In his landmark talk at MoMath, Jim Simons recounts one of the most thrilling breakthroughs in modern mathematics. It is the story of how he, along with a few brilliant colleagues, discovered a “crack” in the universe of geometry—proving that in higher dimensions, the most efficient shapes aren’t always smooth.
1. The Starting Point: Nature’s Obsession with Efficiency
To understand the Simons Cone, you first have to understand how nature behaves. Simons explains this through the Plateau Problem, named after a scientist who played with soap films.
If you take a wire frame and dip it into soapy water, the soap film that forms is a “minimal surface.” It stretches across the frame using the absolute minimum amount of area possible. For hundreds of years, mathematicians believed these efficient surfaces were always “smooth”—meaning they were perfectly rounded, like a ball or a wave, with no sharp corners or jagged spikes.
2. The Great Debate: Does “Smoothness” Last Forever?
Simons explains that in the world we can see (3D), these soap films are always smooth. Mathematicians eventually proved that this rule holds true even as you move into invisible dimensions:
- In 2D, 3D, 4D, 5D, and 6D, every “minimal surface” is perfectly smooth.
- Finally, it was proven true for the 7th dimension as well.
At this point in history, most mathematicians assumed this was a universal law: no matter how many dimensions you have, the most efficient shape will always be smooth.
3. The Breakthrough: The “Simons Cone”
Jim Simons decided to look at what happened when you stepped into the 8th dimension. He discovered something that shocked the mathematical world. He found a specific shape—now known as the Simons Cone—that was perfectly efficient (it used the least possible area) but it was not smooth.
Instead of being rounded, this shape had a “singularity” right in the middle. Imagine two infinitely sharp cones meeting perfectly at their tips.
Jim Simons’ Quote: > “It turned out that seven was the magic number. Up to seven dimensions, the area-minimizing variety is smooth. But in eight dimensions, there’s a thing called the Simons Cone… it has a singularity at the origin, and it is area-minimizing.” [00:05:10]
4. Why This Discovery Was So Special
This wasn’t just a “math fact.” It was a counterexample that destroyed a 50-year-old assumption. Here is why it matters:
- The “Jagged” Truth: Simons proved that as the universe gets more complex (more dimensions/variables), “efficiency” no longer requires “smoothness.” In high-dimensional space, the most perfect, most efficient shape can actually be sharp and jagged.
- The Phase Transition: He discovered a “cliff” in mathematics. Everything works one way up to 7 dimensions, but at the 8th dimension, the rules of the game completely change.
- A New Tool for Science: This discovery gave mathematicians and physicists a new way to study “singularities”—points where the math “breaks” or becomes infinite. This ended up being crucial for things like Black Hole physics and String Theory.
5. Simons’ Interpretation: Beauty in the Breakdown
Simons reflects on this as a moment of pure, aesthetic discovery. He wasn’t trying to make money or crack a code; he was looking for a beautiful truth in geometry. He describes the excitement of finding that “seven was the magic number,” noting that nature has these strange, hidden boundaries.
He interprets this discovery as a lesson in not following the pack. While others were trying to prove smoothness lasted forever, Simons looked where no one else was looking and found the exact spot where the theory fell apart.
Summary: The Legacy of the Cone
The Simons Cone remains one of the most famous objects in geometry. It represents the moment we realized that high-dimensional space is much “weirder” than we ever imagined. For Simons, it was the ultimate proof that if you dig deep enough into the math, you will find structures that are both terrifyingly complex and undeniably beautiful.
Source Video: Jim Simons at MoMath: My Life in Mathematics
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