A lot of our work then becomes counting how many of each kind of object there are, and discussing how those building blocks are put together. We're going to think of objects as being built out of lower-dimensional objects as if they were building blocks.
For this section, we're going to be taking what's called a combinatorial approach. This can make it easier to follow certain transformations.As a way of proving you learned some things about 4D, I'm asking you to understand two 4D objects a lot: a 4D cube and a 4D triangle (or simplex). Scaling shows a sweep across the 4D hypervolume (in the same way scaling a 3D cube along one axis shows the sweep across its internal 3D volume).Ĭell Visibility: Hide or show individual cells. Scale: Scale the tesseract along any of the 4 axes in 4D space. The first 3 rotations include the W-axis, and thus will affect the projection more directly.Īutorotate: Animate rotation of the tesseract along any of the 6 planes formed by pairs of axes in 4D space. Rotate: Rotate the tesseract along any of the 6 planes formed by pairs of axes in 4D space. Unfold: Rotate the cells of the tesseract into 3D space in the form of a Dalí cross. The coloring of the pair of cells on an axis will be light and dark shades of their axis coloring, with the positive cell having the darker shade and the negative cell, the lighter one. cell: Each cell is colored separately.axis: Cells are colored by axis, with the X-axis pair colored red, the Y-axis pair colored green, the Z-axis pair colored blue, and W-axis pair colored yellow.A cell-first view of a tesseract will project orthographically to a 3D cube. orthographic: Projection flattens the 4D scene to 3D without any scaling due to distance.Cells at an angle to the hyperplane of projection appear as distorted cubes (or frustums). Cells further from camera project to smaller cubes than nearer cells. perspective: A "camera" in 4D space is placed at some distance from the origin along the W-axis.solid: Opaque rendering of the projection.cutout: Opaque rendering of the projection with holes cut out of each face to expose the internal cells.transparent: Transparent rendering of the projection.Click and drag to orbit the camera, and use the mouse wheel to zoom in and out. The 3D projection of the tesseract can be explored using the mouse. The tesseract can be manipulated in 4D space, and its projection into 3D space is then rendered in the browser using WebGL 2. The Tesseract Explorer provides a variety of tools for visualizing the projections of a 4D tesseract into 3D space. I highly recommend this site for a primer on visualizing 4D shapes. Obviously, we can't directly visualize a tesseract, since we live in only 3 dimensions, but we can project its form into 3D space, essentially taking "photographs" from 4 dimensions onto the 3D "film" of our universe (in the same way we photograph our 3D universe onto the 2D film of a camera). This is analogous to the 4 edges of each square face of a 3D cube being flush with the edges of 4 of its other faces, an impossible formation in 2D space, but possible with the folding into 3D. A way to imagine the shape of the tesseract is that space is folded in such a way that each of the six faces of each cell are flush with one face of 6 other cells, with the only cell left out being the opposite one on the same axis. These cells enclose the 4D hypervolume of the tesseract. Its 3D "surface" is composed of 8 cubes, called cells, 2 along each of the 4 axes, X, Y, Z, and W. Welcome to the 4 th dimension! Visit the live application here!Ī tesseract, also known as a hypercube or 8-cell, is the 4D analog to the 2D square and the 3D cube.
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