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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 mathemati-
     cal  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 geom-
     view 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 indi-
     cating 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 correspond-
     ing 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 dis-
     tant from the red axis, but it's not a  geodesic  in  hyper-
     bolic space.


SEE ALSO
     geomview(1), geomview(5), oogl(5),	 Not  Knot  (mathematical
     animation available from Jones and	Bartlett publishers, Bos-
     ton, MA).

AUTHORS
     Charlie Gunn   (geometry and flight paths)	  [email protected]
     Tamara Munzner (interactive interface)	  [email protected]
     Stuart Levy    (3D	diagram)		  [email protected]

     Copyright (c) 1993
     The Geometry Center
     1300 South	Second Street, Suite 500
     Minneapolis, MN 55454
     email: [email protected]

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