March 15, 2018 at 03:50AM

Today I learned: 1) You know how gravity falls off with the square of the distance between two objects? As does electromagnetism? That's because we live in three spatial dimensions (for the same reason that the surface area of a 3D sphere increases with the square of its radius). If we lived in 4D space, then forces would fall off with the cube of distance; if we lived in N-D space, forces would fall off to the power of 1/(distance^N). That also means that the inverse-R-squared law is strong evidence that we do, indeed, live in a 3D space, and not, for example, a 3D slice of a higher-dimensional space. Wouldn't that immediately falsify string theory, which posits lots and lots of dimensions? Well... no. There's a caveat to the "forces fall off to the power of 1/(distance^N)" law, which is that it only holds as long as all of the spatial dimensions have the same characteristic length scale. Now, I must admit, I don't fully understand what the "length scale" of a dimension is. Nevertheless, if a dimension is "small", then not much force will leak out into it, and the force falloff will remain very close to the 1/R^2 law. As of 2005, gravity had not been measured to a high enough precision to distinguish between a 1/R^2 falloff and an almost-1/R^2 falloff, leaving room for the possibility of other, smaller dimensions. As far as I know, that fact hasn't changed in the last decade. 2) ...how to return Amazon packages. It's ridiculuosly easy. First, you go to your orders on Amazon, find the thing you want to return, and click some relatively obvious button that says something about returning the item. Follow the instructions. If you can get it to a Kohls or an Amazon locker, they'll package it and label it and ship it to you for free. Otherwise, if you get it to a UPS store, they'll package it, label it, and ship it for some (not always outrageous) application of money. Also, to keep things snappy, if you ask for an item replacement from Amazon, they will immediately ship you the new thing. If you don't return the original item postmarked before some date, they'll automatically charge you for it again. I do wonder if you could abuse this somehow by, say, buying a ton of things at once, ordering up replacements for all of them, then shutting down your Amazon account (or your credit card) before the due date. 3) I somehow got it into my head that you could decompose any linear transformation a rotation and a scaling, possibly with a reflection. I was wrong -- I'm pretty sure scalings *don't* account for shearings, and even including shearings doesn't give you all of the linear transformations. As a side note, I *did* find the conditions under which you *can* write a linear transformation as a rotation plus a scaling -- for a linear transformation with matrix [[a, b], [c, d]], you can do that decomposition iff b = -ac/d. You're welcome? I guess?