Dec 112017

3-D Animation! Good. | |||
---|---|---|---|

File Name | File Size | Zip Size | Zip Type |

TUMBLE.DOC | 2924 | 1376 | deflated |

TUMBLE16.EXE | 4273 | 1637 | deflated |

# Download File HYPER.ZIP Here

## Contents of the TUMBLE.DOC file

A hypercube (or tesseract) is the four dimensional "solid" analagous to the

cube. Just as you can construct a cube by lifting a square out of its

plane in a direction perpendicular to the plane, so you can construct a

hypercube by lifting a cube out of its "hyperplane" (which is our normal,

three dimensional world) in a fourth direction perpendicular to the

hyperplane.

The result is hard to visualize, as our minds have evolved to perceive and

understand a three dimensional world. But, just as a camera or an artist

can project three dimensional objects into a flat picture as a sort of

shadow, so one can project a four dimensional object into our three

dimensional world by simply ignoring the fourth dimension. (An

"orthographic" projection.) The resulting three dimensional object can

then be drawn using conventional perspective techniques, leaving one with a

mass of lines purportedly representing a very strange and unfamiliar three

dimensional object. The projection can be made much clearer by

simultaneously drawing two such perspective drawings side by side in the

manner of an old fashioned stereopticon. This is exactly what HYPERTUMBLE

does, except that the stereo pair is rotated in the first and fourth

dimension (not the first and third) so that your normal perception of depth

also includes some of the fourth dimension.

If you sit back from the monitor and relax your eyes (as if looking into

the distance) the two drawings will seem to diverge, and you will

momentarily see four separate images. Focus on the middle two, and they

will merge into a single, three dimensional object. This will feel very

strange at first, but (no matter what your mother says!) won't hurt your

eyes and will become quite easy with practice.

Initially, you will see a cube absolutely head on, so that it will appear

as a square. The program will select one of six planes and begin rotating

the hypercube. Periodically it will select a new plane and increase or

decrease the component of the hypercubes rotation in that plane. Rotations

involving the fourth dimension may produce radical distortions, while

merely spatial rotations will appear more familiar.

A somewhat simpler version of HYPERCUBE is described in great detail in A.

K. Dewdney's Computer Recreation solumn in the April 1986 issue of

Scientific American. This implementation is done in assembly language

using scaled integers as fixed decimal place numbers: TUMBLE16 uses two

byte integers for four fractional digits while TUMBLE32, somewhat more

conservatively, uses four byte integers and seven fractional digits.

Although either has more than enough accuracy for the screen's resolution,

cumulative rounding error will cause the drawings to shrink and skew and

eventually disappear; press R to reset the hypercube to its original

rotation and "canonical" orientation.

--- JDS, 4/17/86

cube. Just as you can construct a cube by lifting a square out of its

plane in a direction perpendicular to the plane, so you can construct a

hypercube by lifting a cube out of its "hyperplane" (which is our normal,

three dimensional world) in a fourth direction perpendicular to the

hyperplane.

The result is hard to visualize, as our minds have evolved to perceive and

understand a three dimensional world. But, just as a camera or an artist

can project three dimensional objects into a flat picture as a sort of

shadow, so one can project a four dimensional object into our three

dimensional world by simply ignoring the fourth dimension. (An

"orthographic" projection.) The resulting three dimensional object can

then be drawn using conventional perspective techniques, leaving one with a

mass of lines purportedly representing a very strange and unfamiliar three

dimensional object. The projection can be made much clearer by

simultaneously drawing two such perspective drawings side by side in the

manner of an old fashioned stereopticon. This is exactly what HYPERTUMBLE

does, except that the stereo pair is rotated in the first and fourth

dimension (not the first and third) so that your normal perception of depth

also includes some of the fourth dimension.

If you sit back from the monitor and relax your eyes (as if looking into

the distance) the two drawings will seem to diverge, and you will

momentarily see four separate images. Focus on the middle two, and they

will merge into a single, three dimensional object. This will feel very

strange at first, but (no matter what your mother says!) won't hurt your

eyes and will become quite easy with practice.

Initially, you will see a cube absolutely head on, so that it will appear

as a square. The program will select one of six planes and begin rotating

the hypercube. Periodically it will select a new plane and increase or

decrease the component of the hypercubes rotation in that plane. Rotations

involving the fourth dimension may produce radical distortions, while

merely spatial rotations will appear more familiar.

A somewhat simpler version of HYPERCUBE is described in great detail in A.

K. Dewdney's Computer Recreation solumn in the April 1986 issue of

Scientific American. This implementation is done in assembly language

using scaled integers as fixed decimal place numbers: TUMBLE16 uses two

byte integers for four fractional digits while TUMBLE32, somewhat more

conservatively, uses four byte integers and seven fractional digits.

Although either has more than enough accuracy for the screen's resolution,

cumulative rounding error will cause the drawings to shrink and skew and

eventually disappear; press R to reset the hypercube to its original

rotation and "canonical" orientation.

--- JDS, 4/17/86

December 11, 2017
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