I see some of the patterns now as I look at your well organized data. I do know that I would have never come up with your mathematical explanation as an answer to Dr. To be honest, I don’t see how you got your faces and edges either. Still, I don’t believe that is where the “2” came from if considering the “4” coming from 4 dimensions. From looking at hypercubes on the internet, something I could not visualize myself, I see that a 4D cube is often drawn as 2 3D cubes. Strang’s book, the answer to the problem says the number of 3D faces is “4 x 2 = 8.” I have no idea where he came up with that solution. I guess that is why I earned a BS in chemical engineering instead of a PhD. Although I have taken much math as a chemical engineer, problems such as this, fundamental problems in math, have always given me great difficulty. I looked at the problem for 60 minutes and only solved the number of corners. Problem: “How many corners does a cube have in 4 dimensions? How many 3D faces? How many edges? A typical corner is (0,0,1,0). Starting with a 0 dimensional cube (a point) you can safely define. So, if you want to figure out how many “square pieces” you have in a D-dimensional cube you’d take the number of squares in a D-1 dimensional cube, double it (2 copies), and then add the number of lines in a D-1 dimensional cube (from sliding). Also, you find that you’ll have two copies of the original shape (picture above). The “slide, connect, and fill in” technique can be though of like this: when you slide a point it creates a line, when you slide a line it creates a square, when you slide a square it creates a cube, etc. That is, a square (2D cube) has four corners, four edges, and one square. Now define as the number of N-dimensional surfaces in a D-dimensional cube.įor example, by looking at the square (picture above) you’ll notice that, , and. So, is a point, is a line, is a square, is a cube, and so on. that a hyper-cube will have in more than 3 dimensions.ĭefine as an N dimensional “surface”. In this case, the number of lines, faces, etc. However, mathematically speaking, it’s nothing special.Īnswer gravy: This isn’t more of an answer, it’s just an example of how, starting from a pattern in lower dimensions, you can talk about the properties of something in higher dimensions. The only difference between this and all the previous times is that we can no longer picture the process. To go to 4D, same thing: slide the cube in the new (4th) direction. To go to 3D, you’d slide the square in a new direction (the 3rd dimension) and pick up all the points the square covers. To go to 2D, you’d slide the line in a new direction (the 2nd dimension) and pick up all the points the line covers. The 4D picture (being 4D) should be difficult to understand.įor example, to describe a hypercube you start with a line ( all shapes are lines in 1D). all follow from each other pretty naturally. Lines, squares, cubes, hyper-cubes, hyper-hyper-cubes, etc. However, by describing things mathematically, and then following the calculations to their conclusions, we can get a lot farther than our puny minds might otherwise allow. If we had to completely understand modern physics to use it, we’d be up shit creek. It’s necessary to use math to describe things that can’t be otherwise pictured or understood directly. A 4D-to-2D projection, like in the picture above, would involve 2 “camera/eyeball like” projections, so it’s not as simple as “seeing” a 4D object.Īs for knowing what a 4D, 5D, … shape is, we just describe its properties mathematically, and solve. We’re used to a 3D-to-2D projection (it’s what our eyeballs do). This is akin to what a 2D camera would see, photographing from below. Bottom: By projecting out the y axis (up/down) the object is collapsed again into 1 dimension. Middle: By “projecting out” the z axis (toward/away) the object is collapsed into two dimensions. To see it, cross your eyes by looking “through” the screen until the two images line up.
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