Both the original square and the ending square share all of them, but each has a different, fixed, z coordinate. Likewise, the cube-creating square has a set of xy-coordinates. But the original line has one y-coordinate (for all its points) and the new line has another. When swept through y to make a square, the new line has the same x-coordinates. The only difference is that they differ in the new coordinate - which is fixed to a single, different, value in each.įor example, a line has a set of x-coordinates. One observation is that, with regard to the sweep object, both the old and new share the same dimensional coordinates. ![]() The diagrams above try to illustrate the sweep through the purple dimension w, but what really happens? Understanding a tesseract requires some imagination. Remember that the blue and red cubes are actually in the same 3D location! Only w separates them. It also shades in the 12 “squares” created by sweeping the blue lines. This version shows all 8 lines (purple) created by sweeping the points of the blue cube through the w dimension. The idea of an axis or center point has no meaning. It has no dimensions and, thus, no coordinates. Next, I considered how to get from a point to a line, from a line to a square, from a square to a cube, and from a cube to a tesseract ( and on to higher-dimensional objects!). Turns out there is, but I didn’t find it until much later. I gave up, because it was clear the sequences were due to geometry and increasing dimensions, so maybe there wasn’t a simple formula describing the sequences. Regular sequences, sure, but not well-related. Made me think I was on to something!īut the progression in the next diagonal (light yellow) is 2, 3, 4, which is nice and regular, but how did we jump from 1, 1, 1, 1 to that? The next diagonal (light red) was worse: 4, 8. That also seemed to give each column a base number (2, 4, 6, 8). When I factored the numbers as shown I found another diagonal of identities (light green). Obviously, it takes one square to make a square. I noticed the diagonal of identities (light blue). But lines? What formula gives you 1, 4, 12, 32? Squares are even worse: 1, 6, 24? I’m not an expert mathematician, so I never came up with a simple formula that explains the column sequences. 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. 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.
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