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> No, the geometry is fundamental, and the rest is just bookkeeping. I'd push back on this, but I think it's probably a foundational disagreement (perhaps betw

by dvt 6y ago

> No, the geometry is fundamental, and the rest is just bookkeeping. I'd push back on this, but I think it's probably a foundational disagreement (perhaps between pure and applied mathematicians). For example, a key behavior of number systems as we move up in dimensionality is the loss of "nice" properties†: when going from the reals to the complex numbers, we lose total ordering. I've seen proofs of this done visually[1], and some are pretty clever, but it's much easier to simply "do the algebra" and end up with a contradiction. Getting to 1 ≺ 0 ≺ 1 is literally a two-line proof, while geometrically you might have various intuitions that would prove to be wrong: for example, why can't we use a spiral[2] to achieve total ordering on a plane? Not to mention that going to the quaternions, we lose commutativity of multiplication -- I don't think anyone even bothers trying to prove this visually because it would probably end up looking like spaghetti. † This is, right off the bat, not very intuitive: why would more degrees of freedom minimize nice properties? This is in the same vein as the volume of an n-dimensional ball tending to zero as n tends to infinity. To see why this happens, you need to put away the shapes, and look at the math. [1] https://math.stackexchange.com/questions/487997/total-ordering-on-complex-numbers https://math.stackexchange.com/questions/487997/total-orderi... [2] https://mathworld.wolfram.com/RationalSpiral.html https://mathworld.wolfram.com/RationalSpiral.html

jacobolus 6y ago
Commutativity of multiplication is not “lost”. It’s something you shouldn’t generally expect to have. It would be good if people would spend more time learning about non-commutative multiplications at a younger age, which should be expected in general. Mathematicians’ bias towards commutativity is often so culturally ingrained from a lifetime of commutative algebraic structures that it’s very hard for them to break away from. If they spent more time on the geometry earlier on it could help quite a bit. That multiplication of a vector by a scalar + bivector oriented in a plane containing the vector on the left happens to have the same result as multiplication by the reverse on the right Zv = vZ† comes down to the identity a(bc) = (bc)a for any three coplanar vectors a, b, c, which happens because vectors commute with both scalars and coplanar bivectors. This doesn’t work more generally because the trivector part of the product doesn’t vanish when the vectors are not coplanar. This convenient feature results in many nice results for plane geometry / complex numbers which don’t hold in more general settings. It would be good if students would start with the general version (in high school or early in undergraduate science/engineering programs) so they can separate out the special features in their minds and not take them for granted.
- jacobolus 6y ago
  P.S. you can “graphically” demonstrate the non-commutativity of 3-dimensional rotation to a layperson (including children) in a few minutes by turning around an arbitrary 3-dimensional object in your hands. P.P.S. The hypervolume of an n-dimensional ball of unit radius only tends to zero as n tends to infinity if you use an n-dimensional hypercube with unit side length as your unit for hypervolume. If you use a standard simplex as your unit for hypervolume, then the volume of the ball tends to infinity. If you use the volume of the ball as your unit, then the volume of the ball tends to 1 by definition. Using a unit-length n-cube as your unit hypervolume is a convenient but arbitrary cultural convention. One main reason this ratio is ”counterintuitive” is because we conventionally use a unit-side-length square or cube but a unit-radius disk or ball, and the extraneous factors of 2 caused by the discrepancy cause confusion in low dimensions; people weirdly think of disks as being bigger than squares. If we used a unit-diameter disk/ball instead, we would see how the ratio of hypervolume of ball/cube goes strictly down as dimension goes up: dimension 1: 1 dimension 2: π/4 ≈ 0.79 dimension 3: π/6 ≈ 0.52 dimension 4: π²/32 ≈ 0.31 dimension 5: π²/60 ≈ 0.16 dimension 6: π³/384 ≈ 0.08 dimension 7: π³/840 ≈ 0.04 ...

by dvt 6y ago

jacobolus 6y ago

Commutativity of multiplication is not “lost”. It’s something you shouldn’t generally expect to have. It would be good if people would spend more time learning about non-commutative multiplications at a younger age, which should be expected in general. Mathematicians’ bias towards commutativity is often so culturally ingrained from a lifetime of commutative algebraic structures that it’s very hard for them to break away from. If they spent more time on the geometry earlier on it could help quite a bit. That multiplication of a vector by a scalar + bivector oriented in a plane containing the vector on the left happens to have the same result as multiplication by the reverse on the right Zv = vZ† comes down to the identity a(bc) = (bc)a for any three coplanar vectors a, b, c, which happens because vectors commute with both scalars and coplanar bivectors. This doesn’t work more generally because the trivector part of the product doesn’t vanish when the vectors are not coplanar. This convenient feature results in many nice results for plane geometry / complex numbers which don’t hold in more general settings. It would be good if students would start with the general version (in high school or early in undergraduate science/engineering programs) so they can separate out the special features in their minds and not take them for granted.

jacobolus 6y ago
P.S. you can “graphically” demonstrate the non-commutativity of 3-dimensional rotation to a layperson (including children) in a few minutes by turning around an arbitrary 3-dimensional object in your hands. P.P.S. The hypervolume of an n-dimensional ball of unit radius only tends to zero as n tends to infinity if you use an n-dimensional hypercube with unit side length as your unit for hypervolume. If you use a standard simplex as your unit for hypervolume, then the volume of the ball tends to infinity. If you use the volume of the ball as your unit, then the volume of the ball tends to 1 by definition. Using a unit-length n-cube as your unit hypervolume is a convenient but arbitrary cultural convention. One main reason this ratio is ”counterintuitive” is because we conventionally use a unit-side-length square or cube but a unit-radius disk or ball, and the extraneous factors of 2 caused by the discrepancy cause confusion in low dimensions; people weirdly think of disks as being bigger than squares. If we used a unit-diameter disk/ball instead, we would see how the ratio of hypervolume of ball/cube goes strictly down as dimension goes up: dimension 1: 1 dimension 2: π/4 ≈ 0.79 dimension 3: π/6 ≈ 0.52 dimension 4: π²/32 ≈ 0.31 dimension 5: π²/60 ≈ 0.16 dimension 6: π³/384 ≈ 0.08 dimension 7: π³/840 ≈ 0.04 ...