Welcome to Part Two of the Higher Dimensions Series, where we explore some of the strange and delightful curiosities of higher dimensional space. Currently, I have completed Parts One, Two, Three, Four, and Five, with hopefully more to come.
If you have not already done so, I encourage you to read Part One before continuing on with Part Two, especially if you consider yourself to be unsure of what I mean when I say higher dimensional space. We build up some concepts and tools there that will be very helpful for our journey today. Anyway, in Part One, we saw what happens to the distance between the origin and randomly selected points inside a fixed-radius ball as we move into high dimensions, and we had our minds splendidly blown. Today we will explore what I consider to be one of the most peculiar phenomena that we observe when we move from low to high dimensions, the fantastic case of the cube inside the ball! In fact, when I meet someone who conveys any amount of nonchalance regarding the wonders of higher dimensional space, the cube inside the ball is the ace up my sleeve, the sermon that leaves none unconverted. Onward it is!
Our Two Shapes
For today, we need only two shapes: a cube and a ball.
The ball was our hero in Part One of the series, so I’ll only give a quick refresher here. A ball is the set of all points that reside within a fixed distance from a given center. That may sound obscure, but stick with me. In two dimensions, a ball is just a circle along with all the points (or area) inside that circle; In three dimensions, a ball is a sphere along with all the points (or volume) inside that sphere. Each ball has a center and a radius. To keep things simple, we will always take the center to be the origin, which has coordinates of zero for all dimensions, as in (0, 0) in two dimensions or (0, 0, 0) in three dimensions. The radius is the distance from the center to the outer boundary of the ball; for example, if a ball has a radius of one unit (which will be our default for today), all points inside the ball are within one unit from the center. As we move into dimensions higher than three, we can no longer visualize the ball, but its definition remains exactly the same: all points in space within a fixed distance (defined by the radius) from the center of the ball. We use the term n-ball to refer to a ball in n-dimensional space.
What about a cube? Well, as usual, let’s start with the scenarios we can most easily relate to: two and three dimensions. In two dimensions, a cube is just a square: a closed shape with four equal-length sides and four equal angles. We call the sides edges and, for now, we will say that the edges have a length of d units. Also, as with balls, we use the term n-cube to refer to a cube in n-dimensional space. n-cubes are also called hypercubes, which is just awesome. Here is a 2-cube with edges of length d units.
In three dimensions, we have what we typically think of as a “cube”, which is a closed shape bounded by six square (2-cube) faces. Again, all the edges in the cube are the same length, d.
What about a 4-cube, you say? Or even a 10-cube?? Have no worries, I completely understand your eagerness to travel forth into the higher dimensions! However, let’s first take a bit more time to consider the lower dimensional cases, and perhaps that will inform how we can think about the higher dimensional cases. Let’s explicitly lay out the definitions of cubes with edges of length d in the lower dimensions:
- 0-cube: a single point in space (actually, any zero-dimensional object is just a point).
- 1-cube: a line of length d (again, any one-dimensional object is a line, unless it’s just a point).
- 2-cube: a square with edges of length d.
- 3-cube: a cube with edges of length d.
I trust that you are are with me on the definitions of the 2- and 3-cubes, but perhaps think the definitions of the 0- and 1-cubes came out of nowhere. However, bear with me and let’s think about the relationships between each of these cubes to see what they might have in common; specifically, how do we get from a 0-cube to a 1-cube? Then from a 1-cube to a 2-cube? Then from a 2-cube to a 3-cube? Take a long look at the following illustration before reading on (credit for the illustration goes to the awesome but possibly defunct website, Some Physics Insights):
As you can see, if we start with a 0-cube, a single point at the origin, and “sweep” though space in a single direction for a length of d units, we have ourselves a 1-cube, which a line of length d! Then, if we take our 1-cube (a line) and sweep through space in a new (second) direction for a length of d units, we have ourselves a 2-cube, which is a square with edges of length d! Are you starting to catch on, yet? Lastly, if we take our 2-cube (a square) and sweep through space in a new (third) direction for a length of d units, we’ve got our 3-cube. Incredible!
So, what comes next? How do we go from a 3-cube to a 4-cube? Well, let’s do exactly what we did for the previous cases: we take our 3-cube and sweep through space in a new (fourth!!) direction for a length of d units, and that gives us a 4-cube. Wow! Let’s allow that to soak for a minute. I imagine there are more than a few skeptics out there wondering, “what is a new fourth direction?” We all have a pretty strong sense of three dimensions (or directions): left/right, forward/back, up/down. And perhaps we’ve heard of time being referred to as the fourth dimension. But what does it mean move d units in a fourth direction? Honestly, I have no idea. But that is exactly why this is so much fun! For all who are even remotely tickled by this idea, I must refer you to the short but exceptional book, Flatland: A Romance in Many Dimensions, written by Edwin Abbot Abbot in 1884, and offer you the words of its two-dimensional protagonist, A. Square, in his attempt to articulate how to move in a new unobservable third dimension, “upward, not Northward!”.
Anyway, although we’ve migrated into the amazing world of spaces that we cannot visualize, remember that we mustn’t lose hope. We must simply equip ourselves with mathematical tools and proceed onward into the darkness! For now, our main tool is just the simple fact that an n-cube with edges of length d is formed by sweeping an (n-1)-cube in a new direction/dimension through space for a length of d units.
Where Is The Corner of The Cube?
Now that we have a crisp understanding of n-balls and n-cubes, how are we going to blow our minds? Let’s start with a very simple 2-dimensional scenario, a 2-cube with edges of length 1 unit inside a 2-ball with radius of length 1 unit, both centered at the origin. This is just a square inside a circle, like this:
By definition, we know that the distance from the center of the 2-ball to the edge of the 2-ball is 1 unit for every single point on the edge of the 2-ball, right? However, for the 2-cube, the distance from the center of the edge depends on the direction. For example, we know that the distance along either the horizontal or vertical axes is 0.5 units (half the length of the edges):
What about the distance to the corner of the 2-cube? If you have retained the tools from Part One of the series, then you know that this is a quick and easy job for the Pythagorean Theorem! Specifically, the distance to the corner of this 2-cube is the square root of (0.5² + 0.5²), or approximately 0.707 units. The corner is of particular interest here because it is the 2-cube’s farthest point from the center.
So, what have we learned? We know that every point on the outer boundary of the 2-ball is the same distance from the center, 1 unit. We also know that the distance between the center and the outer boundary of the 2-cube varies, and that the shortest distance (in the direction parallel to any of the cube’s edges) is 0.5 units and the longest distance (in the direction of the corner) is 0.707 units. Alternatively, every single point in the 2-cube is inside the 2-ball, as is very clear from the plots above.
What changes when we move into three dimensions? By definition, and as belabored in Part One of the series, we know that every point on the outer boundary of the 3-ball is the same distance from the center, 1 unit. What about for the 3-cube? Well, because the edges only have a length of 1 and the cube is centered at the origin, we know that the shortest distance (in the direction parallel to any of the cube’s edges) remains 0.5 units. How far is the 3-cube’s corner from the center? Again, this is a job for the higher dimensional generalization of the Pythagorean Theorem that we learned in Part One! Specifically, if we have a point located at the three-dimensional coordinates (x1, x2, x3), then we can calculate the distance between this point and the origin as follows:
What are the three-dimensional coordinates for the corner of the 3-cube? That’s easy! They’re just (0.5, 0.5, 0.5). If that’s not clear, take a look at the 2-cube above, and think about the three-dimensional analog. The distance from the center to the corner of the 3-cube ends up being approximately 0.866. Wow! Every point in the 3-cube is still inside the 3-ball, but it looks like the corner of the cube is closer to the edge of the ball than it was in two dimensions. Given that we are using constant definitions of our objects (an n-ball of radius 1 and an n-cube with edges of length 1) and just playing with the dimension, that is an awesome observation! Perhaps you’re not impressed. After all, this makes perfect mathematical sense: we are adding an extra dimension and shifting the coordinates of the corner 0.5 units in the direction of that new dimension, so it makes sense that the corner would be a bit farther from the center when we add a dimension. Plus, the definition of an n-ball with a radius of 1 unit requires that no point is farther than 1 unit away from the center. So what’s the big deal? When do things get weird? Let’s find out…
Below, I am going to plot the distance from the center to the corner of the n-cube for each dimension n from 2 to 100, which we can calculate using the generalization of the Pythagorean Theorem:
Let this soak for a minute. The length of every single edge in any n-cube is 1, yet the distance from the center to the corner gets farther and farther from the center as we increase dimension. Regardless of the dimension, the distance between the center to the mid-point of any single edge remains at 0.5, which is well within the boundary of the ball with a radius of 1; however, the corners of the cube are OUTSIDE the boundary of the ball as we get into higher dimensions. In fact, in four dimensions, the corner of the cube is ON the boundary of the ball (I encourage you to do the math and see this for yourself!). I’ve heard higher dimensional cubes characterized as being “spiky”, which kind of fits. The corners are protruding out into the distance ad infinitum as we increase dimension, while the midpoints of the edges remain 0.5 units from the center, something like this:
Isn’t this incredible? We have simple and constant definitions of a cube and a ball. Specifically, we start with a square that is completely inside of a circle, then a cube that is completely inside of a ball. But as we venture into the abyss of higher dimensional space, the cube is no longer completely inside the ball!
The Finishing Touch
The above images are helpful in that they help illustrate the “spikiness” of higher dimensional n-cubes, but they are also misleading. Obviously, they are attempting to illustrate higher dimensional objects in only two-dimensions, so they are fundamentally inaccurate. However, that’s not exactly what I’m talking about when I say that they are misleading. I want you to think about the “spikiness” of these higher dimensional cubes, the fact that the corners protrude outside the boundary of the ball while other points, notably the mid-points of the edges, remain inside. If your mind is not yet sufficiently blown by that observation alone, I am going to tell you something that will surely finish the job. Every single point within an n-cube is within eyesight of every single other point in the cube; in other words, you can draw a line segment between any two points inside the n-cube and the entire line segment will be inside the cube. Think about a two- or three-dimensional cube and this is clear. If you are inside a cube-shaped room, no part of the room is out of sight, you can throw a dart and hit any point on the walls, floor, or ceiling from any point inside the room. This is true for n-cubes of any dimension!
I challenge you to reconcile the fact that n-cubes are “spiky”, in that their corners stick out farther from the center than the nearest points on their boundary (as we attempt to illustrate with the illustrations above), yet every pair of points can be connected by a line that falls entirely within the cube. Go ahead, give it a try!
If you are like me, you will most certainly fail, but merely trying will elicit a sense of absolute mystique and wonder. Indeed, it is this feeling that we are after and that keeps bringing us back to the abyss of higher dimensional space.
Wrapping Up and Looking Forward
In Part One, we saw what happens to the density of random points within a ball as we treaded forth into higher dimensions; today, we discovered the bizarre but sublime surprises of the hypercube. What’s next? I have not yet decided, but the bag of higher dimensional wonders is vast and full. Until then, I encourage you to reflect on where we have been and perhaps even explore a bit of this new landscape on your own. I look forward to our next journey together!
UPDATE: Part Three is available.