Holographic Fifteen

Here’s a pretty good, simple discussion of the holographic principle. Its main failing is that it conveys the impression that the holographic principle is true! And maybe it is, but it’s only been proved for certain particular cases… not for the universe we live in.

For more on the concept of duality, think about the following game which I’ll call “Fifteen”. (I’ve forgotten where I first read about this. Anyone know of where this has been written up before?) We have nine cards bearing the numbers 1 through 9. Deal them out face up, and then you and I take turns picking up one card at a time. If at any point one of us has three cards in hand that add to 15, that person wins. (If we pick up all nine cards and neither of us has three that add to 15, it’s a draw.)

So for instance:

  1. I take 5.
  2. You take 1.
  3. I take 6.
  4. You take 4.
  5. I take 7.
  6. You take 2.
  7. I take 3 and win, because I have 5 + 7 + 3 = 15.

Is this game a win for one player or the other, and what’s the optimum strategy? Think it over before continuing.

Time’s up!

Okay… think about it like this. I said “Deal [the cards] out face up” but didn’t specify how. It doesn’t matter how, but suppose you do it like this:

8 1 6
3 5 7
4 9 2

That of course is a magic square of order 3. Each row, each column, and each diagonal sums to 15. In fact, as you can easily figure out, those are the only eight sets of three distinct integers in the range 1 to 9 that sum to 15.

So when you’re alternating picking up cards, the first to take all the cards a line of three — row, column, or diagonal — wins. Sound familiar? Right: Tic-Tac-Toe (or Naughts and Crosses, or, as someone once told me they call it in Norway, Swedish Chess). Fifteen and Tic-Tac-Toe are not the same game, but there’s a duality between them: The rules of either one map onto the rules of the other, and given a situation in one game, there’s a corresponding situation in the other. For example, the penultimate situation in the example game above (I hold 5, 6, and 7, you hold 1, 2, and 4, and it’s my turn) corresponds to the Tic-Tac-Toe position

   | O | X
-----------
   | X | X
-----------
 O |   | O

and the winning Tic-Tac-Toe move corresponds to my taking 3. So there’s a winning strategy in Fifteen for the first player, which is dual to the winning strategy in Tic-Tac-Toe.

Likewise the holographic principle asserts that a three dimensional universe is dual to its two dimensional surface (let’s not get into what “surface of the universe” actually means… good question but not for here and now); different mathematics describes the physics of the volume and the surface, but there is a mapping from either onto the other, and given a configuration of the volume, there’s a corresponding configuration of the surface, and vice versa. Knowing what happens on the surface, you can map that onto the volume to find what happens there, and vice versa.

 

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