flythrough(1) Geometry Center (January 29, 1993) flythrough(1)
NAME
flythrough - Geomview external module to fly through Not
Knot hyperbolic dodecahedral tesselation
SYNOPSIS
flythrough [-t] [-h]
DESCRIPTION
Flythrough is a geomview external module that lets you fly
through the tesselation of hyperbolic space by a right-
angled regular dodecahedron which appeared in the
mathematical animation "Not Knot" produced by the Geometry
Center. You can either pick a pre-computed flight path or
fly around interactively. Click on "Not Knot Flythrough" in
the geomview Applications browser to start the program.
OPTIONS
-t Turbo mode: send commands off as fast as possible
without waiting for geomview to catch up.
-h Display help window on startup.
WHAT'S GOING ON
When you hit the "What's Going On?" button (or start up the
module with the -h option), you get a text help window with
most of the information in this man page. There is also a 3D
diagram of a single dodecahedron with color-coded arcs
indicating the pre-computed flight paths. You can drag the
left mouse button in the window to spin this diagram around.
It's easier to see what's going on in the Euclidean diagram,
while the hyperbolic version is more similar to what you see
in the flythrough.
CONTROL PANEL
You can either choose one of four flight paths through the
tesselation or stop the automatic flight by hitting the
"Stop" button and fly around yourself. For interactive
flight, hit the "Cam Fly" button on the geomview Tools
panel: then dragging the mouse with the middle button down
moves you forwards or backwards, and dragging with the left
button down is like turning your head. When you hit "Go",
the automatic flight will continue.
You can choose one of four tesselation levels: level 0 is a
single dodecahedron, level 1 adds a layer of 12 dodecahedra
(one for each face of the original dodecahedron), level 2
tesselates two layers deep, and level 3 has three layers.
The more layers you have the slower the update rate: level 3
is glacially slow, but each frame looks pretty impressive.
You can change the size of the dodecahedra with the "Scale
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flythrough(1) Geometry Center (January 29, 1993) flythrough(1)
Dodecahedra" slider: at 1.0 they fit together exactly. The
"Steps" buttons control the smoothness of the flight path:
you can set the number of steps to 10 (jerky but fast), 20,
40, or 80 (smooth but slow).
FLIGHT PATHS
All 30 edges of the base dodecahedron are white except the
three pairs of edges colored green, blue and red
corresponding to the three loops of the Borromean rings.
Every face of the dodecahedron has exactly one non-white
edge, so we can color the face by this color.
All flight paths begin and end at the center of a green
face. There are three other green faces: one adjacent to
this one, at right angles along the green beam; and a pair
which border the other green beam, on the other side of the
dodecahedron.
The light blue "Direct" path is the simplest to understand:
we go straight through to the green face directly opposite
from the original face.
The yellow "Quarter Turn" path, which goes to the adjacent
green face, simply circles around the green axis which the
two faces share.
The "Full Loop" path is also yellow: it repeats this quarter
turn four times so that we start and finish in the same
place. The three other paths just jump back to the starting
place when they reach the end.
The magenta "Equidistant" path, which goes to the other
green face which doesn't border the original face, is the
most interesting. It follows a so-called equidistant curve:
in this case, one that is equidistant to the red axis that
connects the two green faces in question. This curve is like
a parallel line in Euclidean space: it stays a constant
distant from the red axis, but it's not a geodesic in
hyperbolic space.
SEE ALSO
geomview(1), geomview(5), oogl(5), Not Knot (mathematical
animation available from Jones and Bartlett publishers,
Boston, MA).
AUTHORS
Charlie Gunn (geometry and flight paths) gunn@geom.umn.edu
Tamara Munzner (interactive interface) munzner@geom.umn.edu
Stuart Levy (3D diagram) levy@geom.umn.edu
Copyright (c) 1993
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flythrough(1) Geometry Center (January 29, 1993) flythrough(1)
The Geometry Center
1300 South Second Street, Suite 500
Minneapolis, MN 55454
email: software@geom.umn.edu
Page 3 (printed 12/22/98)
