Vandalism and Mathematics

PROTIP: You can get around the Shannon-Hartley limit by setting your font size to 0.

Image: Randall Munroe.

Regarding the comic

This xkcd comic has two points.  The first is understandable without any context. If the writer had in fact discovered a proof that information is infinitely compressible, then ANY amount of space would be sufficient to contain it. The second point refers to Fermat’s famous statement “I have discovered a truly remarkable proof of this theorem which this margin is too small to contain,” which was, of course, referring to Fermat’s Last Theorem, a topic which we discussed extensively in class.

Liberal use of others property

It is now often believed that Fermat did not actually have a correct proof of this theorem. This minor detail did not, however, deter the great Fermat from writing it as fact in the margin of his copy Arithmetica to be discovered posthumously and baffle mathematicians for centuries to come. This, however, is not the only case of mathematicians writing statements in strange places. Another mathematician who did this was William Rowan Hamilton. Unlike Fermat, Hamilton decided to actually carve in his answer to a question, as opposed to carving in a claim that he has an answer. To be fair though, Fermat did own his book, while Hamilton didn’t actually own the bridge. This occurred in 1843, while taking a walk, he had a flash of brilliance during which he discovered Quaternions. Lacking a proper way with which to write down the result, Hamilton instead chose to carve his answer in the side of a bridge.

But what are quaternions?

Hamilton knew how to add and multiply complex numbers in a plane. However, he did not know how to multiply them in space. Quaternions were his solution to this problem, because while he could not figure out how to multiply complex points in a 3-dimensional space, he could figure out how to do it in a 4-dimensional space. In fact there is now a theorem which says the only normed division algebras which are number systems where we can add, subtract, multiply, and divide, and which have a norm satisfying |zw|=|z||w| have dimension 1, 2, 4, or 8. Quaternions can be thought of as a 4-dimensional space and are often denoted by H or ℍ. They are a noncommutative number system over the complex space, which just means that a*b does not necessarily equal b*a. They are defined as ℍ ={a+bi+cj+dk} where a, b, c, and d all belong to the real numbers. Note in particular that ij = k = -ji, jk = i = -kj, and ki = j = -jk. This eventually leads to what Hamilton engraved on the Brougham Bridge: i2 = j2 = k2 = ijk = -1, which means that ij, and are all equal to square root of -1.

Utility of quaternions

The quaternions can be used to do rotations in 3 dimensions, which may seem unintuitive given that quaternions describe a 4-dimensional space. To better explain this we need the concept of real and pure quaternions. A real quaternion is one which contains only a real part, while a pure quaternion is one which does not contain a real part. This is the equivalent of partioning a complex number into its real and imaginary parts. The difference between these two scenarios is that the pure portion of a quaternion is a vector in 3-space instead of a single number. Thus a real quaternion will take the form [a, 0] where 0 is the zero vector and a pure quaternion will take the form [0, v] where v is a vector of the form v=bi+cj+dj. Note that this means that the set of all pure quaternions define a 3-space. Thus the process of rotating in three dimensions is accomplished by starting with a pure quaternion, called p. This quaternion is then multiplied by the rotor, a second quaternion, called q, of the form [cos(Θ), sin(Θ)*v] where v  is a vector of the form v=bi+cj+dj and Θ is the angle by which we are rotating. If p happens to be perpendicular to q then the result will be a pure quaternion and the process is complete. However, if it is not the resulting quaternion will not be pure and the magnitude will be off. We can, however, multiply this new result by the inverse of q which will result in a pure quaternion of the desired length. Note that this means that the object should start and end in the 3-dimensional space as defined by the set of all pure quaternions with the real portion being used as an intermediary. I should also mention that the inverse is the conjugate of the quaternion divide by its normalization squared, where the conjugate is computed by negating the vector v and the normalization by dividing by the magnitude of the quaternion. Quaterions, however, don’t just allow for rotation in 3 dimensions, they also help avoid certain problems such as gimbal lock. Gimbal lock occurs when two out of the three rotational axes align. When this happens, the aligned axes both rotate the object in the same way. While you can still get out of the gimbal lock, it does force you to do some additional rotations. Quaternions circumvent this problem by having that intermediary 4th rotational axis.

Conclusion

If you want to commit vandalism, all you have to do is discover something brilliant which will be used for quite some time after its discovery in technologies which have yet to exist and engrave it in the side of a bridge or scribble it within the margins of a book. You might even get a plaque commemorating your vandalism.

Image: JP, via Wikimedia Commons.

Sources

http://math.ucr.edu/~huerta/introquaternions.pdf

http://www.cprogramming.com/tutorial/3d/quaternions.html

http://www.euclideanspace.com/maths/algebra/realNormedAlgebra/quaternions/

http://en.wikipedia.org/wiki/Quaternion

http://www.gamedev.net/page/resources/_/technical/math-and-physics/do-we-really-need-quaternions-r1199

http://www.gamedev.net/topic/25314-why-diana-grubers-wrong-about-quats/

http://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation

http://3dgep.com/understanding-quaternions/

https://www.youtube.com/watch?v=zc8b2Jo7mno

A History of Mathematics, Uta C. Merzbach and Boyer.

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