eli5: What is the difference between non-Euclidean Geometry and map projection?

18 views
0

Bolyai, Riemann and others – so I read – devised a new branch of geometry in the 19th Century by discarding Euclid’s axiom that parallel lines never meet. How was this different from the geometry that cartographers doing map projections had done for centuries, or indeed, that ancient Greeks not long after Euclid had used to discover the size and shape of the Earth?

In: 3

There are 5 Euclid’s axioms. There are several formulations of these axioms so I’ll choose the one I’m familiar with

* For any two distinct points, there exists a unique line segment between these two points
* Given a straight line segment, one can extend it indefinitely into a straight line
* Given a line segment, there exists a circle whose radius it that segment centered on one of the endpoint
* All right angles are congruent.
* [The parallel postulate]: If a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than two right angles.

(Note that the last axiom is NOT that “parallel lines never meet” ! Because “not meeting” is actually the definition of parallelism. The last axiom can be reformulated as “given a line and a point not on that line, there is a unique line that does not meet the first line and that is passing through the point”)

The geometry that cartographers used is spherical geometry. In spherical geometry, the parallel postulate fails for sure, because all lines necessarily meet. But some previous axioms also fail. The first one for example is not true anymore as there exists infinitely many line segment between the North pole and the South pole on the sphere. And you cannot extend a line segment between the North Pole and the South pole.

So even if spherical geometry could be considered as “non-Euclidean” because it doesn’t satisfy Euclid’s axioms, it’s very different from the one that we usually decribe as “non-euclidean”.

The non-euclidean geometry discovered by mathematicians in the 19th century, the so-called hyperbolic geometry, respects the first four axioms of Euclidean Geometry, and only the last one fails. And it fails in a completely different way than in spherical geometry. In hyperbolic geometry, given a line and a point, there exists infinitely many other lines not meeting the first line and passing through the point.

(To sum up : in Euclidean geometry, a unique parallel. In spherical geometry, no parallel. In hyperbolic geometry, infinitely many parallel).

That was a surprise, because for a long time, mathematicians were questioning if the fifth axiom of Euclidean geometry was really necessary or not. They didn’t expect to find a perfectly consistent model of geometry where not only that axioms fails, but that turned out to be an extremely rich world where a lot of new results could be proven and used in other domains.

Let’s talk about the history and development of geometry. If you don’t want the gory details, the short version is they had neither the tools nor the motivation to generalize and make-rigorous the idea and implications of “parallel lines not meeting”, and that pondering on an idea is very different from building a theory.

As you point out, people obviously knew that you could map a surface to a plane in such a way that distorted lengths however you wish. The foundations of non-Euclidian geometry arose long before calculus or infinitesimals were even conceived of, and even ancient Greek mathematicians were well aware that the shortest path between two points on a sphere is a great circle segment.

Around the 1600s there was a paradigm shift in the study of geometry, when Descartes introduced the idea of studying geometry with *coordinates*, rather than simply following the direct consequences of axioms. This allowed us to study more complicated shapes, like curves in a plane. Utilizing the concurrent and revolutionary development of calculus, mathematicians of the era realized that at any point on a smooth curve you can fit a “tangent circle” ([pictured here](https://www.google.com/imgres?imgurl=https%3A%2F%2Fupload.wikimedia.org%2Fwikipedia%2Fcommons%2Fthumb%2F8%2F84%2FOsculating_circle.svg%2F1200px-Osculating_circle.svg.png&imgrefurl=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FOsculating_circle&tbnid=haSyPqQpdsXkdM&vet=1&docid=m_hiqdsEa71I6M&w=1200&h=900&source=sh%2Fx%2Fim)), and hence arrived at the first rigorous notion of *curvature*.

Curvature is a deeply fundamental concept in modern (post-1800s) geometry, but the philosophy of the time that mathematics should be grounded in the physical universe (i.e. 3 dimensional and “flat”) didn’t allow the generalization of this concept.

When things really started to kick off is when Euler, in the mid-1700s, using the tools of calculus and curvature, started thinking about geodesics on a surface more broadly. In his path to this, he introduced the idea of *intrinsic geometry*, with local coordinates. More on that in a second, but to motivate it, consider a two dimensional curved surface, such as a hilly terrain. Clearly you can specify a point on this surface with *three* coordinates (x, y, z), but there should be a way to specify a point or a direction on a 2 dimensional surface with only two coordinates. but have a think about how you would actually do that.

Fast forward to 1836, the idea of mathematics that did not follow the perceived rules of the physical universe was already growing in popularity. Freeing himself from this restriction, Gauss published his revolutionary research on the theory of curved surfaces, which his student Riemann later extended to higher dimensional manifolds. (It’s a little ironic that this would turn out to describe the most accurate model of the universe we’ve ever produced)

This brings us back around to spherical geometry: Gauss didn’t necessarily “discover” non-Euclidian geometry. What he did do is show that spherical geometry and “flat” geometry are both tiny puzzle pieces in the ocean of theory that is non-Euclidian (“Riemannian”, specifically) geometry. He also developed the tools to prove that no transformation can uniformly map a sphere onto a plane, and therefore that no perfect map projection was possible (in a way laying the groundwork for topology)

The study of geometry and its interaction with other fields absolutely exploded from there, but there are a couple of things I want to note: (A **TL;DR** if you will)

– First, what Gauss and the other pioneers of the theory did wasn’t just “discard the parallel postulate”. Pondering on an idea is the easy part. The hard part was building up the vast theory that *results* when you discard the parallel postulate, which 200 years later gave us one of the most powerful and beautiful collection of fields in mathematics.

– What this represented wasn’t just a new theory, but a phase shift in the philosophy of maths. After Gauss and Riemann, mathematicians lost their inhibition about “unrealistic” maths.

– Like all maths. Gauss and others’ achievements were built over previous knowledge. Without calculus, or the notion of a coordinate space, or the notion of orthogonality, or many other findings, The revolutionary work done in the 19th century wouldn’t have been possible.