It would help if the article included at least one real world application in layman's terms.
Not all work has to have a real world application ($$$). It would help if the you included at least one real world application of writing your comment.
Around 13-14 years old, I got interested in building mods for Quake. When I decided to create a bot, I finally understood at least how trigonometry was useful. I won the annual mathematics prize at school that year, and I'm sure it was only because real-world use had gotten me interested.
The only way I know of is to just get after school tuition and ignore the school. This is difficult though - why do I have to do maths after school? no one else does.
But I remember my peers asking for applications from math teachers when we were in school, but I never saw them ask the same from art teachers. Somehow, everyone understood that drawing and coloring were just for pleasure and stroking our aesthetic sensibilities, and an application was not needed. But people rarely think of math the same way. Yet, it's all pattern recognition and creation.
To phrase it another way: practicing scales can be boring. But playing a walking bassline is a whole lot more similar to scales than it is anything else, and one of the most fun things to do with an instrument is to jam with other people using the fundamentals you all share.
I never felt the same way with math, because I never felt like math allowed me to put anything unique and of my own into the mix. Learning the same proof from a book that a million other kids taking geometry learned was much less interesting to me than transcribing a solo some musician played so that I could try to riff on the style they put into the world.
Sure, yeah, there's rules in music theory too. You're expected to learn them, and get graded on them in tests if you study music academically. But I don't know of any famous mathematician who ever said something like the various Duke Ellington quotes around "if it sounds good, it IS good". In the art world, similar riffs on an existing idea are everything. In math, they're just... wrong. Or at least thats the understanding I had as a student which led me to only care about the parts of math that seemed to be directly applicable to me, or really intuitive.
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-aig...
> I didn't enjoy the endless calculations of long division in grade school...
Sure, everyone has their own cup of coffee. But drummers, for instance, practice the same beats over and over again for dozens of hours to perfect them. Not very different from doing long division over and over again.
Imagine if you started to learn art by doing shading drills, or practising how to use a paint brush by making the same stroke over and over again, but never actually painting a picture.