CSU Pueblo Math Professor Builds “Impossible” Geometric Sculpture to Make Math Accessible

Tom Edgar was only months into his position at CSU Pueblo when he found himself coordinating something different. After 16 years at Pacific Lutheran University in Washington and a stint at the Air Force Academy in Colorado Springs, he was settling into his new role while organizing a community sculpture build with a colleague.

The collaboration came about almost by accident. When Edgar mentioned his new position in Pueblo to his friend Glen Whitney, Whitney said he’d be in Colorado in late September. “I emailed the (math) chair and I said, ‘I’ve got a friend and he likes to do these builds. Do you think it’s something we could do?’” Edgar recalled.

On September 30th, under a bright autumn sun, they constructed an eight-foot sculpture that embodies one of mathematics’ most intriguing unsolved problems. Students joined them, assembling the colorful geometric structure piece by piece.

Building Math You Can Touch

A mathematical build is exactly what it sounds like. Large-scale construction of geometric sculptures that make abstract mathematical concepts physical. Students and faculty work together, fitting polygons and connecting vertices, creating something that exists in three dimensions rather than on a chalkboard.

Whitney has done similar community builds across the country. The hands-on nature makes theoretical problems tangible. People can see where shapes fit together easily and where they resist, where the math works, and where it breaks down.

For this build, participants assembled various geometric shapes, trying different configurations of pentagons, squares, and other regular polygons. The goal was to construct what Whitney called an “impossible solid,” a structure that challenges one of geometry’s unresolved questions.

A Puzzle That Can’t Be Solved

Whitney, who founded the Museum of Mathematics in New York City, and is now a Trustee for the Seattle Universal Math Museum (SUMM), the project’s sponsor, has spent countless hours wrestling with a particular geometric question. Is it possible to create a three-dimensional shape using only regular polygons where two pentagons and one square meet at a single vertex?

The museum itself represents Whitney’s conviction that math should excite people. A former algorithms specialist at Renaissance Technologies, the famously math-loving hedge fund, Whitney raised more than $22 million to open the museum. Barron’s captured his mission in 2011: “Math excites Glen Whitney. He believes it should excite you, too.”

That belief drives these community builds. Whitney wants people to touch mathematics, not just calculate it.

“The whole purpose of the build is to see if we can create a shape that’s going to have a pentagon, a hexagon, and a square meeting at one point,” Whitney explained before the event. The problem is that the angle where they meet is too narrow. Most toys and construction materials aren’t designed to click together at such tight angles.

Making the Abstract Tangible

For Edgar, this uncertainty represents mathematics at its most appealing. His enthusiasm is evident when he talks about the discipline, though he understands the barriers it faces.

If you talk to anybody (they’ll say), ‘I hated math,’” Edgar said. “If you probe longer, it’s because they had some traumatic experience, usually in high school, sometimes in middle school.”

The sculpture represents Edgar’s attempt to change that internal narrative. Instead of presenting mathematics as rules to memorize, he wants to show it as a source of wonder. Something you can touch and build with your hands.

Edgar brings this same philosophy to his YouTube channel, Mathematical Visual Proofs, where he animates classic and newer “proofs without words.” These are typically diagrams that indicate how a theorem could be proved without any written explanation. He often includes animations without narration, only dramatic music, paying homage to the visual proof tradition.

One of his videos explores the Thue-Morse sequence using turtle graphics, showing how a surprising shape eventually arises from what seems like a random pattern. It’s a mathematical beauty that reveals itself slowly, the way the sculpture reveals its impossibility only through careful calculation.

“So much of the time spent on math, as we go through our grade school, is about a method,” Whitney explained. “There’s always a method. It just gives you the impression that for any (logic) question, you just have to find the person who knows the way to do it. And that makes it feel very sterile and dry.”

Close Enough to Fool the Eye

What makes the sculpture particularly intriguing is how close it comes to solving the unsolvable. Whitney used a design that achieves the desired configuration with an error smaller than the width of a human hair.

“The amount that our shapes are off from the mathematical ideal is going to be under 30 microns,” Whitney said. The error rate is less than a hundredth of a percent.

To observers walking past the math building, it appears to solve the pentagon-square problem successfully. Only mathematical calculation can reveal the hidden impossibility.

“I hope that they see that math can be an unexpected source of beauty or curiosity in their lives,” Whitney said. “If a bunch of people just say, ‘Wow, what’s going on there? Oh, there’s more to math than meets the eye,’ then job well done.”

A Monument to Beautiful Uncertainty

The sculpture now stands near the entrance to the math and physics building, challenging viewers to reconsider their relationship with mathematical uncertainty. It will remain here for the near future, visible to students and faculty as they pass through campus.

The structure won’t solve the pentagon-square problem. But it might solve something else. Helping people see mathematics as a source of beauty rather than anxiety. Mystery rather than certainty.

“Beauty and creativity, to me, are among the hallmarks of mathematics,” Whitney reflected,

Whether the sculpture inspires wonder, curiosity, or simply a moment’s pause on the way to class remains to be seen. For Edgar and Whitney, and the students who helped assemble it on that bright September day, the attempt itself represents something essential about why mathematics matters.