Mathematics

Questions tagged [noneuclidean-geometry]

For questions about non-Euclidean geometries, such as elliptic or hyperbolic.

Non-Euclidean geometry is the study of geometries with different versions of the parallel postulate. Intuitively this is geometry on surfaces other than the Euclidean plane.

Known branches include:

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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…
Mindwin Remember Monica
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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…
xaxa
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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…
Amir Asghari
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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…
MvG
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Closed form for $\int_{0}^\infty e^{-ax^2}\operatorname{erf}(px)\operatorname{erf}(qx)\operatorname{erf}(rx)\operatorname{erf}(sx)\,dx$?

Is there a closed form for this integral? $$\int_{0}^\infty e^{-ax^2}\operatorname{erf}(px)\operatorname{erf}(qx)\operatorname{erf}(rx)\operatorname{erf}(sx)\,dx$$ Recall the definition of the error function: $$\operatorname{erf}(x) =…
Blue
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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…
Ron Shvartsman
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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"…
Joseph Weissman
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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…
ZKG
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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…
timax
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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…
Ren
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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…
lEm
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Tarski-like axiomatization of spherical or elliptic geometry

Preamble Tarski's axioms formalize Euclidean geometry in a first-order theory where the variables range over the points of the space and the primitive notions are betweenness $Bxyz$ (meaning $y$ is on the line segment between $x$ and $z$, inclusive)…
Matt Dickau
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Equiconsistency of euclidean, hyperbolic, and elliptic geometry

Pretty much every text about non-euclidean geometries talks about the various models by Beltrami, Riemann, Poincaré, Klein, and others which demonstrate that if euclidean geometry is consistent, then hyperbolic and elliptic geometry also are…
Frunobulax
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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…
Han de Bruijn
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Can elliptic space be infinite?

The go-to example of elliptic space is a sphere where geodesics turn into great circles of finite length. But is it possible to have an elliptic space which doesn't 'merge' with itself once it's made a full turn? ie. infinite, unbounded,…
David Rutten
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