# Message #2731

From: Roice Nelson <roice3@gmail.com>

Subject: Re: [MC4D] Re: Puzzle in Minkowski Space?

Date: Thu, 02 May 2013 22:46:20 -0500

On Thu, May 2, 2013 at 5:45 PM, schuma wrote:

> My question is more about, can we define 3D puzzles that fill a 3D region

> in a Minkowski 2+1 space?

>

Maybe a starting point to get to a "solid" object in the space would be to

change the expression for the surface of a constant radius to an

inequality, i.e. rather than:

x^2 + y^2 - t^2 = -1 (surface of hyperboloid of two sheets)

do this:

x^2 + y^2 - t^2 >= -1 (see 2D slice of

this<http://www.wolframalpha.com/input/?i=x%5E2+-+y%5E2+%3E%3D+-1>in

wolfram alpha. Also, it’s interesting that the "radius" is negative.)

Then you could slice up that solid with planes. So it seems like a good

thing to understand is what is a plane in M3 (2+1 Minkowski space). The

following article, "Hyperbolic Geometry on a Hyperboloid" looks to offer

lots of good information about M3:

http://www.jstor.org/stable/2324297

M3 planes through the origin result in geodesics on the hyperboloid

surface. Planes not through the origin results in circles (or horocycles

or curves equidistant to geodesics) on the surface. This all seems to

suggest that the puzzle result might be functionally the same as MagicTile

puzzles though. Even if a solid object, the slicing of the stickers on the

boundary might end up the same with the approach I’m describing.

But maybe one could build up some solid objects in M3 with planes, rather

than starting with this "imaginary sphere" surface. And perhaps there

could be new effects from that, especially say, if the object boundary

moved into the area of the Minkowski space outside the "light cone". For

example, what would be the meaning of a dodecahedron plopped straight

inside M3?

I hope you can come up with some new and unique puzzle concepts and puzzles!

Cheers,

Roice