Message #2728

From: Roice Nelson <roice3@gmail.com>
Subject: Re: [MC4D] Puzzle in Minkowski Space?
Date: Thu, 02 May 2013 16:21:11 -0500

This is a cool topic, and I plan to follow any developing insights. Here
are a couple thoughts I’ve had on this:

On Jeff Week’s Hyperbolic Games
page<http://geometrygames.org/HyperbolicGames/index.html>,
he makes the following relevant remark:

Hyperbolic Games also highlight the under-appreciated fact that two
> traditional models of the hyperbolic plane are simply different views of
> the same fixed-radius surface in Minkowski space: the Beltrami-Klein model
> corresponds to a viewpoint at the origin (central projection) while the
> Poincaré disk model corresponds to a viewpoint one radian further back
> (stereographic projection).

In a sense (for 2D tilings), hyperbolic tilings are the
regular tessellations/polytopes of a Minkowski 2+1 space. E.g. one can
think of the {7,3} living on a constant radius surface in Minkowski space,
just as one can think of the spherical tilings living in Euclidean 3+0
space, and the Euclidean tilings living in Euclidean 2+0 space. (Of
course, one doesn’t have to think of all these objects being embedded in
any of these spaces - they can be looked at just from the "intrinsic
geometry" perspective.)

I purposefully didn’t write Minkowski space*time* above by the way. One
can still think of Minkowski 2+1 space without thinking of time. The
"distance" between points is just calculated in a weird way, with one
component having a negative contribution. This makes me wonder though…
What would a {7,3} tiling look like as an animation, where that special
component was plotted along the time dimensions? Would the regular
heptagons even be recognizable?

Roice

On Thu, May 2, 2013 at 3:42 PM, schuma <mananself@gmail.com> wrote:

> When some people try to explain the fourth dimension, they talk about
> time. Personally I don’t like it, because it causes a lot of confusion. And
> the four-dimensional puzzles here don’t have anything to do with time.
>
> However, if one really wants the fourth dimension to be time, the space
> (spacetime) should be Minkowski rather than Euclidean.
>
> http://en.wikipedia.org/wiki/Minkowski_space
>
> In this space, spatial rotation and Lorentz boost are allowed. Because of
> relativity, in some sense Minkowski space is a more complete model than the
> common 3D space.
>
> Since Minkowski space is so cool, my question is: can we define twisty
> puzzles in Minkowski space? A related question I don’t have an answer is
> that, what are the "regular polytopes or tessellations" in Minkowski space?
> I know that Minkowski space lacks the full symmetry as in Euclidean space:
> time and space dimensions are different. But is that a reasonable
> relaxation, under which there are nontrivial regular polytopes or
> tessellations? I did some searching, but I haven’t got any answer.
>
> The traditional Minkowski space has 3 spatial dimensions and one time
> dimension (3+1). But for simplicity we may focus on 2+1 or even 1+1
> dimensions. But I really don’t know how to think of regular shapes or
> puzzles there.
>
> For clarity, the hyperboloid model of hyperbolic geometry is like a 2D
> manifold imbedded in a Minkowski space. But I don’t think a puzzle in
> hyperbolic geometry with an underlying hyperboloid model is what I want.
>
> Any thought?
>
> Nan
>