flythrough(1) General Commands Manual 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 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 The Geometry Center 1300 South Second Street, Suite 500 Minneapolis, MN 55454 email: software@geom.umn.edu Geometry Center January 29, 1993 flythrough(1)