Hyperbolic Geometry: Euclidean v 4D in 3D Space

[Praxisseiz]

Greetings PF, I do very much love lurking these forums for countless hours of leisure brain twisting. Infinite thanks for that.

A very simple question for you all. I believe the answer is 'true', however, I'm not formally educated in mathematics, so I feel a bit like I'm grabbing at straws here.

Is this statement true?

Euclidean geometry helps describe 3 dimensions in a 2 dimensional space whereas hyperbolic geometry helps describe 4 dimensions in a 3 dimensional space.

Can anyone off the top of their heads suggest further readings to develop a basic grasp of hyperbolic math? Especially as it relates to classical geometry such as, trig, calc and their included theorems. I'm finding it difficult to connect the dots myself especially in the sense of how sin relates to sinh, cos to cosh, et. al.

TIA
P!

 

[g_edgar]

False. Or maybe nonsensical.

 

[Coin]

Euclidean geometry helps describe 3 dimensions in a 2 dimensional space whereas hyperbolic geometry helps describe 4 dimensions in a 3 dimensional space.

I don't think I understand what you mean. There is an n-dimensional hyperbolic geometry for any n >= 1, just as there is an n-dimensional euclidean geometry for any n >= 1.

In general though n-dimensional hyperbolic geometry can be thought of as geometry on the boundary of a hyperboloid (a generalized hyperbola) in (n+1)-dimensional euclidean space.

Can anyone off the top of their heads suggest further readings to develop a basic grasp of hyperbolic math?

Here is a free online textbook I found helpful. (It is one of the references on the wikipedia page for "hyperbolic geometry".) If you read this book you may want to be aware that when they refer to the "interior of the disc model" they are talking about what other people would call the "Poincare disc".

Especially as it relates to classical geometry such as, trig, calc and their included theorems. I'm finding it difficult to connect the dots myself especially in the sense of how sin relates to sinh, cos to cosh, et. al.

Have you tried reading the material on wikipedia and Wolfram Mathworld? Some of it is very good. Wikipedia describes the relationship between sin/sinh etc using the analogy of a parametric function:

Just as the points (cos t, sin t) form a circle with a unit radius, the points (cosh t, sinh t) form the right half of the equilateral hyperbola.