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Our role playing group got stuck in a hypercube.

I recognized it as a hypercube, when after travelling "south" 4 times, I ended up in a loop, and counted 8 rooms.

After the game the GM showed me this map from /tg/

If I swapped the NS EW UP pairs, for NE WU PS pairs, would the resulting shape/network still represent a hypercube? if not, what would the new shape be called? Or would it no longer make any sense geometrically?

Each room we are in had an exit to the NSEWUP .

Ryan Leach
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    I don't think "hypercube" is the best word to describe this. Instead it looks like a cube of the usual three dimensions with "wrap-around" connections. After all you have only eight nodes, like the corners of a cube, and each node has six neighbors. – hardmath Jul 07 '17 at 15:39
  • I thought it was a hypercube in terms of the connections, similar to that of faces of a 3d cube being able to be mapped to 4 directions of exits?

    e.g. interacting inside the 3d cube rooms, is really just the 'surface' of the hypercube.

     – Ryan Leach Jul 07 '17 at 15:52
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    I see. You are thinking of the rooms as the cubes that make up facets of a tesseract, so that each "room" has six neighbors. – hardmath Jul 08 '17 at 22:46

1 Answers1

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The short answer is yes, you would still be in a hypercube.

How can you verify this? Well, the only way you can ever verify anything about the fourth dimension... by using analogies with other dimensions. Imagine now that you were a $2D$ creature walking around on a cube.

enter image description here

As you can see, you would have only four directions in this case, and there would be only $6$ rooms.

The problem is, you can't really make a graph with "directions" labelling each edge, because your orientation is constantly changing. For example, if our quadrilateral friend travelled North and then East from there, he would end up in the same place as if he had just travelled East... but he would be facing in different directions when he arrived.

enter image description here

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So when you're a being messing around in a solid of a higher dimension, it doesn't make sense to think about "directions" anymore. For this reason, your graph isn't accurate, because it is static - in reality, the "direction" in which you would travel from one room to the other would constantly change, and so the labelling of the edges would constantly change. Thus, if you made the swap of labelling that you are asking about, you would still be in the same solid - in fact, you could probably achieve that very orientation just by walking around for a little bit inside of it.

Franklin Pezzuti Dyer
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