September 28, 2016

Mapping the Tesseract

I thought it might be fun to make a dungeon built inside a tesseract, but mapping the thing is a bit of a challenge.  I eventually made this model that helps me to visualize travelling within the tesseract, so I thought I'd go ahead and share it.  I'm sure that I don't yet fully understand tesseracts but I'll go ahead and attempt to explain them anyhow.  Wish me luck.

A tesseract is the analog of a cube brought into four-dimensional Euclidean space; a tesseract is to a cube as a cube is to a square.

It might help to think of a tesseract as 8 different 3-D cubes that are attached in an extra-dimensional way.  Each cube shares a side with 6 of the other cubes (4 walls, floor, and ceiling).  However each cube also has an opposite cube on the other end of the tesseract that it can't touch directly, but both of these will touch all six of the others.

If someone were to be teleported into a tesseract the space within might seem to be perfectly ordinary at first.  There don't need to be any obvious distortions or bending of space, the person likely wouldn't notice anything amiss until they started travelling through multiple cubes.  Passing from one cube to another might be obvious, but it could also be something as innocuous as a doorway.  If the person inside picks any direction and walks in a straight line then they would end up passing through three new cubes, but the fourth cube they enter would be the original cube that they started from.  They are effectively trapped until they find some extra-dimensional means of escape.

My handy diagram is below.  The thick black lines show how the sides of the zones actually touch each other.  I've colored each of the cubes/zones to make reference easier.  It might help to think of the Green Zone as the starting point because the six directions (up, down, north, south, east, west) are all from the perspective of a stick figure person who is standing in that green cube facing the same direction as the viewer of the diagram (north).  Below the diagram is a list to help me navigate travel from one zone to another.  For example, "Green Up - White Down" means that taking stairs up through the ceiling of the green zone will bring you up through the floor of the white zone.



1. White Zone - Up
White Up - Yellow Up
White North - Orange Up
White South - Purple Up
White East - Red Up
White West - Blue Up
White Down - Green Up
8. Black Zone - Down (hidden center)
Black Up - Green Down
Black North - Orange Down
Black South - Purple Down
Black East - Red Down
Black West - Blue Down
Black Down - Yellow Down
2. Green Zone - Home Perspective
Green Up - White Down
Green North - Orange South
Green South - Purple North
Green East - Red West
Green West - Blue East
Green Down - Black Up
7. Yellow Zone - Opposite
Yellow Up - White Up
Yellow North - Purple South
Yellow South - Orange North
Yellow East - Blue West
Yellow West - Red East
Yellow Down - Black Down
3. Blue Zone - West
Blue Up - White West
Blue North - Orange West
Blue South - Purple West
Blue East - Green West
Blue West - Yellow East
Blue Down - Black West
6. Red Zone - East
Red Up - White East
Red North - Orange East
Red South - Purple East
Red East - Yellow West
Red West - Green East
Red Down - Black East
4. Purple Zone - South
Purple Up - White South
Purple North - Green South
Purple South - Yellow North
Purple East - Red South
Purple West - Blue South
Purple Down - Black South
5. Orange Zone - North
Orange Up - White North
Orange North - Yellow South
Orange South - Green North
Orange East - Red North
Orange West - Blue North
Orange Down - Black North


The Black Zone is hidden from view, it's in the center of the diagram surrounded by zones on all sides.  From the perspective of the other zones, the Black Zone is always down.  It might help to think of the Black Zone as being like the core of a planet, it's always down from the perspective of the surface.  That said, my understanding of how gravity functions in the tesseract is not great.  Does the Black Zone still have a floor that things fall toward?  Does everything fall toward the center of the Black Zone?  I'm not sure, to be honest.

I think of the White Zone as being "above" the other zones, while the Black Zone as "below".  This way the six middle zones will all seem to have the same gravity.  The white and black zones are going to have some kind of gravitational weirdness though.  I'm thinking of making them both zero gravity zones and maybe filling the black zone with water, but I'm still not sure.

I've labeled the Yellow Zone as "Opposite", but really that is just from the perspective of someone standing in the Green Zone.  Someone walking into the Yellow Zone wouldn't experience reverse gravity or anything, and their compass needles wouldn't suddenly move; I only meant that it's orientation is the reverse of the Green Zone.

I hope this helps someone.  It could make for a neat little eight room dungeon (Hut of Baba Yaga?), but I think I'm going to try for something larger where each zone is a different dungeon level with it's own particular theme.

2 comments:

  1. Robert A. Heinlein wrote "--And He Built a Crooked House" in 1941 about just such a shape.

    http://www.math.union.edu/~dpvc/courses/2010-11/MTH053-FA10/assignments/crooked-house.pdf

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  2. I think I'll come back to this after I finish my coffee...

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