As a mathematician, I would claim that math at its best is exactly about riffing and jamming on shared fundamentals and adding your own insights. But it is ofte

As a mathematician, I would claim that math at its best is exactly about riffing and jamming on shared fundamentals and adding your own insights. But it is often misunderstood what these fundamentals are: not the proofs or the formulas, but the elements of logical reasoning that make it possible to say with absolute certainty that given some conditions, some statement must be absolutely true. A riff on a proof in geometry isn't changing a few of the letters around to see if still works - it might instead be about changing the assumptions. In plane geometry, the angles of a triangle sum to 180 degrees. But what if we are not in the plane, but on the surface of a sphere, such as the earth? Does a triangle connecting the north pole to two points on the equator still have the sum 180 degrees? If not, can we prove something else about it? In the context of the original article, it might be something like "Can we use any parts of our proof about pentagons (5-cycles) for some other shapes? What about hexagons (6-cycles)? Or is there even any insight we can generalize so that it becomes a statement about cycles of any length?" Sadly, this is way too seldom the way math is taught. I didn't enjoy the endless calculations of long division in grade school, or the memorization of different tricks for solving trigonometric integrals in college, either. For a famous mathematician quote, how about Paul Erdös concept of "a proof from the Book" - said about math proofs that are so perfect and clear that they must be in God's celestial collection. Often, there is more than one way to prove something true - checking every example by brute force would be the most extreme - but sometimes you can discover an elegant argument that just convinces everyone who reads it that it simply must be true. That could also be thought of as riffing on a proof: Ok, you convinced me that this is true, after many boring pages of calculations - can I find a simpler way to convince myself of the same thing, and improve both our understandings?" Quanta magazine had a nice article about this as well: https://www.quantamagazine.org/gunter-ziegler-and-martin-aigner-seek-gods-perfect-math-proofs-20180319/ https://www.quantamagazine.org/gunter-ziegler-and-martin-aig...