[extropy-chat] Fractal Escher Tesseract Maze

Eliezer Yudkowsky sentience at pobox.com
Tue Jul 20 02:29:15 UTC 2004

Spike wrote:

> Oh man this is wicked cool, thanks Eli!  spike
>> http://www.mathpuzzle.com/18Nov2003.html

Imagine the Fractal Escher Tesseract Maze.

The basic template is a 4th-dimensional hypercube, with 8 component cubes, 
each component cube filled with pathways and staircases and ladders.  Some 
pathways and staircases re-enter a cube of the maze.  For simplification 
(wicked grin) any given cube might contain re-entrant cubes of only the 
same room.  3D analogy:  Imagine the basic fractal maze as a cube with six 
sides; any given square might contain re-entrances only into that same 
square of the fractal cube.

As a 3D human being walks around the corners of a 4D tesseract, the 
apparent orientation of the rooms changes.  If you carry your personal 
gravity with you (that is, you experience no apparent sudden changes of 
gravity as you walk through the tesseract) then you can walk out of a room 
through one corridor, and seemingly reenter from what used to be the 
ceiling.  Imagine a Flatlander walking along the square sides of a cube. 
The Flatlander can exit a square through a ceiling going up, make 2 
90-degree turns, and instead of re-entering that square from the ceiling 
going down (as he expects), re-enter that square from the side going down.

Thus, as you walked through the Fractal Escher Tesseract Maze, the paths 
you could take and Escher staircases you could traverse would depend on 
your apparent direction of gravity.

I'd like to live in *that* crooked house.  It wouldn't just be a question 
of getting to the bathroom, it would be a question of getting to the 
bathroom using a path that put the toilet on the apparent floor.  You might 
have to cross through your bathroom with the toilet on a side wall in a 
fractal subcube on your way to getting to your bathroom on the top level 
with the toilet on the floor.

Eliezer S. Yudkowsky                          http://singinst.org/
Research Fellow, Singularity Institute for Artificial Intelligence

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