Tuesday, June 30, 2015

Belated Tau Day

I realize I missed it by two days, but happy belated Tau Day everyone! What's Tau Day, you ask? It's like Pi Day (3/14), but for Tau (6/28). "But what's tau," you're probably asking.

Simply put, tau is twice pi, or \(\tau=2\pi\). Like pi, tau is irrational and transcendental. But what's so special about \(2\pi\)? Why not \(3\pi\), or \(4\pi\), or any other integer multiple?

The reason has to do with circles, geometry, and the ancient Greeks. Everyone who's taken geometry is familiar with the formula for the circumference of a circle, \(C=2\pi r\), where C is the circumference and r is the radius. This, however, is the modern formulation. The ancient Greek geometers who first came up with the concept were thinking in slightly different terms: namely, that two times the radius of a circle equals its diameter. Thus, to them, the formula would be expressed as \(C=\pi d\) where d is the diameter (\(d=2r\)).

That formulation looks nice and compact, just four symbols total (although that itself is a modern artifact, as symbolic algebra as we know it wasn't invented until the Middle Ages). However, it's not quite as fundamental as it could be, because the diameter of a circle is a less fundamental concept than its radius. A circle can be—indeed is—defined as the set of all points whose distance from a single central point is exactly equal to the radius. There's no such simple way to define a circle using a diameter other than to cut it in two and use the radius anyway. Hence, the modern formulation uses radius rather than diameter.

Unfortunately, using the radius introduces that factor of two into the picture. You see, the modern formulation goes only halfway. The ancient Greeks liked their version because it was clean and simple: take any line segment to be the diameter of a circle, multiply it by this one single, simple constant (\(\pi\)), and boom, you have the circumference. And you can see the simplicity there that all mathematicians strive for.

So let's say we want to remake the modern formulation to get rid of that ugly extra 2 in there. We can simply multiply pi by it and get tau, to make our new formula \(C=\tau r\). It's simple, elegant, and—importantly—more fundamental than using pi. Take any line segment as the radius of a circle, multiply by tau, and bam—you've got your circumference.

Pi was a good first attempt at getting a very important number in math and physics, the circle constant that shows up everywhere cyclical processes are involved. It's just the wrong constant. It's the ratio of the circumference to the diameter, when the more fundamental constant is the ratio of the circumference to the radius—tau.

It's getting late and I should wrap this post up, so I will end by providing a link to The Tau Manifesto for anyone who's still interested. It's where I first encountered the idea of using tau instead of pi, and shows in detail how tau simply makes more sense to use in many areas of math and physics. I was reflexively against the idea when I first read it, but over time I've come to see how much sense it makes—to the point I'm even writing about it here on my blog. Funny how these things happen. A hui hou!

Wednesday, June 17, 2015

Panorama Extravaganza

In my outings and explorations, I often find myself drawn to taking pictures with an eye to stitching them together into a panorama after getting home. I don't know why; I never really set out with the intention of making one, the muse just comes sometimes. The problem is that after spending time carefully hand-stitching the resulting panorama together and admiring the result they go off to live on my hard drive because it's hard to find a good way to show an image that's much wider than it is tall.

Last year I realized that with my dual-monitor setup I actually have a pretty good way to display some of my panoramas: as wallpaper for my 3,840-by-1,200 desktop. I created a wallpaper from some pictures of Polulū Valley which has adorned my desktop until today. However, I didn't get around to making more wallpapers from my old panoramas until today when I went through and created a few more from those that would stand it, then took pictures of them so I can show them off.

My Pololū Valley wallpaper.
Some panoramas just didn't work, being much too wide (although I just realized I could probably cut those into pieces and make wallpapers from them). None of them fit perfectly, of course, and all required a little cropping, but I found another four that were close enough in aspect ratio to make work with minimal cropping.

A lovely view from Laupāhoehoe Point. I love the dynamism of the waves coming in and the poignantly deep sapphire blue of the ocean.

A beautiful sunset from Pu‘u Kalepeamoa, near Hale Pōhaku, with Mauna Loa (left) and Hualalai (just visible through the clouds right of the sun).

A tranquil vista of Lake Waiau (minus the northernmost bit). This is the wallpaper I have right now, as it's just so...irenic. Idyllic, even.

I was a bit surprised to find that some of my older panoramas were actually too small to be made a proper wallpaper at this resolution, as they were too short either vertically or horizontally. I used to take photos with smaller resolutions back before I had a terabyte of hard drive space to store them on, and it definitely shows (my lovely Crater Lake panorama wasn't wide enough to cover both screens). I was equally surprised to find one of my oldest panoramas was big enough to use:

This panorama, from sometime early 2008, is a not-particularly clear shot of the lower tall of Tall El-Hammam, an archaeological site in Jordan north-east of the Dead Sea that I had the pleasure of excavating at for parts of two seasons in 2007 and 2008. It's beyond doubt the site of Sodom from the Early Bronze through Middle Bronze periods, then several other cities through Iron, Roman, and (I think) even Byzantine periods (it may be the site of the Roman-period city Livias, mentioned in texts, but whose location was otherwise unknown). As you can see, the soil in the plain of the Jordan down in the Dead Sea Valley is quite fertile and the lower tall had a lot of crops being grown on it at that time. I know they've excavating quite a lot in the lower tall in the years since I went so I don't know what it looks like now, though I hear this coming season (next January) they'll be going back to the upper tall where I dug and where this panorama was taken from.

It's so nice to finally have a place to show off my panoramas rather than letting them languish on my hard drive, and I hope you enjoyed seeing so many in one post. A hui hou!