Questions tagged [hyperbolic-geometry]

Questions on hyperbolic geometry, the geometry on manifolds with negative curvature. For questions on hyperbolas in planar geometry, use the tag conic-sections.

The prototypical example of hyperbolic geometry in two dimensions of Gauss-Lobachevsky-Bolyai in which the parallel postulate of Euclidean geometry is replaced by a new postulate of at least 2 parallel lines through an external point not on the given line with sum of interior angles of a geodesic triangle smaller than $\pi$ radians.

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Studying Euclidean geometry using hyperbolic criteria

You've spent your whole life in the hyperbolic plane. It's second nature to you that the area of a triangle depends only on its angles, and it seems absurd to suggest that it could ever be otherwise. But recently a good friend named Euclid has…
Zach Conn
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How many "super imaginary" numbers are there?

How many "super imaginary" numbers are there? Numbers like $i$? I always wanted to come up with a number like $i$ but it seemed like it was impossible, until I thought about the relation of $i$ and rotation, but what about hyperbolic rotation? Like…
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What are the interesting applications of hyperbolic geometry?

I am aware that, historically, hyperbolic geometry was useful in showing that there can be consistent geometries that satisfy the first 4 axioms of Euclid's elements but not the fifth, the infamous parallel lines postulate, putting an end to…
Vincent L.
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How to create mazes on the hyperbolic plane?

I'm interested in building maze-like structures on the [5, 4] tiling of the hyperbolic plane, where by maze-like I mean something akin to a spanning tree of the underlying lattice: a subgraph of the lattice that's still connected (every cell can…
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What hyperbolic space *really* looks like

There are several models of hyperbolic space that are embedded in Euclidean space. For example, the following image depicts the Beltrami-Klein model of a hyperbolic plane: where geodesics are represented by straight lines. The following image, on…
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What is the relationship between hyperbolic geometry and Einstein's special relativity?

I am a third year math student writing a term paper on hyperbolic geometry and I would like to understand its relationship with special relativity. I have read that the hyperboloid model of hyperbolic geometry, also known as the Minkowski model,…
Sid
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Is hyperbolic rotation really a rotation?

We define a $2\times 2$ Givens rotation matrix as: $${\bf G}(\theta) = \begin{bmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) &\cos(\theta) \end{bmatrix}.$$ On the other hand, we define a $2\times 2$ hyperbolic rotation matrix as: $${\bf…
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how to generate tessellation cells using the Poincare disk model?

I'm a computer programmer, and while I like math, this is an area where my understanding of math falls short of what I need in order to apply it successfully. I've been looking at M.C. Escher's "Circle Limit" drawings, which use a Poincare disk…
LarsH
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Minkowski plane vs. hyperbolic plane

As a physics student, I have studied some elements about hyperbolic geometry in many different contexts. In linear algebra, I was told that equipping $\mathbb{R}^2$ with a non-degenerate symmetric bilinear form gives us a space isometric to the…
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Magnifying glass in hyperbolic space

My grandmother used to read with a magnifying glass. What (an ideal) magnifying glass does, is basically a homothety: it scales the picture by some factor. Now, in a hyperbolic space there is no such thing as homothety. So, what a person living in a…
liaombro
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How did Beltrami show the consistency of hyperbolic geometry in his 1868 papers?

This is in response to comments and the answer by user studiosus to this question: As for Beltrami's work: Consistency of a geometry from (post) Hilbert viewpoint has nothing to do with existence of an (isometric) embedding in a particular…
Sid
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How is this matrix called, and does it have a purpose?

I stumbled upon the 2d rotation matrix $$R(\theta)=\begin{pmatrix} \cos(\theta) & -\sin(\theta)\\ \sin(\theta) & \cos(\theta) \end{pmatrix}$$ which has determinant 1 because $$ \cos^2(\theta) + \sin^2(\theta) =1$$ So I thought what would happen if I…
user736166
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The Teichmüller space $T_g$ of a closed riemann surface $S_g$ of genus $g \geq 2$ can't be parametrized by $6g-6$ geodesic length functions

Setting: It is well known that the Teichmüller space $T_{g,b}$ of an oriented Riemann surface $S_{g,b}$ of genus $g \geq 2$ with $b \geq 1$ boundary components (satisfying $2g + b \geq 3$) can be globally parametrized by $6g - 6 + 3b$ geodesic…
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Symbolic coordinates for a hyperbolic grid?

Rephrasing     (one year later)    (original question is below) Apparently the original question wasn't clear, or nobody knows an answer (or both). So I will try to rephrase it. Look at your favorite hyperbolic grid. This question asks for a…
Matt
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Trigonometric integral related to Gieseking's constant

This question at MathOverflow https://mathoverflow.net/questions/302982/how-to-prove-the-identity-l2-frac-cdot3-frac215-sum-limits-k-1-inf conjectures certain relation between fast converging hypergeometric series and Gieseking's constant…
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