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Physics Found a Geometric Structure for Math to Play With
- kens 6y agoIf you're interested in William Ronan Hamilton's life, I recommend this YouTube video, a clever takeoff on Hamilton: https://youtu.be/SZXHoWwBcDc https://youtu.be/SZXHoWwBcDc
- Koshkin 6y agocomplex numbers. These numbers involve i, the square root of −1, and they take the form a + bi, where a is the real part and b is the imaginary part. This is the shortest explanation I've seen of what a complex number is! (Reminds me of when a piece on some advancement in quantum physics in a pop-science magazine tries to explain to the reader what an atom is.) Edit: OK, let me try then to explain what homology is: it is a way of discovering (and counting) interesting features of the space in consideration (like holes in a topological space or fixed points of a symplectic transformation) by looking at a series of algebraic structures, such as Abelian groups or vector spaces, which one can often conjure up from objects of the space (points, paths, etc.)
- golergka 6y agoIt's short, but I don't think that it's very helpful for someone who hears about it for the first time. When you see a + bi your first reaction, as a 12 year old, is to look at the plus sign and wonder - what's the actual result here is? How do these things add up? I think that explaining it through geometry and vectors first is far more natural to begin with, and gives a solid intuition of what happens with complex numbers when they go through simpler operations.
- Koshkin 6y agoThis is perhaps true for the complex numbers, but I am not sure if any field extension can be explained geometrically in a sensible way. (For example, think about extending the rationals by adding the square root of 2.)
- jacobolus 6y agoAdding the square root of 2 gives you a way of describing coordinates in a symmetry system where you start with a square grid and then allow yourself to make 1/8 turn rotations.
- golergka 6y agoThat's why you first develop a very good intuition of what complex numbers really are, understand Euler's formula (and probably asked to come up with a proof for it), and only then move forward. In mathematics, it's really easy and dangerous to think that you understand a concept only because you can memorize it's definitions and properties; any math education should fight this pseudo-understanding and make sure that you really grok it.
- Koshkin 6y agoWhat you are saying is true of course, except nobody in their right mind should think that algebra lacks intuition and that learning it consists of rote memorization.
- nine_k 6y agoExplaining complex numbers without a picture loses like 90% of the "aha!" moment.
- dvt 6y agoI kind of disagree with this. The geometry implications are incidental (extremely interesting and useful, but incidental nonetheless). If you only understand complex numbers from a visual standpoint, you're going to be very confused when we go to (hypercomplex) number systems with > 3 dimensions.
- jacobolus 6y ago> geometry implications are incidental No, the geometry is fundamental, and the rest is just bookkeeping. If you go through life only ever doing the bookkeeping without understanding its meaning you’re going to be very confused (as indeed many students are today, and many mathematicians were historically). The proper way to understand complex numbers is as ratios of planar vectors. (Other interpretations can be layered on top of that one.) A complex number z = v/u is the object which we can multiply the Euclidean vector u by to get the vector v, via scaling and rotation. That is, zu = (v/u)u = v (u\u) = v. Notice we have no coordinates here. If we like, we can split the complex number as a sum z = x + yI into a scalar part x = ½(v/u + u\v) and a bivector part yI = x = ½(v/u – u\v), where x and y are scalars and I is a bivector representing a quarter turn in the u–v plane. This version generalizes beautifully to higher (or lower) dimensions, to pseudo-Euclidean space, and even to non-metrical settings. For more, http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf http://geocalc.clas.asu.edu/pdf/GrassmannsVision.pdf http://geocalc.clas.asu.edu/pdf/GrassmannsVision.pdf
- Koshkin 6y agoSure, a useful representation is important, and it demonstrates the universal power of the complex numbers. (Same with groups vs. linear maps vs. matrices.) But the proper way to understand the field of complex numbers is indeed to do it in all its abstract generality, i.e. simply as the algebraically closed extension of the field of reals.
- alexpetralia 6y agoI know Eric Weinstein has introduced a (fairly undocumented) "theory of everything" which pins the fundamental structure of the universe to be geometric in nature (11 dimensions). Does anyone know if this is at all related? Or perhaps the geometric approach in physics is fairly typical?
- Koshkin 6y agoI do not believe it's related. Symplectic spaces are useful in classical mechanics. Not sure about "typical", but yes, geometry in general is the most important part of the conceptual mathematical framework of modern theoretical physics.
- alexpetralia 6y agoGood to know - thanks for clarifying.
- wwarner 6y agoThat was a great read!
- andi999 6y ago'Relatively new...', well at least 35 years..