Poincaré disk model

[geordief]

I believe this is a projection of the tiling of a hyperboloid onto an x,y plane. Although this is far beyond my level of understanding , I have a question as to whether it is possible to use these Poincaré disk models in the plane to build a 3-dimensional model by perhaps rotating the disk /planearound one of its axes?

 

If this is possible can this be seen as a projection of 4-D spacetime onto a solid,3-D "globe of the world* type" ball ?

 

Can this be seen as a possible representation of the universe in a purely special relativistic way (without accounting for mass or energy) - kind of like a blank page?

 

This is a link ,although I imagine it is obvious what I am referring to:

 

https://en.wikipedia.org/wiki/Poincar%C3%A9_disk_model

 

 

 

* this kind of thing http://www.toysrus.com/buy/globes-telescopes/edu-science-world-globe-12-inch-diameter-globemaster-30524-2303946 although it would be transparent and the 3-D tiling would be visible into its interior.

[studiot]

The idea is not magic, it is similar to something you may have used in secondary school.

 

Graph paper where the scale is not linear

logarithmic graph paper.

 

Here is a simple log-log graph paper.

Note that the grid lines becomes closer and closer together as we move away from the origin.

So that because of the nature of the logarithm we can compress infinity into this space.

 

 

Of course with Poincare we are talking about polar coordinates.

 

Here are a linear, uncompressed version.

Note the grilines are evenly spaced.

 

 

and here is a compressed version along the same lines

Note the grid lines get closer as we approach the edge of the disk.

 

 

We can also offset the origin as in the Smith chart

 

Edited April 20, 2016 by studiot

[geordief]

The idea is not magic, it is similar to something you may have used in secondary school.

 

 

 

But this disk is supposed to be a projected model of the set of 4-d spacetime events onto a plane ,isn't it?

 

If so can this set of events also be projected onto a 3 -dimensional space?

 

I am happy with the way the origin can be offset to any particular event.

 

I am guessing that the event that represents the "Big Bang" singularity cannot be represented ......

[Strange]

I can't answer your questions but oddly (or maybe not) I came across a very similar representation when reading about modular functions: https://www.quantamagazine.org/20150312-mathematicians-chase-moonshines-shadow/