The Hypercube Projections & Slicing + others

This tape features two distinct pieces: "Hypercube Projects" and "This Is Early Bird." At the end of the video there are a series of loading screens and LANDSAT images of San Francisco Bay and the Seattle area. "Hypercube" is an instructional video that describes geometrical transformations--rotations, projections, and slicings--within two, three, and four-dimensional space. The catalog contains a full transcription of the narration. "This Is Early Bird" was produced for CBS as part of a documentary/educational show about Intelsat1, the world's first commercial communications satellite. The show also includes basic educational introductions about how geosynchronous satellites function and enter their orbits.

0:00Copy video clip URL Static and color bars.

0:23Copy video clip URL Title card: “The Hypercube Projections and Slicing.” Zoom in through the computer generated text. Zoom back out through different text: “Copyright 1978 Banchoff/Strauss Productions.” The block text rotates counter-clockwise. When the text is at a right angle to the camera lens, the camera zooms “through” the text.

1:23Copy video clip URL An orange square. Narrator: “We start with a square, which is really a head-on view of a cube, which begins to rotate. Our first impression of a rotating three-cube is that of squares sliding. This is the Revolving Door Illusion. If we rotate the three dimensional cube about a different axis, we get parallelograms and rectangles, images that we readily interpret as the shadows of a rotating three dimensional cube. But now we rotate in four dimensional state. Our first view of a rotating four dimensional cube gives us the impression of a three dimensional cube sliding: The Revolving Door Illusion in one higher dimension.”

2:15Copy video clip URL Narrator: “If we stop the four dimensional rotation and revolve around an axis in three space, we get something which resembles a wire frame model, like the frame of a box kite. But now we rotate in a different plane in four dimensional space. This is our first really good view of a four dimensional cube. It has sixteen corners: Eight on the red cube, eight on the green cube, with corresponding corners connected. But as we rotate in three space we see that one three dimensional face is still quite flat. We rotate again in four dimensional space. This time leading to a truly general position. At almost any instant the four edges that come out of one of the corners have no three lines lying in a plane. Stopping our four dimensional rotation we show a three dimensional spin around an axis where we see the red cube and the green cube interlaced. At the end of our four dimensional rotation we fold all the way back, come back to three space, continue back to the plane, and end at a square.”

3:44Copy video clip URL Narrator: “Viewing objects in three dimensional space we are aided by the devices of perspective. As we rotate a three cube in perspective we see each of the six bounding squares as the smallest, farthest away square and then again as the largest, closest square. This is not the top view of some two dimensional creature swimming in on itself. It’s a rotating three dimensional cube. And we are quite used to these perspective distortions. But now we go to four space. We change our perspective and we proceed to walk around that four dimensional cube. As we walk around we see each of the eight cubical faces as the smallest cube when it’s farthest away or the closest cube when it’s largest. This is not a three dimensional creature swimming through itself, but it’s the shadows of a rotating four dimensional cube. This is the best cube where the red cube–now largest–now flattened out–now turned inside out as the smallest, farthest away cube. And if we become comfortable with rotations, every once in a while we see a rotation which confuses us once more. Rotating directly toward us and away from us we find another strange view of the rotating four dimensional cube in perspective. At the end we go back and end at our three dimensional cube.”

5:55Copy video clip URL Narrator: “In addition to projections and rotations, we can use the device of slicing. Slicing a square by an edge parallel to one side gives us a segment for a while. Corner first gives us segments which grow and come back off ending at a corner. We can use the same technique to slice a three dimensional cube. Parallel to one face we get a square for a while. Edge first, we get rectangles which grow and come back off ending at an edge. Most interesting is corner first. What do we get half way through? We can see that a bit more slowly: Triangles which become cut off, so that half way through we cut each of the six bounding squares in precisely the same way. We obtain a perfect regular hexagon half way through the cube.”

7:16Copy video clip URL Narrator: “We can use the same techniques to slice a four dimensional cube. We position our three dimensional knife parallel to one face to get a cube for a while. Square first, our slices are square prisms that come back off ending at a square. There are two more ways of slicing a four dimensional cube. Edge first, gives us triangular prisms; half way through hexagonal prisms coming all the way back off ending at an edge. Most interesting, most challenging, what do we get when the slice comes through corner first along the long diagonal? We can see that more slowly: A small tetrahedral pyramid which grows until it hits the vertices and becomes truncated–cut off–so that three eighths of the way through we have four equilateral triangles, four regular hexagons, one hexagon exactly half-way through that red cube. This is a semi-regular polyhedron already known to Archimedes. Half-way through, we cut each of the eight bounding three dimensional cubes precisely the same way. We get eight equilateral triangles fitting together to form a perfect regular octahedron–a Platonic solid.”

9:09Copy video clip URL Narrator: “We complete our tour of the four dimensional cube by showing slices which go off at a point not situated at the center of the screen. One last pass: Tetrahedra, tetrahedra cut off, in the middle–the octahedron, tetrahedra cut off the other way, coming back off ending at a point, completing our tour of the four dimensional cube.”

9:45Copy video clip URL Zoom out through a logarithmic spiral and then rotation within the spiral. Zoom all the way out to show a longer helical structure.

11:41Copy video clip URL View from within a rotating torus.

14:36Copy video clip URL End of segment.

14:40Copy video clip URL Count down. Open to the flight control center for Intelsat 1 (Early Bird), the first communications satellite put into geosynchronous orbit. The segment is called “This Is Early Bird” for CBS. Men inside flight control speak with each other. Cut to a Soviet sailor dancing. Cut back to inside flight control: “We have Mexico!…” Title cards: “Communications Satellite Corporation presents: Live Via Early Bird.”

16:09Copy video clip URL Computer graphic representation of the Early Bird satellite orbiting Earth. Narrator: “The first satellite ever to maintain constant, 24 hour communications between North America and Europe is in orbital position. Its inaugural transmission was a public transmission between two great continents. Now men living and working on the ground have sent their minds and their skills into the three dimensional world of orbital flight 22,300 miles in space. ” Video of various broadcast satellite transmitters, people working in flight control, and the computer graphics of Early Bird. The narrator gives a brief history of telecommunications and space travel. Archival photographs and film of the first telephone lines. He explains how direct line-of-sight microwave communication systems were augmented by satellite communications.

18:38Copy video clip URL Recording of the first human voice transmitted by satellite to Earth: President Eisenhower’s Christmas message in 1958 (carried by Project SCORE). The narrator explains the basics of synchronous equatorial orbit: “Three such satellites could cover the entire globe.” Graphical representation of TelStar 1, which provided the first (intermittent) live video, intercontinental space transmission. Syncom1, Syncom2, and Syncom3 are discussed. Snycom3 provided live coverage of the 1964 Tokyo Olympics to the United States. Narrator describes the Satellite Act of U.S. Congress (August 31, 1962), which authorized the creation of the Communications Satellite Corporation (COMSAT). Archival footage from within COMSAT meetings.

21:04Copy video clip URL Archival film from April 1964 outside the Federal Communications Commission. Shot of the modification of the Thor Delta Vehicle (rocket), created by the Douglas Aircraft Company. Archival film of the actual Early Bird satellite, the Goonhilly Downs Satellite Earth Station Downs (United Kingdom), the Pucino Satellite Earth Station (Italy), the Pleumeur-Bodou Station (France), and the Reistig Station (Germany). Archival film of the Andover, Maine Earth Satellite Station. Map of all the nations part of the satellite communications consortium, managed by COMSAT. Narrator describes technical management and launch operations of the Early Bird system: “Reliability has a special meaning in space.”

25:23Copy video clip URL Archival footage of the Early Bird launch. The narrator describes the stages from launch into a transfer orbit and then a geosynchronous equatorial orbit.

29:09Copy video clip URL Archival film from the various ground stations. The narrator describes how telephone operators and television antennae will be the means through which we receive satellite data transmissions. Narrator: “On May 2, 1965, the promise of a global commercial satellite system, of which Early Bird is the forerunner, was presented for the first time by an intercontinental live satellite transmission.” Archival film from that broadcast, which included transmissions from all over the world. A different narrator describes how medical knowledge gets advanced through satellite-based education.

33:49Copy video clip URL Original broadcast narrator: “And so this is the conclusion of our broadcast inaugurating the transmission of regular live television between Europe and North America. Proceeding from the assumption that men are good and decent and that their troubles grow from untruths and misunderstandings. This communication facility will serve mankind well. It will serve by making truth and understanding between people easier to come by. Nations know this little thing or that about each other, but essentially they are strangers. Their judgments about one another are formed by a caricature  a trait, a look of a face, or the half-told story of a thing that never happened. If Early Bird and the satellites yet unlaunched can help reveal nations to nations and people to people, this will have been a day.” The show’s narrator continues the timeline of various kinds of satellite communications.

35:10Copy video clip URL Archival film of President Johnson’s first satellite-based telephone call. Cut to computer graphic representation of Early Bird. Credits.

36:13Copy video clip URL Beginning of a sequence of loading screens and LANDSAT images. A baboon; LANDSAT images of the San Francisco Bay area; the Golden Gate Bridge; a ship; calibration images (Sine, vectors, noice, ramp, gause) from the De Anza Systems Library of Image Processing Software; a fountain; another LANDSAT image of San Francisco Bay; LANDSAT images of Seattle; a San Francisco trolley; LANDSAT images of San Francisco.

39:20Copy video clip URL MRI scans of the brain. Photographs of the ship docked in San Francisco Bay; the Golden Gate Bridge with various image processing techniques overlaid; intensity and Triangulated Irregular Network graphs of terrain; computer generated shapes; computer generated image of planets; processed image of the Golden Gate Bridge.

43:48Copy video clip URL End of tape.

 

1 Comment

  1. Phaedrus Bùxiu says:

    Wow, OUTTA SIGHT!!!
    (In both the 1970s, and 4th dimensional senses.)

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