Robert McNees
The mathematician Nikolai Ivanovich Lobachevsky was born #OTD in 1792. He developed a non-Euclidean geometry that shows up in areas ranging from relativity to the designs of M.C. Escher.
In Lobachevsky's geometry there exists more than one line parallel to a given line and passing through a point not on that line. Although the term "Lobachevsky geometry" is still used, we more frequently refer to it as "hyperbolic geometry" – a term introduced by Felix Klein.
Portrait: Lev Kryukov (wikimedia)
Robert McNees
In Euclid's geometry, if you have a line L and a point P that is not on L, then there is exactly one line through P that does not intersect L. That line is parallel to L.
Robert McNees
On a curved surface this isn't necessarily true. There, the generalization of a straight line is a "geodesic."
A geodesic is "locally straight," in the sense that a geodesic joining two very nearby points is the shortest path between them.
Two geodesics on a curved surface can satisfy Euclid's notion of "parallel" over a short distance, but may eventually intersect when you extend them over longer distances.
Robert McNees
For example, "straight lines" (geodesics) on a sphere are "great circles" – circles with the same circumference as the equator.
Any two great circles intersect. So on the curved surface of a sphere if you pick a straight line L, and a point P not on L, then any straight line through P will intersect L.
Robert McNees
(The shortest path between two points on a sphere will lie on a segment of a great circle, so we use them for flight paths. They no longer look like the shortest path when projected onto a plane representation like a map, because on a flat plane we expect the "shortest distance path between two points" to look like our usual notion of a straight line.)
Robert McNees
A curved surface appears flat over very short distances – you don't notice the curvature of the Earth standing in your front yard.
Over those "local" scales two geodesics may resemble Euclid's parallel lines. Measure the separation with a ruler and it seems constant. But this isn't quite true; there are small deviations due to the curvature.
On longer scales – for a sphere, distances comparable to its radius – the effect of curvature accumulates and these initially parallel lines intersect.
Robert McNees
The type of geometry studied by Lobachevsky is actually the opposite. In Lobachevsky's geometry, given a straight line L (a geodesic) and a point P not on L, there are an infinite number of geodesics that pass through P but never intersect L.
There is a simple (but very important) example of this geometry known as the Poincaré Upper Half-Plane. Despite the name, it was (afaik) first studied by the Italian mathematician Eugenio Beltrami.
Robert McNees
We can express the actual / proper / physical distance between two nearby points in terms of their separation on the sort of grid we would usually use to describe locations and distances on a plane.
In a little local patch where the space looks approximately flat, the coordinates of two nearby points differ by dx along the x-axis and dy along the y-axis. The proper distance between them, which we call ds, is:
Robert McNees
First you take the top half of the x-y plane: all the points -∞ < x < ∞ and 0 < y < ∞. Then you introduce a special rule for the distance between points.
Here's where it gets weird.
As you get closer to the x-axis (y=0) the rulers get shorter and shorter, so that the distances measured with them get longer and longer.
The "proper distance" between two points is measured with these rulers. It is not the same as "coordinate distance" – the number of squares separating those points on a grid.
Robert McNees
With this notion of distance, you get two kinds of geodesics:
1) Straight vertical lines that take you between two points with the same x coordinate.
2) Semi-circles centered on the x-axis, which are the shortest paths between two points with different x coordinates.
This makes sense: Geodesics are smoothly stitched together from locally shortest paths. So to go from A to B you should always veer away from the x-axis towards larger values of y where your rulers aren't so short.
Robert McNees
And this means that given a straight line L like the vertical geodesic shown here, and a point P not on L, there are an infinite number of semicircle geodesics that pass through P but never intersect L.
It's an unusual geometry!
Source code for this diagram, made in LaTeX with pgf/tikz, can be found here: https://gist.github.com/mcnees/c8e9840c39c5d9202a509c46c2ad4c91
Robert McNees
You have probably seen this geometry before. It can be mapped onto a disk by a Möbius transformation, which preserves relationships between straight lines. The x-axis becomes the edge of the disk.
The geometry of this "Poincaré disk" inspired M.C. Escher's "Circle Limit" series of drawings. In this case the rulers are the fish-lengths. See how they get small as you approach the edge?
Image: M.C. Escher, "Circle Limit III"
Robert McNees
Lobachevsky's hyperbolic geometry (also studied by János Bolyai) shows up throughout physics, especially in relativity.
The Poincaré models described above are relevant for the AdS/CFT correspondence, as they describe a patch of (Euclidean) anti de Sitter space.
If you are curious about the relationship between AdS and hyperbolic geometry, and you've got some math under your belt, I strongly recommend Ingemar Bengtsson's ~beautiful~ lectures on Anti de Sitter space.
http://3dhouse.se/ingemar/Kurs.pdf
Robert McNees
Thank you for reading this thread, and thank you especially to everyone who did so without making a Tom Lehrer reference.
Robert McNees
@dralgaegirl Ma'am.
DrAlgaeGirl🦠💧
@mcnees But I have this friend in Minsk....
Robert McNees
Glenn Fleishman
@mcnees he made it into Freud's index because he loved his mother
Robert McNees
By the way, I think I included alt-text for all the images and diagrams. I've been using LaTeX as a replacement for math in alt-text, but don't know if that's a good idea since not everyone uses it. Most of what I've used so far is pretty translatable, though – \frac and \sqrt, ^ for powers, etc.
Greg Whitehead
@mcnees I don't know that this would be any better, but I was curious and found this MathJax tool for converting LaTeX/MathML to natural language text
Robert McNees
@nico_forgot Not at all!
nico?
@mcnees did I just ignoramusplain non-euclidean geometry to an effin physicist? Shit
No! I can't explain anything to anyone. I'm an open idiot
nico?
@mcnees non-euclidean geometry (hyperbolic and elliptic) are so fascinating like so many other things that are logical but so hard to visualize. Math is a mind bender that way, to me
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