Hyperbolic Geometry: Euclidean v 4D in 3D Space

In summary, a forum user expresses their love for lurking and thanks the forum for providing mental stimulation. They ask if a statement about Euclidean and hyperbolic geometry is true and if there are any recommended resources for learning more about hyperbolic math. Another user responds with a link to a helpful online textbook and suggests looking into resources like Wikipedia and Wolfram Mathworld.
  • #1
Praxisseiz
3
0
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!
 
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  • #2
False. Or maybe nonsensical.
 
  • #3
Praxisseiz said:
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.
 

1. What is hyperbolic geometry?

Hyperbolic geometry is a non-Euclidean geometry that deals with the properties of space and figures in which the parallel postulate of Euclidean geometry does not hold.

2. How does hyperbolic geometry differ from Euclidean geometry?

In hyperbolic geometry, the parallel postulate is replaced with the hyperbolic parallel postulate, which states that through a point not on a given line, there can be drawn multiple lines parallel to the given line. This leads to the existence of multiple parallel lines through a point in hyperbolic geometry, unlike in Euclidean geometry where only one parallel line can be drawn through a point.

3. What is the significance of 4D in 3D space in hyperbolic geometry?

In hyperbolic geometry, 4D in 3D space refers to the use of 4-dimensional space to represent and visualize the properties of hyperbolic geometry. This allows for a better understanding of hyperbolic figures and their properties, which cannot be accurately represented in 3-dimensional space.

4. How is hyperbolic geometry applicable in real-life situations?

Hyperbolic geometry has applications in various fields such as physics, engineering, and computer science. It is used in the study of curved space-time in Einstein's theory of relativity, in the design of curved mirrors and lenses, and in the development of algorithms for computer graphics and simulations.

5. Is hyperbolic geometry a recent discovery?

No, hyperbolic geometry was first introduced by the mathematician Nikolai Lobachevsky in the early 19th century. It was further developed by other mathematicians such as János Bolyai and Bernhard Riemann. However, it was not widely accepted until the late 19th and early 20th century with the work of Henri Poincaré and David Hilbert.

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