Thinking Beyond the Third Dimension
Thinking Beyond the Third Dimension
A mathematician's talk on reality.
On May 1st, 2016, I co-curated a roundtable discussion entitled,
“While You Were Texting” in New York City. The thirteen panelists
presented subjects for discussion, but it was John Kiehl who provoked
the room and really got people talking. It seems the implications of
this reality challenge the meanings of our existence and consciousness.
Below is an adaptation of the transcript from his presentation.
The average person is kind of up on the latest thing in science. But when I started to learn what mathematicians know to be true, it's astounding, and none of it informs any of the discussions we're having. That’s because the average man in the street doesn't know about Gödel's Incompleteness Theorem, which is actually as close as a mathematician gets to a joke. It says, “If you can form a system of logic that includes addition, you can say things in that language that are not provable in that system.”
The two things that I've tuned into in this awareness I have about mathematics are complexity and higher dimensional space. Stephen Wolfram is a scientist who, in 1980, when he was trying to figure out what he was going to do with his career, got fascinated by the fact that 20th century physics was very good at asking new questions, but with no new answers coming in. He said, “When that happens historically, that means the tools you are using are no longer good enough.” The tool that he decided wasn't good enough for 20th century physics was mathematics. So he said, “If I throw mathematics out, where do I start?” The minute he turned that corner, he started making some amazing discoveries, and a language of patterns.
He started playing with pixels, little patterns. All of his patterns sort of ended up looking like veins in leaves and leopard spots and things from the natural world.
But he realized, “I don't think this is going to lead anywhere successful if I start trying to catalogue patterns.” So he took a step back and said, “I'm going to catalogue the algorithms, or the computations that go into these patterns. That actually might be something that I can work with as a scientist.”
From there, he played. He used pixel patterns that have only eight different ways of dealing with their neighbors, creating 256 possible patterns. This is also completely deterministic, meaning we know everything about the system. There’s no quantum mechanics magic sauce here; it's one plus one equals two, every time, all the time.
His work is so maniacal.
He just goes about things very simply. To start, rule one does (almost) absolutely nothing. Rule two makes a diagonal line. Rule three makes a vertical line. But he got to rule 30, and rule 30 creates chaos. Rule 30, a completely deterministic system just like the other 29 rules before that, explores every possibility that can ever happen in this universe.
I think Stephen discovered what we've always wanted to be true, that from nothing comes something, and from that differentiation comes the 10,000 things. He discovered how the universe can have insanely simple, fundamental grounding, and yet all of this complexity can come from that. It is not a hindrance to be born into a universe that is completely deterministic at some level.
The other thing, higher dimensional space, is not as much fun to talk about. A tabletop is a two-dimensional space. Glasses are three-dimensional objects. But—bear with me for just a second—if I take a square, and I put a circle inside that square, there's a distance between the circle and the corner of the square. It turns out that that diagonal is the square root of two. The circle has a radius of one. So this “little distance” is 1.414 minus one, some little number.
If we go into three dimensions, that diagonal is now the square root of three. But the radius of our circle is still one. So that distance has gotten a little bigger, right? Now if we go into nine dimensions—in nine dimensions,
this thing that is a square now has an inconceivable number of sides and vertices and what have you—the diagonal, the “little distance,” now is the square root of nine, which is three. The circle, however, still has radius one. It's sitting inside a square, but somehow its diagonal has gotten longer and longer and longer because of the additional “elbowroom.”
That means the distance from the sphere to the vertices is two, which means we can surround the circle with another circle and still be inside the square. That's something we can't even begin to do in that first, two-dimensional square.
So every time I look at some stockbroker show me his two-dimensional chart, which is some weird shadowed projection from a nine-dimensional space or a 50 dimensional space or a 100-dimensional space—they don't have a clue about the elbowroom they’re swimming in, and that’s why their stock market and their projections and our banking systems and our connectivity are going to constantly surprise us. It’s because our minds can't navigate anything but three-dimensional space.
Over the years, I've been trying to collect these weird things that happen in higher-dimensional space just to remind me that if we're going to solve these problems, we've got to stop kidding ourselves that we can look at them and take them on the surface.
Here’s another example: When you see a fruit stand where they stack up oranges, you can think of one orange as a three-dimensional circle, a sphere. If you could peer into this pyramid of oranges and pick out one orange, you would find that in three dimensions, 12 or 13 oranges surround it. In four dimensions, the next dimension, they still don't know. Even at this late date, they still argue—is it 23 or 24 oranges that surround that circle? That's how mysterious higher dimensional spaces are.
To help you understand how mysterious higher dimensional spaces are, let’s talk about infinity. Even acknowledging the work of the ancient Greeks,
who used infinity to solve problems of volume, and people like Newton and Leibniz, who invented calculus and manipulated infinity in the late 1600s, infinity itself was not put on solid ground until 1890. That's how recent in the history of mankind’s awareness of his universe that infinity was put on solid ground: around 1890. That scientist, Poincare, in thinking about shapes and infinity and higher-dimensional spaces, said, “You know what? I bet you spheres are so simple that in four dimensions, if it looks and smells like a sphere, it is a sphere.”
So what makes a sphere look and smell like a sphere? Well, if you stand on a sphere, no matter which way you look, the sphere kind of bends away from you with the same amount of curvature. That's what he meant, right? He said, “I can't prove it, but I'm pretty sure as we go into these higher-dimensional spaces, if we set ourselves down on these higher-dimensional spheres and notice how they're curving, if it looks and smells like a sphere, it is a sphere.”
This was just proven a few years ago by a Russian mathematician named Grigori Perelman. Here’s the interesting thing: it was actually proven for dimensions eight and higher in the 60s. Then, a few years went by and someone proved it for seven dimensions. Then, someone proved it for six dimensions, then five dimensions, and then, finally, four dimensions. The very next thing from our world, the thing that Poincare was looking at, was the very last to be solved.
There's something magical about the jump from three dimensions to four dimensions, and this happens all the time in mathematics. You can prove something for one, two, and three dimensions. You can also prove it for five dimensions and up, but damn, solving it for four dimensions is a bitch. I think in this universe in which live, we are all the phenomena that can't bust through to four dimensions—that somehow, whatever is the fundamental thing that makes this universe click, it has the same problem that mathematicians have. It just can't bust through the four dimensions.
Below is an adaptation of the transcript from his presentation.
Source: John Kiehl
John Kiehl is a mathematician, technologist, and music producer who has collaborated with Stephen Wolfram.
The average person is kind of up on the latest thing in science. But when I started to learn what mathematicians know to be true, it's astounding, and none of it informs any of the discussions we're having. That’s because the average man in the street doesn't know about Gödel's Incompleteness Theorem, which is actually as close as a mathematician gets to a joke. It says, “If you can form a system of logic that includes addition, you can say things in that language that are not provable in that system.”
The two things that I've tuned into in this awareness I have about mathematics are complexity and higher dimensional space. Stephen Wolfram is a scientist who, in 1980, when he was trying to figure out what he was going to do with his career, got fascinated by the fact that 20th century physics was very good at asking new questions, but with no new answers coming in. He said, “When that happens historically, that means the tools you are using are no longer good enough.” The tool that he decided wasn't good enough for 20th century physics was mathematics. So he said, “If I throw mathematics out, where do I start?” The minute he turned that corner, he started making some amazing discoveries, and a language of patterns.
He started playing with pixels, little patterns. All of his patterns sort of ended up looking like veins in leaves and leopard spots and things from the natural world.
But he realized, “I don't think this is going to lead anywhere successful if I start trying to catalogue patterns.” So he took a step back and said, “I'm going to catalogue the algorithms, or the computations that go into these patterns. That actually might be something that I can work with as a scientist.”
From there, he played. He used pixel patterns that have only eight different ways of dealing with their neighbors, creating 256 possible patterns. This is also completely deterministic, meaning we know everything about the system. There’s no quantum mechanics magic sauce here; it's one plus one equals two, every time, all the time.
His work is so maniacal.
He just goes about things very simply. To start, rule one does (almost) absolutely nothing. Rule two makes a diagonal line. Rule three makes a vertical line. But he got to rule 30, and rule 30 creates chaos. Rule 30, a completely deterministic system just like the other 29 rules before that, explores every possibility that can ever happen in this universe.
I think Stephen discovered what we've always wanted to be true, that from nothing comes something, and from that differentiation comes the 10,000 things. He discovered how the universe can have insanely simple, fundamental grounding, and yet all of this complexity can come from that. It is not a hindrance to be born into a universe that is completely deterministic at some level.
The other thing, higher dimensional space, is not as much fun to talk about. A tabletop is a two-dimensional space. Glasses are three-dimensional objects. But—bear with me for just a second—if I take a square, and I put a circle inside that square, there's a distance between the circle and the corner of the square. It turns out that that diagonal is the square root of two. The circle has a radius of one. So this “little distance” is 1.414 minus one, some little number.
If we go into three dimensions, that diagonal is now the square root of three. But the radius of our circle is still one. So that distance has gotten a little bigger, right? Now if we go into nine dimensions—in nine dimensions,
this thing that is a square now has an inconceivable number of sides and vertices and what have you—the diagonal, the “little distance,” now is the square root of nine, which is three. The circle, however, still has radius one. It's sitting inside a square, but somehow its diagonal has gotten longer and longer and longer because of the additional “elbowroom.”
That means the distance from the sphere to the vertices is two, which means we can surround the circle with another circle and still be inside the square. That's something we can't even begin to do in that first, two-dimensional square.
So every time I look at some stockbroker show me his two-dimensional chart, which is some weird shadowed projection from a nine-dimensional space or a 50 dimensional space or a 100-dimensional space—they don't have a clue about the elbowroom they’re swimming in, and that’s why their stock market and their projections and our banking systems and our connectivity are going to constantly surprise us. It’s because our minds can't navigate anything but three-dimensional space.
Over the years, I've been trying to collect these weird things that happen in higher-dimensional space just to remind me that if we're going to solve these problems, we've got to stop kidding ourselves that we can look at them and take them on the surface.
Here’s another example: When you see a fruit stand where they stack up oranges, you can think of one orange as a three-dimensional circle, a sphere. If you could peer into this pyramid of oranges and pick out one orange, you would find that in three dimensions, 12 or 13 oranges surround it. In four dimensions, the next dimension, they still don't know. Even at this late date, they still argue—is it 23 or 24 oranges that surround that circle? That's how mysterious higher dimensional spaces are.
To help you understand how mysterious higher dimensional spaces are, let’s talk about infinity. Even acknowledging the work of the ancient Greeks,
who used infinity to solve problems of volume, and people like Newton and Leibniz, who invented calculus and manipulated infinity in the late 1600s, infinity itself was not put on solid ground until 1890. That's how recent in the history of mankind’s awareness of his universe that infinity was put on solid ground: around 1890. That scientist, Poincare, in thinking about shapes and infinity and higher-dimensional spaces, said, “You know what? I bet you spheres are so simple that in four dimensions, if it looks and smells like a sphere, it is a sphere.”
So what makes a sphere look and smell like a sphere? Well, if you stand on a sphere, no matter which way you look, the sphere kind of bends away from you with the same amount of curvature. That's what he meant, right? He said, “I can't prove it, but I'm pretty sure as we go into these higher-dimensional spaces, if we set ourselves down on these higher-dimensional spheres and notice how they're curving, if it looks and smells like a sphere, it is a sphere.”
It took 100 years.
This was just proven a few years ago by a Russian mathematician named Grigori Perelman. Here’s the interesting thing: it was actually proven for dimensions eight and higher in the 60s. Then, a few years went by and someone proved it for seven dimensions. Then, someone proved it for six dimensions, then five dimensions, and then, finally, four dimensions. The very next thing from our world, the thing that Poincare was looking at, was the very last to be solved.
There's something magical about the jump from three dimensions to four dimensions, and this happens all the time in mathematics. You can prove something for one, two, and three dimensions. You can also prove it for five dimensions and up, but damn, solving it for four dimensions is a bitch. I think in this universe in which live, we are all the phenomena that can't bust through to four dimensions—that somehow, whatever is the fundamental thing that makes this universe click, it has the same problem that mathematicians have. It just can't bust through the four dimensions.
