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.world
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    1 year 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.

    • cll7793@lemmy.worldOP
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      1 year ago

      Thanks for the comment! It is cool and also pretty aesthetically pleasing!

      • Reliant1087@lemmy.world
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        1 year 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@lemmy.world
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      1 year 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?

      • Afrazzle
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        1 year 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.

        • SgtAStrawberry@lemmy.world
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          1 year ago

          Looks to be that way one of the examples given on the wiki page. It is still however an interesting theory, if four countries touching at a corner, are the diagonal countries neighbouring each other or not. It honestly feels like a question that will start a war somewhere at sometime, probably already has.

          • Vegasimov@reddthat.com
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            1 year ago

            In graph theory there are vertices and edges, two shapes are adjacent if and only if they share an edge, vertices are not relevant to adjacency. As long as all countries subscribe to graph theory we should be safe

            • SgtAStrawberry@lemmy.world
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              1 year ago

              The only problem with that it that it requires all countries to agree to something, and that seems to become harder and harder nowadays.

        • Blyfh@lemmy.world
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          1 year ago

          But each small country has three neighbors! Two small ones, and always the big donut country. I attached a picture to my previous comment to make it more clear.

          • Pronell@lemmy.world
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            1 year ago

            In your example the blue country could be yellow and that leaves the other yellow to be blue. Now no identical colors touch.

          • Afrazzle
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            1 year ago

            Whoops I should’ve been clearer I meant two neighbours within the donut. So the inside ones could be 2 or 3 colours and then the donut is one of the other 2 or the 1 remaining colour.

              • Afrazzle
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                1 year ago

                That map is actually still quite similar to the earlier example where all 4 donut hole countries are the same. Once again on the right is the adjacency graph for the countries where I’ve also used a dashed line to show the only difference in adjacency.

      • BitSound@lemmy.world
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        1 year 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.

      • Kogasa@programming.dev
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        1 year ago

        There are some rules about the kind of map this applies to. One of them is “no countries inside other countries.”

        • atimholt@lemmy.world
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          1 year ago

          Not true, see @BitSound’s comment.

          It does have to be topologically planar (may not be the technical term), though. No donut worlds.

          • Kogasa@programming.dev
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            1 year ago

            The regions need to be contiguous and intersect at a nontrivial boundary curve. This type of map can be identified uniquely with a planar graph by placing a vertex inside each region and drawing an edge from one point to another in each adjacent region through the bounding curve.

    • SanguinePar@lemmy.world
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      1 year 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!

    • Artisian@lemmy.world
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      1 year 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.

      • rycee@lemmy.world
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        1 year ago

        I suspect that the Belgium-Netherlands border defies any mathematical description.

    • Dandroid@dandroid.app
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      1 year ago

      If you had a 3 dimensional map, would you need more colors to achieve the same results?

      Edit: it was explained in your link. It looks like for surfaces in 3D space, this can’t be generalized.

    • clumsyninza@lemmy.world
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      1 year 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.