Put on your arithmetic caps, dear readers. Also your math mittens, geometry galoshes and cosine coats. Today we’re venturing after numeric prey that lurks down among the lines and angles.
There’s no danger, at least not to life or limb, but I can’t promise some ideas won’t take root in your brain. There’s a very real danger of learning something when you venture into dark territory such as this. Even the strongest sometimes succumb, so hang on to your hats (and galoshes and mittens and coats and brains).
Today we’re going after vectors and scalars (and some other game)!
Last time I focused on how it was possible for Al to see — even enclose in a tunnel — a train that appears shorter to him due to its motion. It turns out that the train Al sees is a stack of time slices of the train at different moments. As we’ve seen, lots of things look different in a moving frame.
Today I want to say a little about Em’s point of view, run some numbers, and take you through a little math (just one equation, I promise). Then, because it’s Friday (when I try to write about light), I’ll introduce you to light cones.
They’re not actually necessary, but they’re kinda cool.
The main topic this week was how simultaneity is relative to your frame of reference. How there are (virtual) lines of simultaneity where all points on some line — at all distances from you — share the same moment in time. For any instant you pick, that instant — that snapshot — includes all points in your space.
A line of simultaneity freezes the relative positions of objects at a given moment — which enables distance measurements. Simple example: When their watches both read 12 noon, Al and Em were 30 miles apart. A more mathematical example uses x, y, & z (& t), but it amounts to the same thing: a coordinate system.
The gotcha is that simultaneity and coordinate systems are relative when motion is involved!
A couple of readers have asked about the diagrams in this series of Special Relativity posts. I created them with the freeware 3D ray tracing application, POV-Ray. The diagrams are actually three-dimensional “scenes” designed to be viewed as flat pieces. If some of the “dots” look more like little spheres, that’s because they are!
I wrote some introductory posts a while ago (here, here, and here). You can read those if you want more details about the application.
For a little (optional!) Friday fun, I thought I’d share some POV-Ray images that have a bit more “dimension” to them.
3D holograms! Me want!!
Last time I introduced you to the idea of a time-space diagram, which is a kind of map used to describe motion. As with many maps and diagrams, we choose to use a flat, two-dimensional representation. Someday hologram technology may advance to casual use of three-dimensional images, but so long as we use paper and display screens, we’re stuck with two.
Motion is movement in both space and time, so we want to use one of our two dimensions to represent time. That leaves us with only one remaining dimension for space, so our diagrams exist in a reduced one-dimensional world.
Today I’ll explore that world in more detail.
Last week I introduced you to the idea of relative motion between frames of reference. We’ve explored this form of relativity scientifically since Galileo, and it bears his name: Galilean Relativity (or Invariance). Moving objects within a (relatively) moving frame move differently according to those outside that frame.
I also introduced you to the idea that light doesn’t follow that rule; that light moves the same way to all observers. This is what makes Special Relativity different. It turns out that, if a frame is (relatively) moving fast enough, some bizarre things happen.
Time-space diagrams will help us explore that.
As reported earlier, this week got off to a rough start. I let my guard down (foolishly) and got nabbed by the greedy PC rapists. All I wanted was to find a particular font for a project. The next day a more careful search turned up exactly what I needed, the fonts and just the fonts (ma’am).
Monday I mentioned that I planned to share my font-needing project with you. It’s not finished (many of my projects live a long time as I tweak them — some are living things that grow and improve forever). But it turned out so much better than I expected, I just had to share it with you this Science Fiction Saturday.
I’m also going to boldly try a new WordPress blogging trick!
Back on my first Sideband post, I wrote that, “Sideband posts are miscellaneous thoughts that accompany the main thread of posts. Think of them as small paths that meander off the main road. Some branch off, go a short ways and die after a short while. Others are scenic trails that follow along the main road.”
They never quite achieved that vision, so this year one goal is getting Sidebands back on track with that original “mission statement.”
And I’m going to start with fun topic: computer-generated 3D images!
The maps you find in some buildings and malls have a little marker flag that says, “You are here!” The marker connects the physical reality of where you are standing at that moment with a specific point on a little flat map.
Your GPS device provides your current location in terms of longitude and latitude. Those numbers link your physical location with a specific point on any globe or map of the Earth.
But to fully represent our location, longitude and latitude are not quite enough. (We might be high overhead in a hot air balloon!) To fully represent our position, we need a little more ‘tude, but in this case that’s altitude, not attitude.
We need three (and only three) coordinates to completely represent our location in space. This post is about why.
I completely forgot to rant about 3D when I wrote about the Green Lantern movie. Part of the reason I did forget is that I didn’t see it in 3D.
Except for certain special cases—basically occasionally checking to see the state of the art—I will never willingly see any movie in 3D. Number me, along with film critic Roger Ebert, as a hater of 3D.
I’ll write a much longer rant later, but I just wanted to get that out there in case anyone was going to ask about how I liked the 3D in Green Lantern. I have no opinion about the 3D (except to despise it in general), because–thankfully–I didn’t see the 3D. (I hate to think a time could come when I won’t have the choice to see the 2D version.)
On rare occasion, I will see a 3D movie, as I said, to check out the state of the art. Also with the idea of “know thy enemy.” But I would never do that with any live-action film. The only sort of film I would even consider seeing in 3D would be an animated film that was created and rendered in 3D.
The main reason for that is that 3D is necessarily darker. It has to be, because it’s a filthy trick being played on your eyes. The only way it works is to provide each eye with separate images your brain tries to merge into a 3D image. There’s various ways of doing this, but they all involve reducing the light that reaches each eye.
There’s a lot more to this, and I will
write rant about it later, but for now, please, “Just say NO to 3D!”