Thread: "Puzzle in Minkowski Space?"

Thanks for the attention!

OK, a 2D hyperbolic tiling lives on a 2D surface in a Minkowski 2+1 space. =
Here the Minkowski space is just a model that can be replaced by a disk mod=
el, say. So I think Minkowski is not essential in this application.

My question is more about, can we define 3D puzzles that fill a 3D region i=
n a Minkowski 2+1 space?

Nan

--- In 4D_Cubing@yahoogroups.com, Roice Nelson wrote:
> 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.)
>=20
> 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?
>=20
> Roice
>=20

Certainly this is a question whose answer currently is "don't know", but
the question itself is already a great contribution. Time is such a
funny dimension and has been one of the most difficult to deal with in
my career. Naturally people first think of mapping time to animations
over time, but I don't find that terribly interesting. I would much
rather see iso-surfaces built from MRI data for example than to watch 2D
slices moving up and down through a 3D density field. A simple stock
chart is a better use of a time dimension when needing to understand a
given equity than something bouncing up and down during animation. We
therefore naturally roll time into a spatial dimension, which is an
improvement, but still something seems to be missing. The big question
is what exactly is missing? I think so far I may be just restating what
Roice expressed. I would like to reframe the question as "how can we
treat time in a more intrinsic way?" Whatever falls out from that
question, I would like to then ask what might be the most interesting
meanings and visualizations and puzzles that can be designed explicitly
in 0+2 or 0+3 spaces, or really any topologies involving more than one
time dimension.

-Melinda

On 5/2/2013 3:45 PM, schuma wrote:
> Thanks for the attention!
>
> OK, a 2D hyperbolic tiling lives on a 2D surface in a Minkowski 2+1 space. Here the Minkowski space is just a model that can be replaced by a disk model, say. So I think Minkowski is not essential in this application.
>
> My question is more about, can we define 3D puzzles that fill a 3D region in a Minkowski 2+1 space?
>
> Nan
>
> --- In 4D_Cubing@yahoogroups.com, Roice Nelson wrote:
>> 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
>>
>
>
>
> ------------------------------------
>
> Yahoo! Groups Links
>
>
>
>

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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
thisin
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

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