![]() ![]() I looked at that table for a while trying to figure out a formula describing the mathematical progressions. (The numbers in square brackets are factors of the bold numbers.) Each row indicates how many instances of the component shape are in a given object: Square Shapes The table below lists the square shape objects along the left and their component parts across the top. (In this case, “square” has more the “right-angle” meaning than the four-sided shape, although that shape is one of the shapes involved.) That got me wondering what the count table looked like for all those regular square shapes. More to the point, Egan mentions that a tesseract is composed of 8 cubes, 24 squares, 32 lines, and 16 points. The inspiration for this came from a Greg Egan book ( Diaspora) that mentions tesseracts (you run into them in science fiction sometimes one of my childhood SF short story collections had a story featuring a tesseract house). For myself, I find writing (or talking) about a topic helps clarify it, so this is mostly an exercise for the writer. No promises that this will be coherent, useful, or even interesting, but it is long. It was an interesting diversion, and at least I think I understand that image now!įWIW, here’s a post about what I came up with… Recently I spent a bunch of wetware CPU cycles, and made lots of diagrams, trying to wrap my mind around the idea of a tesseract. This is always the case for the boundary of a shape in any dimension – consider the circle, C = 2(pi)r is the derivative of A = (pi)r^2, or the sphere volume & surface area formulae.If you’re anything like me, you’ve probably spent a fair amount of time wondering what is the deal with tesseracts? Just exactly what the heck is a “four-dimension cube” anyway? No doubt you’ve stared curiously at one of those 2D images (like the one here) that fakes a 3D image of an attempt to render a 4D tesseract. You may notice the (n-1)-dimensional surface-volume is the derivative of the n-dimensional volume. When s = 2n (side length = 2 x number of dimensions) these formulae will yield equal results. (n-1) dimensional surface-volume = (2n) x s^(n-1) Generally, an n-dimensional cube with side lengths s has: (I can’t prove this, but a little internet research reveals it!) The number of (n-1)-dimensional boundary elements of an n-dimensional cube is 2n.Ī 4d hypercube has 8 x 3d cubes surrounding The 3-dimensional surface-volume (akin to surface-area) would indeed be 8 x 8^3, so this would be an equable 4d shape. ![]() The 4-dimensional volume of this would be 8^4. ![]() I believe the shape you are referring to is a hypercube or tesseract. ![]() Can anyone help? Oh, and if you could do it in such a way that a 12 year could understand that’d be great. What is 4D space called? It’s fair to say that my 4D shape work has been somewhat lacking in the past and I feel that I’m now at a stage in my life where I’m happy to delve into 4D. I was able to describe that the 4D shape in question would be bounded (somehow) by 8 cubes of size 8x8x8, which I hope is actually true. What I’d like help with is how to determine if that’s true. Now, I have a couple of students who noticed that in 2D, a 4×4 shape is equable and in 3D, a 6圆圆 shape is equable and so, they suggested that in 4D, an 8x8x8x8 shape should be equable with volume = to hypervolume? It’s really nice to get to the point where students are actually asking if they can use algebra to solve something. This is the perfect cue for “You’re right, there is a quicker way…” and demonstrating an algebraic approach. Usually, someone eventual points out that just trying rectangles isn’t very efficient or, more bluntly, calls it tedious. In year 8 (ages 12 and 13), we have a project called Equable Shapes which starts off with trying to find a rectangle where the area and perimeter have the same value (ignoring units). Students often stumble across some through trying various shapes and are more successful if they are systematic in their approach. ![]()
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