Before passing along my copy1 of Flatland to my brother (as he needs it for a class), I reread it on an airplane. It turns out that airplanes are the best environment in which to read this book. It's fortunate that nobody was sitting beside me and asked what I was reading because they would have gotten the following lecture. You, dear reader, are not as lucky2.

Flatland

Firstly a short synopsis of the book. It was written in 1884 by a schoolteacher/theologian Edwin Abbott Abbott3. You're introduced to a geometrical world from the perspective of "A Square", who is a square inhabiting a two-dimensional manifold. He describes life in two-dimensions and then is visited by a three-dimensional sphere who introduces him to a one-dimensional line world (and a zero-dimensional point world) and eventually shows him the third-dimension. There's also a good deal more to it, including commentary on Victorian society and metaphysical musings. It's very short, so there's no reason not to read the entire thing. For those who prefer a movie, apparently such a thing exists and actually has Martin Sheen and Kristen Bell as voice actors; I don't know how well it tracks the book, but the trailer is certainly surreal.

Spherical Geometry

So what does air travel have to do with anything? It's all about geometry. It turns out the spherical geometry is rather strange.. Here's just a few of the "obvious" geometrical intuitions that not necessarily true on a sphere.

The shortest distance between two points is a line.

This first is actually true, but it's misleading. When you draw a line on a map4 it's not actually the shortest path between the two points. In fact, for a sphere the "lines" are great circles. These are just the paths you would take if you kept walking perfectly straight (from your perspective) until you returned to your starting location; any such path is called a geodesic. All shortest paths are geodesics but not all geodesics are shortest paths. You can see this in spherical geometry because while great circles are geodesics, we can choose to go clockwise or counterclockwise around the great circle.

Two points determine a line

Not necessarily. Consider the North and South pole. There are infinitely many lines of longitude (which are great circles) extending between the two. In fact any two points which are antipodal, on the exact opposite side of the sphere, will work. If the points aren't antipodal then only two are required.

There are such things as parallel lines

This is not the case. One common objection is that lines of latitude do not intersect. Looking back at the definition of a line, we see that only great circles are lines and thus lines of latitude are not lines (in the spherical geometry sense).

The sum of the angles of a triangle is 180°

This is where things start getting very interesting. To show that this is not true a simple counter-example will suffice. You might want to obtain a globe and perform this example for it to sink in. 

Start at the intersection of the equator and the prime-meridian. Walk
counter-clockwise along the equator a quarter of the way around the
globe. Turn left. Walk until you reach the North Pole. Turn left. Walk
until you reach the equator at which point you turn left again and
you're facing the same direction you started.

However we turned left three times which results in a sum of interior angles of 270°.

It turns out that this is very interesting. We can measure this as the angular defect defined as the difference between the sum of the angles (in radians) minus the usual value of π.. There are several things that can be proven about this, the first of which is that it's non-negative. All triangles on a sphere have sum of angles greater than 180°.

We can go further and show5. that the area of the triangle is the same as the angular defect (for a sphere of radius 1, you just scale by \(r^{2}\) otherwise) and hence trivially prove the previous non-negativity result.

"Some yet more spacious Space, some more dimensionable Dimensionality"

Perhaps what I most love about Flatland is that it enthusiastically keeps going. You could have gotten a lot of the satire and novelty out of the story of introducing "A Square" to the third dimension. However, Abbot doesn't stop there. Instead he follows the obvious extrapolation and has "A Square" postulate the existence of a 4th and 5th dimension, indeed an entire infinite sequence of dimensions.

And even as we, who are now in Space, look down on Flatland and see the inside of all things, so of a certainly there is yet above us some higher, purer region, whither thou dost surely purpose to lead me - O Thou Whome I shall always call everywhere and in all Dimensions, my Priest, Philosopher, and Friend - some yet more spacious Space, some more dimensionable Dimensionality, from the vantage-ground of which we shall look down together upon the revealed insides of solid things, and where thine own intestines, and those of thy kindred Spheres, will lie exposed to the View of the poor wandering exile from Flatland, to whome so much has already been vouchsafed.”

Taking that example to my spherical geometry case, what would be the next extension? Suppose instead of a sphere, we were flying on a donut or in mathematical terms a torus. The first thing that comes to mind with a torus is that there are infinitely many geodesics which connect any two distinct points. For a simple example take two points on the outer circle of the torus. You can connect them with a straight line along the circumference. Or you can connect them with a looped line that goes around the ring once, or twice, or three times, or however many times you want. All are geodesics and all connect the two points.

Of course, the torus also brings up another interesting point: topology6. Abbot describes Flatland as a piece of paper. What would be another difference if the Flatlanders lived on a torus? For one, they wouldn't have hard edges to their world. They'd also find that walking the same direction is faster in some places than others, namely when they're on the inside or outside of the hole (from our perspective).

This branches out into figuring out what kinds of surfaces can be distinguished by Flatlanders using intrinsic geometry, geometry that uses only measurements that can be performed on the surface, not using the 3rd dimension. This is especially interesting if you consider that we're in a putatively 4-dimensional space and necessarily need to use intrinsic geometry to figure out what kind of space we're in. Experiments have shown that we don't live in a Euclidean space, rather we live in a far more interesting curved space-time.

Footnotes:

1

Don't worry, I found The Annotated Flatland by Ian Stewart on Amazon, so I'll be trading up.

2

Although I suppose you could stop reading.

3

Not a typo, his middle and last name are the same

4

Technically this is not true if you use a Gnomonic projection, but I'd wager good money that if you know what a Gnomonic projection is, you are already aware of this fact.

5

The proof is very fun and I enjoy performing it with oranges. If you need a hint, start with cutting the orange in pieces along the great circles defining the triangle. Also, look up the definition of a spherical lune and think about its area.

6

Apparently this is discussed in Flatterland: Like Flatland, Only More So also by Ian Stewart.