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No Stupid Questions @lemmy.world cll7793 @lemmy.world

What are the most mindblowing things in mathematics?

What concepts or facts do you know from math that is mind blowing, awesome, or simply fascinating?

Here are some I would like to share:

  • Gödel's incompleteness theorems: There are some problems in math so difficult that it can never be solved no matter how much time you put into it.
  • Halting problem: It is impossible to write a program that can figure out whether or not any input program loops forever or finishes running. (Undecidablity)

The Busy Beaver function

Now this is the mind blowing one. What is the largest non-infinite number you know? Graham's Number? TREE(3)? TREE(TREE(3))? This one will beat it easily.

  • The Busy Beaver function produces the fastest growing number that is theoretically possible. These numbers are so large we don't even know if you can compute the function to get the value even with an infinitely powerful PC.
  • In fact, just the mere act of being able to compute the value would mean solving the hardest problems in mathematics.
  • Σ(1) = 1
  • Σ(4) = 13
  • Σ(6) > 101010101010101010101010101010 (10s are stacked on each other)
  • Σ(17) > Graham's Number
  • Σ(27) If you can compute this function the Goldbach conjecture is false.
  • Σ(744) If you can compute this function the Riemann hypothesis is false.

Sources:

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  • One of my favorite little math gizmos is Pascal's triangle modulo 2. The most striking fact about it is that if you put each number in a box and color the 0 and 1 boxes two different colors, you'll find that the image converges to the famous fractal Sierpinski's triangle as you zoom out! Try looking up "pascal's triangle mod 2" in Google images or something so you can see for yourself! https://en.m.wikipedia.org/wiki/Sierpiński_triangle

    But wait, there's more! If you read each row as a binary number, the rows will enumerate all possible products of Fermat numbers (numbers of the form 22n + 1)! But this isn't restricted to just base 2! If you read the rows on the same triangle in base b, you'll get all possible products of the base b generalized Fermat numbers (which have the form b2n + 1)!

    You might recognize the Fermat numbers from their role in compass and straightedge constructions, as prime Fermat numbers help describe which regular polygons are possible to construct with a compass and straightedge. Due to a few unrelated coincidences, this means that the first 32 rows of Pascal's triangle mod 2 read as binary numbers list out all currently known regular n-gons with odd n that are possible to construct with a compass and straightedge alone. Whether or not there are more is an open question!

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