Questions tagged [noneuclidean-geometry]

For general questions about non-Euclidean Geometry. Consider using more specific tags, like (projective-geometry), (hyperbolic-geometry), (spherical-geometry), etc.

Non-Euclidean geometry is the study of geometries with different versions of the parallel postulate. Intuitively you can think of this as doing geometry on surfaces besides the plane.

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Is there a known non-euclidean geometry where two concentric circles of different radii can intersect? (as in the novel "The Universe Between")

From the 1951 novel The Universe Between by Alan E. Nourse. Bob Benedict is one of the few scientists able to make contact with the invisible, dangerous world of The Thresholders and return—sane! For years he has tried to transport—and…
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Is Tolkien's Middle Earth flat?

In the first introductory chapter of his book Gravitation and cosmology: principles and applications of the general theory of relativity Steven Weinberg discusses the origin of non-euclidean geometries and the "inner properties" of surfaces. He…
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Non-Euclidean Geometry for Children

I should've asked this question two years ago when my son (at that time, 9 years old) came to me and said: "Dad, today in school our teacher drew a line on a paper and said this is a straight line, it goes from both directions and doesn't meet…
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The term “elliptic”

There are many things which are called “elliptic” in various branches of mathematics: Elliptic curves Elliptic functions Elliptic geometry Elliptic hyperboloid Elliptic integral Elliptic modulus Elliptic paraboloid Elliptic partial differential…
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If you know that a shape tiles the plane, does it also tile other surfaces?

For instance there is a hexagonal tiling of the plane. There is also one using quadrilaterals. It seems intuitive that both of these tilings also apply on a torus. Is it the case that anything that maps 1-to-1 to a plane surface has a "complete"…
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What is the maximum "alive density" of cells in Conway's Game of Life when played on a torus?

I've read that Conway's Game of Life (CGOL) can have unbounded growth from a finite initial number of alive cells (e.g. a glider gun). However, if CGOL is played on a torus, space (the number of cells) becomes finite, and glider guns are guaranteed…
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What is the most efficient shape for tiling curved spaces?

In a great video by PBS Infinite Series, the mathematician Kelsey Houston-Edwards argues that bees build their honeycombs into hexagonal shapes because that's the most efficient way of tiling two-dimensional euclidean space that "minimizes the…
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Cosine Law Duality in Hyperbolic Trigonometry

From setting up a hyperbolic triangle with hyperbolic side length $a,b,c$ and corresponding angles $A,B,C$, it is not hard to prove the following law of cosine: $$\cos A= \frac{\cosh b \cosh c -\cosh a}{\sinh a \sinh b}$$ Since we have $3$ formula…
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Non-Euclidean Geometrical Algebra for Real times Real?

This question was triggered by a series of others and reading some references: Keshav Srinivasan & Euclid Eclid's Elements As quoted from the last reference: GEOMETRICAL ALGEBRA. We have already seen [ .. ] how the Pythagoreans and later Greek…
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Modular geometry: The parabolas of quadratic residues modulo $p$

[For using the available space better, I rotated the function graphs by 90 degrees.] For the quadratic function $f_1(x) = x^2$ (with $x \in \mathbb{R}$) there is only one parabola all integer points $(k,k^2)$ lie on: But for the quadratic residual…
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How do you convince someone that parallel lines can touch/meet?

When talking to someone who knows basic mathematics but not really in-depth, how would you explain that parallel lines can touch? I am referring to Non-Euclidian/ Projective Geometry. Edit: Why do parallel lines in the Euclidean plane (correspond…
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Why Do The Axioms of Euclidean Geometry Not Need To Include the Definition of Space?

EDIT: update, I found that Euclid's axioms are not considered rigorous. David Hilbert did a full axiomatization of Euclidean Geometry (1899 in his book Grundlagen der Geometrie--tr. The Foundations of Geometry). To do so, he required 6 primitive…
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If proposition $\text{I}, 17$ in Euclid's Elements does not depend on the parallel postulate, how can elliptic geometry be consistent?

I'm from Italy, so I was reading an Italian translation of Euclid's Elements; and while I was doing just that, something about the introduction—written by Attilio Frajese—really caught my attention. Frajese explains how Euclid, not feeling confident…
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Is there a name for an infinite spherical plane?

I was dabbling in hyperbolic/spherical geometry when I had the thought, "Why does an ant walking on a spherical plane have to come back to the same point it started on?" I knew that the answer is because it lives on a sphere(-ical plane), but,…
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Book recommendations for Euclidean/Non-Euclidean Geometry

Request for Book Recommendations: Background introduction Disclaimers: If the tone below is a little arrogant I apologize beforehand, but I'm being very specific here because I want to make sure that I will be learning and accessing the right…
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