We are bound to a three-spatial-dimension world. As you know, that means objects reach out in three directions of motion: forward, side-to-side, and up-and-down. You could also say, "length, width, depth." Our brains are nicely built to think about 3d (or 2d or 1d) spaces and objects, though. That's quite nice for artists, and others who like graphic representations for ideas, and people who just like to think about shapes.
But what about that pesky 4th spatial dimension? What about that speculative, hypothetical, additional dimension that isn't length, width, or depth? I don't know whether or not our brains are fundamentally, physically capable of imagining a shape in four spatial dimensions. But I like to try.
Here's something I thought about recently: half of a square's perimeter is the derivative of its area, with regard to the length of its side, right?
But what about that pesky 4th spatial dimension? What about that speculative, hypothetical, additional dimension that isn't length, width, or depth? I don't know whether or not our brains are fundamentally, physically capable of imagining a shape in four spatial dimensions. But I like to try.
Here's something I thought about recently: half of a square's perimeter is the derivative of its area, with regard to the length of its side, right?
Perimeter = 4 * x.
Area = x^2.
d/dx(x^2) = 2 * x, which is half of the a square's perimeter (4 * x)/2.
Yes. Right.
And half of a cube's surface area is the derivative of its volume, with regard to the length of its side, right?
Area = 6 * x^2.
Yes. Right.
And half of a cube's surface area is the derivative of its volume, with regard to the length of its side, right?
Area = 6 * x^2.
Volume = x^3.
d/dx(x^3) = 3 * x^2, which is half of its area.
Now, picture those two equations in your mind. Why is half of perimeter the derivative of area? Imagine a square is sitting comfortably in a corner. If you increase the length of its side, x, its area will grow outward, along both of the sides that are not adjacent to the corner.
The SAME mental picture applies to the cube. It is sitting in the corner of a room, and when you increase the length of x, it grows in the regions of its three faces that aren't adjacent to the walls.
Terminology attack: as a working definition, let's take the sides (or faces) of a square (or cube) that actually expand when the length of the square or cube increases, and call those the "growth regions." A square has two growth regions that are straight lines. A cube has three growth regions that are perfect squares.
Now, here's my idea. Just try to extrapolate and imagine the derivative of a hypercube's, uh, "inside stuff." (Some call it hyper-dimensional volume. Mathematically, it's the continuation of the square to cube pattern: x^4) Well, the derivative of x^4 = 4 * x^3. That means that the "growth region" is composed of four separate 3d cubes. Also note that in some conceptual way, these cubes must be perpendicular to each other in 4-space just like a square's growth lines are 90 degrees apart and a cube's growth faces are, well, 90 degrees apart...sort of.
It makes sense... a square "grows" through two sides, a cube grows through three faces, and a hypercube grows through four regular cubes.
Now, onto some really cool, cool stuff. Let's start from square 1. Imagine a circle growing inside of a square, and then growing larger than the square. The circle's edge first intersects the square's edge at the very center of each of the square's edges.
But now suddenly imagine you're in three dimensional space. You have a cube, in which a sphere is growing, and the sphere outgrows the cube. The edge of the sphere intersects the cube first at the very center of each of the cube's faces. Now imagine the region where the sphere and cube intersect, as the sphere gets larger. Well, that intersection is a growing circle on each face of the cube. It starts as a tiny dot when the sphere barely touches the edge, and grows until the diameter of the circle exceeds the side-length of the face of the cube.
And now... bang. Imagine what would happen if the equivalent of a four-dimensional sphere, a hypersphere, was birthed in the center of a hypercube, and grew inside of it, and continued to expend its way out of the cube! Remember those four 3d cubes that compose the "growth region" of the hypercube? The intersection of hypersphere against the edge of the hypercube would compose 3d sphere inside of the 3d "edge" cubes, which would start as tiny dots in the center and grow their way out, until they became larger than the cube itself.
There's your intellectual and artistic chewing gum for the day (night?). Tell me if you can see hypercubes in your head now. I'll be really jealous.
d/dx(x^3) = 3 * x^2, which is half of its area.
Now, picture those two equations in your mind. Why is half of perimeter the derivative of area? Imagine a square is sitting comfortably in a corner. If you increase the length of its side, x, its area will grow outward, along both of the sides that are not adjacent to the corner.
The SAME mental picture applies to the cube. It is sitting in the corner of a room, and when you increase the length of x, it grows in the regions of its three faces that aren't adjacent to the walls.
Terminology attack: as a working definition, let's take the sides (or faces) of a square (or cube) that actually expand when the length of the square or cube increases, and call those the "growth regions." A square has two growth regions that are straight lines. A cube has three growth regions that are perfect squares.
Now, here's my idea. Just try to extrapolate and imagine the derivative of a hypercube's, uh, "inside stuff." (Some call it hyper-dimensional volume. Mathematically, it's the continuation of the square to cube pattern: x^4) Well, the derivative of x^4 = 4 * x^3. That means that the "growth region" is composed of four separate 3d cubes. Also note that in some conceptual way, these cubes must be perpendicular to each other in 4-space just like a square's growth lines are 90 degrees apart and a cube's growth faces are, well, 90 degrees apart...sort of.
It makes sense... a square "grows" through two sides, a cube grows through three faces, and a hypercube grows through four regular cubes.
Now, onto some really cool, cool stuff. Let's start from square 1. Imagine a circle growing inside of a square, and then growing larger than the square. The circle's edge first intersects the square's edge at the very center of each of the square's edges.
But now suddenly imagine you're in three dimensional space. You have a cube, in which a sphere is growing, and the sphere outgrows the cube. The edge of the sphere intersects the cube first at the very center of each of the cube's faces. Now imagine the region where the sphere and cube intersect, as the sphere gets larger. Well, that intersection is a growing circle on each face of the cube. It starts as a tiny dot when the sphere barely touches the edge, and grows until the diameter of the circle exceeds the side-length of the face of the cube.
And now... bang. Imagine what would happen if the equivalent of a four-dimensional sphere, a hypersphere, was birthed in the center of a hypercube, and grew inside of it, and continued to expend its way out of the cube! Remember those four 3d cubes that compose the "growth region" of the hypercube? The intersection of hypersphere against the edge of the hypercube would compose 3d sphere inside of the 3d "edge" cubes, which would start as tiny dots in the center and grow their way out, until they became larger than the cube itself.
There's your intellectual and artistic chewing gum for the day (night?). Tell me if you can see hypercubes in your head now. I'll be really jealous.
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