It just occurred to me that Dark Energy is pushing galaxies at an accelerating rate, then why isn’t it pushing all the rest of space apart? Is there something different about the space between galaxies?
I was able to find some great graphing software GeoGebra (http://www.geogebra.org/cms/en/) so I could graph the 3x+1 function. Here are a few samples in pdf
A couple of things to notice:
– The saw-tooth look to the function means that each odd number is followed by an even.
– Once the function has reached its peak y-value, that’s the point at which the sequence begins its descent to 1.
– After plotting some very large seed values, it’s clear that this function is a fractal.
I just head about this problem yesterday from a work colleague. Here’s wikipedia reference: http://en.wikipedia.org/wiki/Collatz_conjecture
I was intrigued so plotted the problem. I plotted 3x+1 and x/2. Then, since we’re using a recursive feature, I plotted y = x.
So, we start with a random number for x, say, 3. The function 3x+1 returns 10, we reflect 10 to the y=x line and because it’s even, we reflect it down to the x/2 line and that returns 5. We reflect 5 to the y=x line but since it’s odd, we reflect it up to the 3x+1 line and that returns 16. We reflect 16 to the y=x line and because it’s even, we reflect it down to the x/2 line. That returns 8. We reflect that to the y=x line and because it’s even, it goes to the x/2 line, which returns 4. All the subsequent numbers are even and so the reflection finally falls to 1. Here’s a quick and dirty graph:
Quite a few years ago, I found this in in a magazine.
I don’t recall which on but on the reverse side are references to Brookhaven, cosmic-rays, neutrino background and American Scientist, so it was some particle physics journal, I guess. Anyway, I’ve had it in my various offices b/c I find it really cleaver. But just the other day, something occurred to me. I was reading a book on XPages and the author keeps using “an xpage”. I know that it should be “a xpage” since “x” is a consonant. But “a xpage” just doesn’t sound right. While “a function f(x)” sounds ok, this doesn’t “a f(x) function”. Same with letters starting with “h”. Then it dawned on me! The reason we WANT to say “an xpage” is because “x” SOUNDS like “ex” and so it’s like we’re writing/saying “an ex”. Same with “f”: “an ef”. “H” too: “a aitch” doesn’t feel right, of course, but it does when we are saying/writing “an aitch”. Mystery solved!!! 🙂
Rudolf Clausius walked into his garage looking for a screwdriver and said, “This doesn’t make sense. I just spent last weekend cleaning out the garage and it’s a mess again! It wasn’t anything I did. I think there’s some universal principle in place that causes garages to go from an ordered to a disordered state all by themselves. hmmmm… wait….maybe this is a UNIVERSAL principle! Maybe this has implications for the arrow of time. Maybe…. “. “Hey, Rudolf! Haven’t you found that screwdriver yet??” “I’ll be right there dear!”….