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:

  • @mookulator@lemmy.worldEnglish
    52•2 years ago

    The four-color theorem is pretty cool.

    You can take any map of anything and color it in using only four colors so that no adjacent “countries” are the same color. Often it can be done with three!

    Maybe not the most mind blowing but it’s neat.

      • @Reliant1087@lemmy.world
        3•2 years ago

        Your map made me think how interesting US would be if there were 4 major political parties. Maybe no one will win the presidential election 🤔

    • Blyfh
      14•2 years ago

      What about a hypothetical country that is shaped like a donut, and the hole is filled with four small countries? One of the countries must have the color of one of its neighbors, no?

      • @BitSound@lemmy.world
        8•2 years ago

        In that image, you could color yellow into purple since it’s not touching purple. Then, you could color the red inner piece to yellow, and have no red in the inner pieces.

      • @Afrazzle@sh.itjust.works
        0•2 years ago

        I think the four small countries inside would each only have 2 neighbours. So you could take 2 that are diagonal and make them the same colour.

    • @Artisian@lemmy.worldEnglish
      5•2 years ago

      Note you’ll need the regions to be connected (or allow yourself to color things differently if they are the same ‘country’ but disconnected). I forget if this causes problems for any world map.

    • SanguinePar
      5•2 years ago

      I read an interesting book about that once, will need to see if I can find the name of it.

      EDIT - well, that was easier than expected!

    • @clumsyninza@lemmy.world
      1•2 years ago

      Isn’t the proof of this theorem like millions of pages long or something (proof done by a computer ) ? I mean how can you even be sure that it is correct ? There might be some error somewhere.