²+z²=1. A heart is something more complex:

(x²+((1+b)y)²+z²-1)³-x²z³-ay²z³=0

Süss — German for “sweet” — is an interactive widget that allows you to tweak the algebra and customize the heart to your souls’s delight. It was created for Valentine’s Day by Imaginary, a nonprofit organization in Berlin that designs open-source mathematics programs and exhibitions.

You can stretch and squeeze the heart by moving the two left-most sliders, which change the “a” and “b” parameters; the right-most slider zooms in and out. Better yet, canoodle directly with Süss’s equation and engage in the dialectical interplay between algebra and geometry. (Change that final z³ to a z² to see the heart in its underwear.)

In the 17th century the French mathematician and philosopher René Descartes built a bridge between the algebraic and geometric realms when he devised the Cartesian system of coordinates. (He also classified six primitive passions: wonder and love, hatred and desire, sadness and joy.)

Granted, some of the entities in the gallery do not seem so singular — “Sphäre,” for instance. But given the right mathematical techniques, Dr. Hauser said, he could squeeze that sphere down into a point. “Or I could sit on it,” he added.

The process of identifying intriguing algebraic surfaces that possessed singularities sometimes took Dr. Hauser and his students days. More often than not the results were incompatible: attractive equations gave rise to shapes with scant mathematical appeal. Dr. Hauser was matchmaking, essentially.

“Sometimes you play completely on the algebraic side,” he said. “You plug in an equation and discover what the geometric object looks like. And sometimes you do the opposite.”

The opposite — reverse engineering, having a figure in mind and then finding the equation that generates it — is much more difficult, almost impossibly so. Nonetheless, ten years ago Valentina Galata, then a high-school student, saw a poster of the Süss heart during a math-class excursion to an exhibit featuring Surfer, rendering software made by Imaginary that allows users to experiment in creating algebraic surfaces. Inspired, Galata created a collection of still lifes such as “Cherries” and “Spoon.”

She did not find the task impossible. “Challenging is a better word for it,” said Ms. Galata, now a Ph.D. student in bioinformatics at Saarland University, in Germany. “It was fun to have a tool, using formulas to approximate shapes you have in your head, rather than just drawing on paper.”

Dr. Hauser, a purist, is not fond of rendering real-life objects. In fact, he is not particularly enamored of Süss. Too many people loved it too much: It became the poster heart for algebraic surfaces — on thousands of actual posters in hundreds of Germans schools, and featured in an installation for a shopping-center car park.

“It’s just kitsch,” he said. “Mathematically, it is not so interesting.

Sebastian Gann, one of Dr. Hauser’s former Ph.D. students on the Gallery of Singularities project, still finds Süss more appealing.

“The intrinsic fascination remains,” he said. “Looking at the image of an emotionally charged symbol, and knowing that the particular shape is defined by some sober maths. How do you perceive it?”

Dr. Hauser prefers “Herz” (German for “heart”), which looks a bit more like an actual human heart:

Herz is alluring for what’s not there, he said: “How do you create a hole in a surface when you choose an equation?”

One of the bigger questions within algebraic geometry is how to “resolve” singularities that is, how to get rid of them. Mathematically, resolving a singularity means smoothing over the problematic peaks in a surface, but this often requires jumping to a higher dimension.

Alas, in resolving matters of the heart, that strategy seldom works.




Sahred From Source link Technology

Leave a Reply

Your email address will not be published. Required fields are marked *