The large blue cube opens toward the top, flattens out, and opens inward to form an incomplete pyramid, while at the same time the small red cube opens toward the bottom to form another incomplete pyramid. The images formed when the hypercube rotates in four-dimensional space resemble those of the ordinary cube rotating in three-space. The analogous sequence for the hypercube (as it is shown in the illustration on the next page) starts with a red cube inside a blue cube joined by black edges stretching between corresponding vertices. As the cube rotates in three-space, its images under central projection change until at one point we have a green square inside a red one, then a red trapezoid next to a green trapezoid, and then a red square inside a green square. In the first sequence, white edges join corresponding vertices of a red square and a green square lying opposite. The second section of that film begins by showing central projections of the three-cube. Color-coding makes it possible to keep track of different parts of the rotating hypercube, as portrayed in the film The Hypercube: Projections and Slicing. Some additional calculations are necessary for central projections since images of parallel segments will no longer be parallel but will lie in lines going through "vanishing points." The computer is quick enough to carry out these computations to produce frames in perspective of an animation as the hypercube rotates in four-dimensional space. Once the images of these points are determined, all the other points and segments can be drawn easily since the images of parallel segments of the same length will be parallel segments of the same length (as described in Chapter 4). To create an animation using parallel projection, the program keeps track of the position of one corner of the cube or hypercube and of all corners attached to it by edges. Dewdney described ways ofĬreating programs for hypercube rotation in the April 1986 " Computer Recreations" column of Scientific American.
The hypercube is not that much more complicated, withġ6 vertices and 32 edges. To give the impression of continuous motion in what is called Machines, a cube, with 8 vertices and 12 edges, can be rotated Production and display depends strongly on the number of Vertices and draws the appropriate segments. Wire-frame object, the computer calculates the positions of the Modern graphics computers can produce images very quickly. Should this window be placed slightly higher? Should that entranceway be longer? A turn of a dial can produce the new view and simultaneously make the changes for a new set of blueprints. As an architect takes her client on a tour of a prospective auditorium, she can alter the different features to create different impressions. We can experience what it would be like to walk along a corridor or down a staircase in a building that has not yet been constructed. Architectural and industrial design become dynamic processes as we look at not just a few views but 30 views per second, each slightly different from its predecessor, giving the impression of continuous motion. We can combine a century and a half of animation experience together with modern computer graphics to create and investigate complicated configurations in three-dimensional space. Slow motion and freeze frame techniques made it possible to analyze the motion of a racehorse or the exertion of muscles in lifting a log. The photographs of Muybridge himself, striding up a ramp in front of his camera, could be placed on a rotary device and flipped over and over so that he walked on and on, in a primitive version of a motion picture. Within a generation of the invention of photography 150 years ago, Eadweard Muybridge had used this new technology to alter our perception of time and space. Show Extra Information Links Hide Extra Information Links Editorial Version Chapter 6 : Perspective and Animation Animating the HypercubeĮadweard Muybridge strides in front of his camera to create one
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