A couple of hours ago I drove back to Ithaca from Saugerties. It was amazing outside. Warm but not hot, a little windy, and the leaves are just starting to catch Autumn flame. I went by way of Route 28 and Route 17, and in the middle of this is a weird little road called Route 30.
No, it’s name is not what is weird about it.
The road twists and turns its way around the Pepacton Reservoir, which is this lovely collection of water surrounded by pillows of hills. (I don’t think I actually knew how to spell “reservoir” until just now.) My favorite part is before you’ve gotten a good look at the water, where you’re mostly encompassed by trees, and suddenly everything opens up and you’re crossing this long, thin bridge across the middle of the open water. To either side the view is stunning. Someday I’m just going to stop in the middle of the bridge and take a good, long look. After the bridge, there is one of the zaniest series of turns I have ever seen a road take. I’d almost swear that some of them turn me around more than 360 degrees, which I would have thought to be very unlikely until I drove on this road.
On my trip I listened to two audiobooks, audiobooks being a secret passion of mine. First I listened to Richard Feynmann’s Six Not So Easy Pieces, which is actually a collection of a couple of recordings of his lectures. He must have been incredible to see in a classroom. I’d actually listened to some of this already, so the remaining bit didn’t fill my entire trip. For the rest of the time, I listed to some of Garrison Keillor’s Tales from Lake Woebegone, which are also amazing. He’s amazing. They’re just great.
What I wanted to write about here, though, is regarding some of Feynmann’s material. It was probably around the fifth piece. He had given an introduction to the idea of spacetime, much of which seemed dedicated to convincing people that the idea of a four dimensional vector space is not so crazy. (I didn’t need much convincing, of course.) He was going to start talking about Einstein’s reinterpretation of gravity as causing curved spacetime. To do this, he wanted to describe what he meant by curved space. He was basically trying to give a very informal, but fairly exact, description of Riemannian manifold.
He explained that as far as four-dimensional space is concerned, our imaginations suck. The only method we have open to us is to work by analogy, imagining how we in three dimensional space would look at someone stuck in two dimensional space. (This is very Flatland, which means it’s very smart.) He then took three examples of two dimensional spaces, and asked us to imagine little bugs who live there. He added later that these are bugs who decided to investigate some geometry. He then paused, and commented that these are very unusual bugs.
The first example was our usual Euclidean plane. The second was the surface of a ball, a sphere. (To many people, a sphere and a ball are the same thing. To mathematicians, of course, a sphere is the surface of a ball.) The way to measure distance in these first two examples is just what you think it is. Now we take a weird turn. His third example was bugs on a hotplate, armed with metal rulers. This requires some explanation, but is actually incredibly cool.
Imagine a hotplate the size of a plane. The temperature is different in different spots. At one spot, which we can think of as the origin of the plane, the temperature is pretty low. The further away from the origin you get, the warmer it gets. Now, the bugs on this hotplate have metal rulers to measure distance, and of course as these metal rulers expand or contract based on the temperature. Near the origin, the rulers stay a pretty normal size, but as the bug carries his ruler away from the origin it expands. This means that when he measures distances far away from the origin, the distances won’t seem as large to him as they do to us. The ruler gets bigger! Do you get it?
What I want to say is that when he’s far away from the origin and he sprints for a second, he hasn’t actually gone as far as he thinks he has. But this is ridiculous. The only way the bug can measure the distance is with his expanded ruler. He doesn’t notice anything! But if we’re watching him from above, we will definitely notice the difference. To us, the further away the bug is from the origin, the faster he can move, because distances which look huge to us are not really that big to the bug anymore. As the bug approaches the origin, he will appear to slow down.
The three spaces that Feynmann was describing were models of the Euclidean plane, the spherical plane, and the hyperbolic plane. He goes on to describe how one can measure curvature in these spaces by studying triangles, or by studying circles. If the curvature isn’t zero, weird stuff starts happening. Mathematical X-Files stuff. An important point he mentions is that the bugs can do this themselves. Curvature is intrinsic. You don’t need some higher dimensional cohort to measure it for you. The Euclidean plane has zero curvature, the spherical plane has positive curvature, and the hyperbolic plane has negative curvature. Feynmann briefly mentions that you could have a space where the curvature changes from place to place, like the space we live in. This is a general Riemannian manifold.
Another cool model of the hyperbolic plane is the Poincare disc. (Poincare should have an accent aigu over the “e”, but I’m afraid it might get lost when I move this text around. Plus I can’t figure out where to find it.) Take the open unit disk in the Euclidean plane, and imagine that the rulers shrink as you stray away from the origin towards the boundary. In fact, they shrink so fast that you move more and more slowly as you get to the boundary - so slowly that you never reach the boundary. This model has a nice complex analysis description, which I don’t remember. (It’s probably not that nice anyway.) You can find very nice pictures of this online, or in Bill Thurston’s book on three dimensional geometry and topology.
I was going to write about my ponderings regarding the origin of the term “hyperbolic” to describe these spaces, but I just looked it up in Wikipedia and found the truth, or at least bread crumbs leading to the truth. Now that I think about it, though, why spoil the suspense? Join me in some speculation.
A hyperbole is a figure of speech whose main feature is gross exaggeration. I will share that I just learned that the root of the word comes from Greek for “overshooting” or “excessive”. (I think I could have reasonably been expected to have deduced this.) The main feature of the hyperbolic models I described above is that weird things started happening as you moved towards the fringes, or away from the middle. The weird things were actually of a very specific type, namely exaggeration of length, or its opposite, understatement of length. I think I’ll stop there. I’ve shared enough of my brilliant insight.
No, it’s name is not what is weird about it.
The road twists and turns its way around the Pepacton Reservoir, which is this lovely collection of water surrounded by pillows of hills. (I don’t think I actually knew how to spell “reservoir” until just now.) My favorite part is before you’ve gotten a good look at the water, where you’re mostly encompassed by trees, and suddenly everything opens up and you’re crossing this long, thin bridge across the middle of the open water. To either side the view is stunning. Someday I’m just going to stop in the middle of the bridge and take a good, long look. After the bridge, there is one of the zaniest series of turns I have ever seen a road take. I’d almost swear that some of them turn me around more than 360 degrees, which I would have thought to be very unlikely until I drove on this road.
On my trip I listened to two audiobooks, audiobooks being a secret passion of mine. First I listened to Richard Feynmann’s Six Not So Easy Pieces, which is actually a collection of a couple of recordings of his lectures. He must have been incredible to see in a classroom. I’d actually listened to some of this already, so the remaining bit didn’t fill my entire trip. For the rest of the time, I listed to some of Garrison Keillor’s Tales from Lake Woebegone, which are also amazing. He’s amazing. They’re just great.
What I wanted to write about here, though, is regarding some of Feynmann’s material. It was probably around the fifth piece. He had given an introduction to the idea of spacetime, much of which seemed dedicated to convincing people that the idea of a four dimensional vector space is not so crazy. (I didn’t need much convincing, of course.) He was going to start talking about Einstein’s reinterpretation of gravity as causing curved spacetime. To do this, he wanted to describe what he meant by curved space. He was basically trying to give a very informal, but fairly exact, description of Riemannian manifold.
He explained that as far as four-dimensional space is concerned, our imaginations suck. The only method we have open to us is to work by analogy, imagining how we in three dimensional space would look at someone stuck in two dimensional space. (This is very Flatland, which means it’s very smart.) He then took three examples of two dimensional spaces, and asked us to imagine little bugs who live there. He added later that these are bugs who decided to investigate some geometry. He then paused, and commented that these are very unusual bugs.
The first example was our usual Euclidean plane. The second was the surface of a ball, a sphere. (To many people, a sphere and a ball are the same thing. To mathematicians, of course, a sphere is the surface of a ball.) The way to measure distance in these first two examples is just what you think it is. Now we take a weird turn. His third example was bugs on a hotplate, armed with metal rulers. This requires some explanation, but is actually incredibly cool.
Imagine a hotplate the size of a plane. The temperature is different in different spots. At one spot, which we can think of as the origin of the plane, the temperature is pretty low. The further away from the origin you get, the warmer it gets. Now, the bugs on this hotplate have metal rulers to measure distance, and of course as these metal rulers expand or contract based on the temperature. Near the origin, the rulers stay a pretty normal size, but as the bug carries his ruler away from the origin it expands. This means that when he measures distances far away from the origin, the distances won’t seem as large to him as they do to us. The ruler gets bigger! Do you get it?
What I want to say is that when he’s far away from the origin and he sprints for a second, he hasn’t actually gone as far as he thinks he has. But this is ridiculous. The only way the bug can measure the distance is with his expanded ruler. He doesn’t notice anything! But if we’re watching him from above, we will definitely notice the difference. To us, the further away the bug is from the origin, the faster he can move, because distances which look huge to us are not really that big to the bug anymore. As the bug approaches the origin, he will appear to slow down.
The three spaces that Feynmann was describing were models of the Euclidean plane, the spherical plane, and the hyperbolic plane. He goes on to describe how one can measure curvature in these spaces by studying triangles, or by studying circles. If the curvature isn’t zero, weird stuff starts happening. Mathematical X-Files stuff. An important point he mentions is that the bugs can do this themselves. Curvature is intrinsic. You don’t need some higher dimensional cohort to measure it for you. The Euclidean plane has zero curvature, the spherical plane has positive curvature, and the hyperbolic plane has negative curvature. Feynmann briefly mentions that you could have a space where the curvature changes from place to place, like the space we live in. This is a general Riemannian manifold.
Another cool model of the hyperbolic plane is the Poincare disc. (Poincare should have an accent aigu over the “e”, but I’m afraid it might get lost when I move this text around. Plus I can’t figure out where to find it.) Take the open unit disk in the Euclidean plane, and imagine that the rulers shrink as you stray away from the origin towards the boundary. In fact, they shrink so fast that you move more and more slowly as you get to the boundary - so slowly that you never reach the boundary. This model has a nice complex analysis description, which I don’t remember. (It’s probably not that nice anyway.) You can find very nice pictures of this online, or in Bill Thurston’s book on three dimensional geometry and topology.
I was going to write about my ponderings regarding the origin of the term “hyperbolic” to describe these spaces, but I just looked it up in Wikipedia and found the truth, or at least bread crumbs leading to the truth. Now that I think about it, though, why spoil the suspense? Join me in some speculation.
A hyperbole is a figure of speech whose main feature is gross exaggeration. I will share that I just learned that the root of the word comes from Greek for “overshooting” or “excessive”. (I think I could have reasonably been expected to have deduced this.) The main feature of the hyperbolic models I described above is that weird things started happening as you moved towards the fringes, or away from the middle. The weird things were actually of a very specific type, namely exaggeration of length, or its opposite, understatement of length. I think I’ll stop there. I’ve shared enough of my brilliant insight.
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