# Non-Euclidean

“Non-Euclidean Geometry” is the field of geometry as applied to non-flat surfaces and non-straight lines.

Euclid was an ancient mathematician who published a work called *The Elements*. This laid out the basic, common principles of geometry that we know today. However, Euclid only dealt with flat, two-dimensional surfaces.

When applying geometry to curved surfaces, many of Euclid’s principles don’t work anymore. There are at least two types of geometry – elliptic and hyperbolic – that are considered “Non-Euclidean.”

As an aphorism, “Non-Euclidian” might refer to something that’s confusing or convoluted, or that breaks from a common norm in such a way that the usual principles and rules don’t apply anymore.

## Why I Looked It Up

I’m kicking myself that I didn’t note the reference, but it was a usage outside of mathematics. It referred to some set of facts or observations that just didn’t add up correctly.

## Postscript

Added on

I found this quote in a conference presentation about mapping.

Maps are never value-free images; except in the narrowest Euclidean sense they are not in themselves either true or false.

I asked on the English Stack Exchange what this meant. Mostly, people said it was a literal reference to the simple map itself, which is necessarily a flat surface. Many argued that this was a one-off usage that didn’t really indicate the existence of a generally-accepted idiom (someone called the usage “pretentious”).

The next day, I found this quote in *The Structure of Scientific Revolutions*:

The laymen who scoffed at Einstein’s general theory of relativity because space could be “curved” … were not simply wrong or mistaken. Nor with the mathematicians, physicists, and philosophers who tried to developed a Euclidean version of Einstein’s theory. What had previously been meant by space was necessarily flat, homogeneous, isotropic, and unaffected by the presence of matter.

Again, lots of references to curves and flatness.

## Postscript

Added on

An image I found on some weird Facebook group.