2d objects rotate about points, 3d objects rotate about axes, while 4d objects rotate about planes. Now, in the fourth dimension, we move up one more level. They only appear so because their z values are changing, which changes their scaling (as explained in ). the x-axis, the x values still appear to vary. One small observation to take note of is that when rotating about an axis, e.g. Similarly, rotation about the z-axis (A/D) means that z-values are constant as x and y vary, while rotation about the x-axis (up/down) means that x values are constant, while y and z values vary. What this rotation means is that the x values and z values vary from -1 to 1, but y values are constant. If you press Left/Right, you rotate the cube about its y-axis. In, we can only truly rotate about a single point (press A/D). The next concept to understand in the fourth dimension is rotation. Things that are further away in the w direction just appear shrunken! So that inner cube in the may look smaller, but it's the same size, just further away in the w direction! To display our points in 3d space, we divide all x, y and z coordinates by their w distance from the camera. That is exactly how we represent the ethereal fourth dimension. That's the essence of projection from 3d space to 2d divide x and y values by the z distance, and we have projection onto our screens! In essence, things that are twice as far away appear half as big. In fact, they are divided by the distance from the camera, and hence the projected x and y values of coordinates with greater z values (where higher z means deeper into the screen) are smaller than those with smaller z values, even though they have the same actual x and y values. In other words, even though it is a unit cube where the x and y values are equivalent, (let the x-axis be along the length of your screen and the y-axis along the height) the projected x and y values of the further vertices are shrunken. Observe how the near points appear "larger" than the further points. To understand how we project the 4d cube into three dimensions, we can draw an analogy from how we visualize 3d objects on our 2d screen. (for this visualization, the unit hypercube has a central w depth of 5, so the nearest vertices have w of 4, and the furthest have w of 6)īut we can't really see this w dimension, since it's in some ethereal space outside of our three dimensions, so how do we visualize it? We do it by projecting the four dimensional hypercube into three dimensions, and project this "3d shadow" on our screens! So things that are in "our 3d space" have a w distance of 0, things further away in the ethereal fourth dimension will have a larger w value, and things closer in the fourth dimension will have a smaller w value. So, we treat coordinates as having a length, depth, height, and "depth in the fourth dimension", which I'll just call "w distance" for short. What this viewer does however, is to visualize a fourth spatial dimension that extends our 3d space, in an analogous way that our three dimensional space extends two dimensions. What is the fourth dimension? A common answer might be "it's time!" That's not wrong when we say "four dimensional", it can refer to anything that requires four numbers to describe an object's size or location. Stick to simple rotations with the arrow keys and A/D first, before doing plane rotationsįor a detailed explanation, press the above tabs for a walkthrough of how to interpret the fourth spatial dimension! it looks wonky to your eyes), reset the view (R) and calibrate your mind by visualizing the cube again, focusing on the fact that the corners with the largest points are the closest to you.
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