“The most common instance of beauty in mathematics is a brilliant step in an otherwise undistinguished proof.”
— Gian-Carlo Rota
“[Mathematics] is the work of the human mind, which is destined rather to study than to know, to seek the truth rather than to find it.”
— Évariste Galois | Mathematician, who by age 20, solved the 350 year old problem of describing which polynomials are solvable by radicals (turns out there are only nice general formulas [like the quadratic formula] up to degree 4)
The Pythagorean theorem can be extended to any number of dimensions. In 2D space, the Pythagorean theorem gives us the length of the diagonal of a rectangle. It turns out, a simple modification (adding another square term) to the formula gives the diagonal of a rectangular prism. In the same way, the formula can be further extended to apply to 4 dimensional situations, and give the diagonal of a hypercube (4D cube) or tesseract (yes that’s where Marvel got the name for that glowing blue cube, one of the infinity stones). Keeping with the pattern, the Pythagorean theorem can be generalized to any arbitrary whole number dimension to apply to finding diagonals of n-dimensional rectangle analogs.
Okay, but is this 4th dimension, hypercube stuff even real? IDK, is any math real? It doesn’t matter. But I can tell you this type of mathematics is useful. For example, these formulas are used to calculate lengths of vectors and are used in statistics to calculate standard deviation. Also Einstein’s theory of general relativity relies on a 4-dimensional mathematical framework. Without knowing this, the GPS in your phone would not work (GPS relies on precise timing, so precise that the effects of earths mass on spacetime become necessary to consider).
See also Weisstein, Eric W. “Tesseract.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/Tesseract.html
New comic is up! You can read it in full here: https://sequentialmath.com/comic/from-the-blog-of-crypto-llama-public-key-encryption
There’s barely any aspect of our modern lives that hasn’t had a mathematical contribution at some point and yet, if you asked the average person, they might think that maths is just difficult, irrelevant and uninteresting.
“Mathematics, in the common lay view, is a static discipline based on formulas…. But outside the public view, mathematics continues to grow at a rapid rate … the guide to this growth is not calculation and formulas, but an open ended search for pattern.”
— Lynn A. Steen