Non-euclidean geometry Definition and 15 Threads

Alright, I've been wondering this for a while now. Say you have an infinite grid of squares in hyperbolic geometry, such that the curvature makes it so each angle of each square is 72° (5 squares at each corner). At the very 'center' of the grid, or the origin, there would be 5 straight rays...

I don't want to post this in a math forum because it's very basic and I just want a straightforward answer, not something math heavy . What's the definition of angle in a cuved space embedded in a higher eucledian space? Like when I have a spherical surface in 3d eucledian space and want to work...

In the book: The Elements Euclid defined 5 postulates: 1) A straight line segment can be drawn joining any two points. 2) Any straight line segment can be extended indefinitely in a straight line 3) Given any straight line segment, a circle can be drawn having the segment as radius and one...

I always tend to get confused when thinking about non-Euclidean geometry and what straight lines and parallel lines are. If I think of a sphere, I get how two people driving north would almost mysteriously intersect at the North Pole and how the angles of a triangle would not add up to 180...

I am not sure that this is the right rubric for this question, as it is historical, but as it is part of the history of Model Theory, I am putting it here. I will not be offended if the moderators decide that it doesn't belong here. In https://arxiv.org/pdf/1008.2667.pdf, the author states...

As the title implies, I'm looking for books on non-euclidean geometry. I'm not looking for very advanced thing, more on some book with a good introduction to this topic.

As I understand it, a Cartesian coordinate map (a coordinate map for which the line element takes the simple form ##ds^{2}=(dx^{1})^{2}+ (dx^{2})^{2}+\cdots +(dx^{n})^{2}##, and for which the coordinate basis ##\lbrace\frac{\partial}{\partial x^{\mu}}\rbrace## is orthonormal) can only be...

A sphere with radius "r" has three points on its surface, the points are A, B, and C and are labelled (xa, ya, za) and so on. What is the general formula to calculate the area on the surface of the sphere defined by these points?

Homework Statement A short paper 12-16 pages I'm also fairly new to this topic Homework EquationsThe Attempt at a Solution I tried to ecplain it's application using the Robertson walker mrttuc but it ended upmlooking too much like a physics psper:/

Homework Statement Consider a universe described by the Friedmann-Robertson-Walker metric which describes an open, closed, or at universe, depending on the value of k: $$ds^2=a^2(t)[\frac{dr^2}{1-kr^2}+r^2(d\theta^2+sin^2\theta d\phi^2)]$$ This problem will involve only the geometry of space at...

If QM is a statistical model to approximate something underlying space time we don't quite understand yet, and there is a complex geometry underlying space time, is it possible to find other ways to simplify molecular optimizations and electron interactions in computational chemistry using...

In introductions to non Euclidean geometry, examples are often given in the form of measuring angles on a 2D surface embedded in a 3D space, such as the surface of a sphere or on a saddle surface. This gave me the initial impression that 3D non Euclidean geometry would have to be embedded in 4...

Are circles considered straight lines in Non-Euclidean Geometry?

I'm trying to prove that the interior angles on a spherical triangle sum to \pi + (A)/(a)^2 where A is the area of the triangle and a is the radius of the spherical space. I think I know how to prove it, but there is one part that has me stumped. I'm using the General Relativity book...

1089 and all that. A Journey into Mathematics. David Acheson Oxford University Press (www.oup.com) ISBN 0-19-851623-1 Introduction: You like maths don't you? Of course you do, because you would be making haste to get out of my sight before I damn your eyes with a strategically and...