The National Museum of Mathematics, which opened in Manhattan in December, doesn’t have a logo. It has an infinite family of logos. And the logos the museum uses for official business are not created by design professionals. They’re designed by the museum’s visitors. The logo is itself an exhibit in the museum.

The museum’s unique meta-logo was conceived and implemented at Wolfram Research. When I say “implemented,” I don’t mean just “calculated” or “rendered,” but actually “programmed.” This is a logo that requires an implementation.

The story of the logo begins in 2009 when Stephen Wolfram joined the museum’s Board of Trustees. Stephen volunteered the services of our award-winning design department to design a logo for the museum. Art director Jeremy Davis and designer Anastasia Tumanova developed an initial color palette, and several variations for the typography and logo form. These were among some of the early concepts:

As nice as many of these ideas are, the group decided that the logo form itself really had to have a more technical connection to math, so the design team called in my colleague Ed Pegg and me to help contribute. Among the many concepts that I explored was one based on one of the very first tests of Mathematica‘s Manipulate function that I ever wrote, and one of my first contributions to the Demonstrations Project: an exploration of the patterns created by symmetric arrangements of letterforms.

My idea was to make a logo mark from a mathematical symbol like ∑ or π repeated with some sort of symmetry. But the question was, which one symbol and which one symmetry would best represent the museum? I realized that I couldn’t find the ideal configuration without seeing how the mark worked in the context of the logotype. So I updated my simple Demonstration so that I could see the logotype combined with the mark as I explored.

Using the logo explorer, I found so many interesting marks that it was nearly impossible to home in on any one of them. It was much more fun to play with the possibilities than to chose one:

While I was avoiding committing to any particular logo, it occurred to me that the museum visitors ought to have as much fun exploring as I was. And that the museum shouldn’t have a logo, but an infinite family of logos. And furthermore: the logo should be an exhibit in the museum, and the logos the museum actually uses should be designed by its visitors!

I pitched that idea to the museum in a proposal that outlined the concept and its possibilities. The museum’s reaction was—understandably—something like, “Oh, that’s, um, … interesting,” in the same way you would tell your aunt that her gift of a hand-knit nose warmer is “interesting.” The idea of not having a logo, let alone letting visitors design logos, was just too out there. But after a few days of discussion they eventually warmed to and embraced the concept. I transformed my initial logo explorer into a production-quality logo designer that the museum could use to produce its logos, and Anastasia drew up a style guide for the logos’ use.

Some of the first official instances of the logo that I saw were on T-shirts, like these held by members of the cast of NCIS Los Angeles:

Soon the logo started appearing in various incarnations on the museum’s web pages:

And museum associates designed personal logos for their business cards:

In August of 2011, I got word that the logo was approved as an opening exhibit in the museum. Thus began the work of translating the logo design software I had created for the museum into an exhibit. As is often the case in software development, 90% of the effort went into dealing with 10% of the functionality. That troublesome 10% was the overlap issue.

It takes a trivial snippet of Mathematica code to render a symmetric arrangement of characters:

But if the characters are not all the same color, naively rendering the characters in a back-to-front stack breaks the symmetry of the mark. In this example, the frontmost yellow π should tuck underneath the backmost red π so that every character has exactly the same relationship to the characters preceding and following it.

The situation can be even more complex when a character’s “limbs” overlap in different ways. It takes some surprisingly sophisticated computational geometry programming to analyze the regions of overlap in a given configuration and figure out how to render them in a consistent way.

The approach I took with the MOMATH logo was suggested by Jade Vinson, a friend of the museum who offered helpful advice. If you could reach into your screen and grab hold of a logo, you’d realize that what you see is a two-dimensional projection of a three-dimensional object.

Each instance of the mathematical symbol is projected onto one of eight intertwined helicoids, screw-like helical surfaces. Projecting that structure along the axis of the helicoid gives the proper overlap. In theory, that is. In practice, it’s not quite that simple.

One revolution of a helicoid has a discontinuity—an abrupt jump—where the surface wraps around on itself. If that discontinuity passes through an area where two or more characters intersect, they’ll appear in projection to intersect, as if slotted together, rather than overlap. Getting the correct appearance requires analyzing the regions of intersection of all pairs of motifs to find a path from the center of rotation to the outside of the configuration that avoids intersections. The discontinuity must be positioned to follow that path.

When I had all the algorithmic pieces in place, the work began of coupling the software with the exhibit hardware created by the museum’s fabricators, Moey Inc. That involved a fair amount of restructuring to convert my design software, a big Manipulate, into software that responds to the low-level serial events generated by the hardware’s controllers. Aided by an advance team at the museum, colleague Michael Sollami, we completed the exhibit just in time for the grand opening on 12/12/12.

The logo is the first exhibit you see upon entering the museum: a sleek black console brimming with joysticks, knobs, and buttons that you can use to explore logo designs. Visitors can designate one of their designs as “their” logo, which then follows them on electronic signage throughout the museum, triggered by an RFID tag that they receive upon entry. On the way out, they can have a one-of-a-kind T-shirt printed with their logo in the museum’s gift shop. And maybe—just maybe—their logo will one day appear on the museum’s letterhead.