| Check out this link from the venerable Ken Perlin: https://mrl.nyu.edu/~perlin/experiments/demox/Hyper.html There's a hard-to-get-running-due-to-age Java applet there which, when it used to run easily, was very powerful for getting me to understand 4D in an intuitive way. It has a variety of stereo viewing modes, ranging from eye-crossed 3D (and wall-eye), to red/blue glasses, and so forth. It also has an excellent option for "thick" lines, that accentuates the position of the 4D object by making lines closer to the camera thicker. The trick is to use your brain's two different depth perception mechanisms in different ways. You have hardware depth perception in the form of parallax difference between your two eyes, and you have software depth perception in the form of your brain's image analysis capabilities that, given a 2D wireframe, can determine a 3D projection of it. This is always fun with optical illusions, where you can see that your brain only really uses small localities for these calculations - observe the traditional "blivet" fork for an example. Anyone who has looked at an isometric rendering of a wireframe 3D cube on paper knows that there are two ways you can perceive this shape, and that's the key to getting 4D intuitive perception going. You should already be accomplished at mentally switching the cube back and forth without even closing your eyes before you move on to another dimension! If you use a hardware stereo mechanism like crossing your eyes to get one depth axis, you can concentrate and use that software depth perception to get the second depth axis. In particular, it's important to note that different mouse buttons and key combos will rotate the shape in different axes, so just make gentle movements with each to get a feeling for the range of motion. One set of axes will rotate the cube as though its 3D projection was one of the two possible software-depth-perception interpretations of it, and the other set of axes will rotate it as though the other interpretation was correct. You can seamlessly move back and forth between those states! Perlin's applet features a variety of shapes, from the classic hypercube, to a simplex with the fewest possible edges (akin to a triangular pyramid in 3D) on to a sort of Klein bottle. It's really tricky to figure that one out, but in the end it's basically a sort of hypertorus where the inside and outside can become one another. Yeah, that's not really conveying very well in text, is it.... |