Thread by @getjonwithit

Moths are attracted to lights because of the same mathematics that underlies twistor theory and compactification in theoretical physics: projective geometry.

It all starts from a simple observation: translations are just rotations whose center is located "at infinity". (1/11)

So if we take an ordinary space (like the 2D plane) and we adjoin a hypothetical point "at infinity", then the distinction between translations and rotations disappears: as the center of rotation moves further and further away from the object being rotated,... (2/11)

...the "translation to rotation" ratio gets larger and larger, until, at infinity, it becomes infinite. So translations reduce to a special case of rotations. We call such a space a "projective space", and it represents a "compactification" of the space we started from. (3/11)

The 2D plane with a point adjoined at infinity becomes a sphere (namely the "Riemann sphere"), meaning that every point on this projective plane can be mapped uniquely to a point on this sphere, via "stereographic projection". Imagine suspending the sphere above the plane. (4/11)

Then, stand at the North Pole of the sphere. If we cast a ray down from the Pole to any point on the surface of the sphere, that ray will intersect some point on the plane below. Hence every point on the sphere can be associated to a point on the plane (and vice versa). (5/11)

The only point for which that is not true is the North Pole itself: that is associated with the point "at infinity". Since this mapping takes us from a 2D plane (which is not compact) to a sphere (which is compact), it is referred to as a "compactification" of the plane. (6/11)

Compactifications appear all over the place in physics, and I plan to talk about them more in future posts. But for now, I want to talk about moths.
Moths fly at night using celestial navigation: they use the stars to ensure that they continue flying in a straight line. (7/11)

They do this by first selecting a star, and then keeping that star at a fixed angle in their vision. Since the star is effectively at "optical infinity", the moth can then exploit the basic feature of projective geometry that translations=rotations. (8/11)

By keeping the star at a fixed angle in their sight, they are rotating themselves around a point at projective infinity, which is equivalent to a straight translation. But what if the "star" that they lock on to is in fact not a star at all, but rather an artificial light? (9/11)

Rotating around a point that is not at infinity is *not* equivalent to a translation, and so they will not maintain a fixed course. Rather, by fixing an artificial light at a constant angle in their vision, the moth will instead orbit the object in a logarithmic spiral. (10/11)

So moths end up inside your house at night because their brains evolved to navigate a world in which geometry is projective, where rotations=translations; yet human lighting breaks that expectation, and places them inside the much uglier world of non-projective spaces. (11/11)