7/12/2023 0 Comments Rotating hypercubeThe 4×4 rotation matrix is continually updated so that the object rotates along that drag path on the S2 boundary. The path of a tracked ViveRemote inside the sphere volume can then be interpreted, while the ViveMote’s trigger is pressed, as a 4D “mouse drag” on the corresponding S2 boundary. To implement the 4D virtual trackball, the interior of a unit sphere volume is lofted to the upper (w >= 0) half of the S2 bounding volume of an S3 hypersphere by mapping to. The orientation of the 4D object is internally represented as a 4×4 rotation matrix. Think of all the 3D objects that can be approximated as just a stretched sphere or cube.This is a project in which a participant can freely walk around four dimensional objects, such as a hypercube, a 24-Cell or an aerochoron, and can rotate those objects via a 4D virtual trackball, implemented as a generalization of the 3D virtual trackball. In fact, you’re probably much more capable of understanding the fourth dimension than you realize. There’s really no limit to the amount of insights you can gain from these tesseract representations, but at this point we’ve covered all the core concepts. The cubes colored and have their faces pointed inwards while the, ,, and cubes have their edges pointed inwards. Look over to the projection image and this is the exact alignment you’ll see. According to what we learned when slicing a 3D cube, this means 2 of the tesseract’s cubes are moving through the 3D universe face first while 4 others are edge first. The slice has 2 square faces and 4 rectangular faces. Look what happens when you rotate on only ZW by 45°. Note: in the projection we can’t color all 8 cubes at once because they share edges, so only 2 are highlighted at a time. Here you’ll want to select “Color Cubes” so you can keep track of the cubes between visualizations. There’s also insight to be gained by comparing the tesseract’s slices directly to its projection. Here are some more direct parallels with the 2D slices of a 3D cube. The 3D cube has two faces that aren’t intersecting with the 2D universe, and the 4D hypercube has two cubes that aren’t intersecting with the 3D universe. This makes sense when you realize the 2D slice of a 3D cube in the same position is just a square with 4 sides, even though the cube has 6 faces. Even though the tesseract has 8 potential cubes to slice, the final image appears to be a normal cube with only 6 faces. Observe the slice created when all the sliders are set to 0. Of course if a cube is too far from us on the w axis, then not even a slice will appear in our 3D universe. The same can be seen one dimension lower where a square rotating into the third dimension instantly turns into a 1D slice. You could give a cube the slightest twist into the fourth dimension and all that would remain in our 3D universe is a 2D slice. This is due to the cubes being situated somewhere along the w axis. But if we're viewing in 3D, why would the 3D cubes get sliced to 2D? The first big realization is that the object you’re looking at is just a bunch of 2D slices of the tesseract's 8 cubes stuck together! This is what a 4D cube would look like if it moved through our 3D universe! □Īgain, just go crazy playing with all the sliders for a bit. Now that we’re familiar with the basic structure of the 4D cube, we're ready to observe its slices. Again, a similar motion can be seen with squares in a 3D cube projection. Rotate on one plane and you see the cubes growing and shrinking as they move in and out of the center. This is all analogous to the 2 squares connected by trapezoids you see in the projection of a 3D cube. The outer cube appears larger because it's near us on the w axis, and the inner cube appears smaller because it's further away.Īll 6 connecting cubes are moving from near to far so they get skewed into trapezoidal prisms along the way. To see how this affects the tesseract’s 8 cubes, set all the rotation sliders to 0°. Remember that due to projection, points that are further away on the w axis will shrink towards the center of the 3D image. To start, just play around with all the rotation sliders to see how the projection morphs around.
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