flythrough(1)               General Commands Manual              flythrough(1)

       flythrough - Geomview external module to fly through Not Knot
       hyperbolic dodecahedral tesselation

       flythrough [-t] [-h]

       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.

       -t     Turbo mode: send commands off as fast as possible without
              waiting for geomview to catch up.

       -h     Display help window on startup.

       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.

       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
       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

       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

       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.

       geomview(1), geomview(5), oogl(5), Not Knot (mathematical animation
       available from Jones and Bartlett publishers, Boston, MA).

       Charlie Gunn   (geometry and flight paths)
       Tamara Munzner (interactive interface)
       Stuart Levy    (3D diagram)        

       Copyright (c) 1993
       The Geometry Center
       1300 South Second Street, Suite 500
       Minneapolis, MN 55454

Geometry Center                January 29, 1993                  flythrough(1)