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+.TH XScreenSaver 1 "" "X Version 11"
+.SH NAME
+klein - Draws a 4d Klein bottle.
+.SH SYNOPSIS
+.B klein
+[\-display \fIhost:display.screen\fP]
+[\-install]
+[\-visual \fIvisual\fP]
+[\-window]
+[\-root]
+[\-delay \fIusecs\fP]
+[\-fps]
+[\-klein-bottle \fIbottle-name\fP]
+[-figure-8]
+[-squeezed-torus]
+[-lawson]
+[\-mode \fIdisplay-mode\fP]
+[\-wireframe]
+[\-surface]
+[\-transparent]
+[\-appearance \fIappearance\fP]
+[\-solid]
+[\-bands]
+[\-colors \fIcolor-scheme\fP]
+[\-twosided]
+[\-rainbow]
+[\-depth]
+[\-view-mode \fIview-mode\fP]
+[\-walk]
+[\-turn]
+[\-walk-turn]
+[\-orientation-marks]
+[\-projection-3d \fImode\fP]
+[\-perspective-3d]
+[\-orthographic-3d]
+[\-projection-4d \fImode\fP]
+[\-perspective-4d]
+[\-orthographic-4d]
+[\-speed-wx \fIfloat\fP]
+[\-speed-wy \fIfloat\fP]
+[\-speed-wz \fIfloat\fP]
+[\-speed-xy \fIfloat\fP]
+[\-speed-xz \fIfloat\fP]
+[\-speed-yz \fIfloat\fP]
+[\-walk-direction \fIfloat\fP]
+[\-walk-speed \fIfloat\fP]
+.SH DESCRIPTION
+The \fIklein\fP program shows three different Klein bottles in 4d: the
+figure-8 Klein bottle, the squeezed torus Klein bottle, or the Lawson
+Klein bottle.  You can walk on the Klein bottle, see it turn in 4d, or
+walk on it while it turns in 4d.  The figure-8 Klein bottle is well
+known in its 3d form.  The 4d form used in this program is an
+extension of the 3d form to 4d that does not intersect itself in 4d
+(which can be seen in the depth colors mode).  The squeezed torus
+Klein bottle also does not intersect itself in 4d (which can be seen
+in the depth colors mode).  The Lawson Klein bottle, on the other
+hand, does intersect itself in 4d.  Its primary use is that it has a
+nice appearance for walking and for turning in 3d.  The Klein bottle
+is a non-orientable surface.  To make this apparent, the two-sided
+color mode can be used.  Alternatively, orientation markers (curling
+arrows) can be drawn as a texture map on the surface of the Klein
+bottle.  While walking on the Klein bottle, you will notice that the
+orientation of the curling arrows changes (which it must because the
+Klein bottle is non-orientable).  The program projects the 4d Klein
+bottle to 3d using either a perspective or an orthographic projection.
+Which of the two alternatives looks more appealing depends on the
+viewing mode and the Klein bottle.  For example, the Lawson Klein
+bottle looks nicest when projected perspectively.  The figure-8 Klein
+bottle, on the other hand, looks nicer while walking when projected
+orthographically from 4d.  For the squeezed torus Klein bottle, both
+projection modes give equally acceptable projections.  The projected
+Klein bottle can then be projected to the screen either perspectively
+or orthographically.  When using the walking modes, perspective
+projection to the screen should be used.  There are three display
+modes for the Klein bottle: mesh (wireframe), solid, or transparent.
+Furthermore, the appearance of the Klein bottle can be as a solid
+object or as a set of see-through bands.  Finally, the colors with
+with the Klein bottle is drawn can be set to two-sided, rainbow, or
+depth.  In the first case, the Klein bottle is drawn with red on one
+"side" and green on the "other side".  Of course, the Klein bottle
+only has one side, so the color jumps from red to green along a curve
+on the surface of the Klein bottle.  This mode enables you to see that
+the Klein bottle is non-orientable.  The second mode draws the Klein
+bottle with fully saturated rainbow colors.  This gives a very nice
+effect when combined with the see-through bands mode or with the
+orientation markers drawn.  The third mode draws the Klein bottle with
+colors that are chosen according to the 4d "depth" of the points.
+This mode enables you to see that the figure-8 and squeezed torus
+Klein bottles do not intersect themselves in 4d, while the Lawson
+Klein bottle does intersect itself.  The rotation speed for each of
+the six planes around which the Klein bottle rotates can be chosen.
+For the walk-and-turn mode, only the rotation speeds around the true
+4d planes are used (the xy, xz, and yz planes).  Furthermore, in the
+walking modes the walking direction in the 2d base square of the Klein
+bottle and the walking speed can be chosen.  This program is somewhat
+inspired by Thomas Banchoff's book "Beyond the Third Dimension:
+Geometry, Computer Graphics, and Higher Dimensions", Scientific
+American Library, 1990.
+.SH OPTIONS
+.I klein
+accepts the following options:
+.TP 8
+.B \-window
+Draw on a newly-created window.  This is the default.
+.TP 8
+.B \-root
+Draw on the root window.
+.TP 8
+.B \-install
+Install a private colormap for the window.
+.TP 8
+.B \-visual \fIvisual\fP
+Specify which visual to use.  Legal values are the name of a visual
+class, or the id number (decimal or hex) of a specific visual.
+.TP 8
+.B \-delay \fImicroseconds\fP
+How much of a delay should be introduced between steps of the
+animation.  Default 10000, or 1/100th second.
+.TP 8
+.B \-fps
+Display the current frame rate, CPU load, and polygon count.
+.PP
+The following three options are mutually exclusive.  They determine
+which Klein bottle is displayed.
+.TP 8
+.B \-klein-bottle random
+Display a random Klein bottle (default).
+.TP 8
+.B \-klein-bottle figure-8 \fP(Shortcut: \fB\-figure-8\fP)
+Display the figure-8 Klein bottle.
+.TP 8
+.B \-klein-bottle squeezed-torus \fP(Shortcut: \fB\-squeezed-torus\fP)
+Display the squeezed torus Klein bottle.
+.TP 8
+.B \-klein-bottle lawson \fP(Shortcut: \fB\-lawson\fP)
+Display the Lawson Klein bottle.
+.PP
+The following four options are mutually exclusive.  They determine
+how the Klein bottle is displayed.
+.TP 8
+.B \-mode random
+Display the Klein bottle in a random display mode (default).
+.TP 8
+.B \-mode wireframe \fP(Shortcut: \fB\-wireframe\fP)
+Display the Klein bottle as a wireframe mesh.
+.TP 8
+.B \-mode surface \fP(Shortcut: \fB\-surface\fP)
+Display the Klein bottle as a solid surface.
+.TP 8
+.B \-mode transparent \fP(Shortcut: \fB\-transparent\fP)
+Display the Klein bottle as a transparent surface.
+.PP
+The following three options are mutually exclusive.  They determine the
+appearance of the Klein bottle.
+.TP 8
+.B \-appearance random
+Display the Klein bottle with a random appearance (default).
+.TP 8
+.B \-appearance solid \fP(Shortcut: \fB\-solid\fP)
+Display the Klein bottle as a solid object.
+.TP 8
+.B \-appearance bands \fP(Shortcut: \fB\-bands\fP)
+Display the Klein bottle as see-through bands.
+.PP
+The following four options are mutually exclusive.  They determine
+how to color the Klein bottle.
+.TP 8
+.B \-colors random
+Display the Klein bottle with a random color scheme (default).
+.TP 8
+.B \-colors twosided \fP(Shortcut: \fB\-twosided\fP)
+Display the Klein bottle with two colors: red on one "side" and green
+on the "other side".
+.TP 8
+.B \-colors rainbow \fP(Shortcut: \fB\-rainbow\fP)
+Display the Klein bottle with fully saturated rainbow colors.  If the
+Klein bottle is displayed as see-through bands, each band will be
+displayed with a different color.
+.TP 8
+.B \-colors depth \fP(Shortcut: \fB\-depth\fP)
+Display the Klein bottle with colors chosen depending on the 4d
+"depth" of the points.
+.PP
+The following four options are mutually exclusive.  They determine
+how to view the Klein bottle.
+.TP 8
+.B \-view-mode random
+View the Klein bottle in a random view mode (default).
+.TP 8
+.B \-view-mode walk \fP(Shortcut: \fB\-walk\fP)
+View the Klein bottle as if walking on its surface.
+.TP 8
+.B \-view-mode turn \fP(Shortcut: \fB\-turn\fP)
+View the Klein bottle while it turns in 4d.
+.TP 8
+.B \-view-mode walk-turn \fP(Shortcut: \fB\-walk-turn\fP)
+View the Klein bottle as if walking on its surface.  Additionally, the
+Klein bottle turns around the true 4d planes (the xy, xz, and yz
+planes).
+.PP
+The following options determine whether orientation marks are shown on
+the Klein bottle.
+.TP 8
+.B \-orientation-marks
+Display orientation marks on the Klein bottle.
+.TP 8
+.B \-no-orientation-marks
+Don't display orientation marks on the Klein bottle (default).
+.PP
+The following three options are mutually exclusive.  They determine
+how the Klein bottle is projected from 3d to 2d (i.e., to the screen).
+.TP 8
+.B \-projection-3d random
+Project the Klein bottle from 3d to 2d using a random projection mode
+(default).
+.TP 8
+.B \-projection-3d perspective \fP(Shortcut: \fB\-perspective-3d\fP)
+Project the Klein bottle from 3d to 2d using a perspective projection.
+.TP 8
+.B \-projection-3d orthographic \fP(Shortcut: \fB\-orthographic-3d\fP)
+Project the Klein bottle from 3d to 2d using an orthographic
+projection.
+.PP
+The following three options are mutually exclusive.  They determine
+how the Klein bottle is projected from 4d to 3d.
+.TP 8
+.B \-projection-4d random
+Project the Klein bottle from 4d to 3d using a random projection mode
+(default).
+.TP 8
+.B \-projection-4d perspective \fP(Shortcut: \fB\-perspective-4d\fP)
+Project the Klein bottle from 4d to 3d using a perspective projection.
+.TP 8
+.B \-projection-4d orthographic \fP(Shortcut: \fB\-orthographic-4d\fP)
+Project the Klein bottle from 4d to 3d using an orthographic
+projection.
+.PP
+The following six options determine the rotation speed of the Klein
+bottle around the six possible hyperplanes.  The rotation speed is
+measured in degrees per frame.  The speeds should be set to relatively
+small values, e.g., less than 4 in magnitude.  In walk mode, all
+speeds are ignored.  In walk-and-turn mode, the 3d rotation speeds are
+ignored (i.e., the wx, wy, and wz speeds).  In walk-and-turn mode,
+smaller speeds must be used than in the turn mode to achieve a nice
+visualization.  Therefore, in walk-and-turn mode the speeds you have
+selected are divided by 5 internally.
+.TP 8
+.B \-speed-wx \fIfloat\fP
+Rotation speed around the wx plane (default: 1.1).
+.TP 8
+.B \-speed-wy \fIfloat\fP
+Rotation speed around the wy plane (default: 1.3).
+.TP 8
+.B \-speed-wz \fIfloat\fP
+Rotation speed around the wz plane (default: 1.5).
+.TP 8
+.B \-speed-xy \fIfloat\fP
+Rotation speed around the xy plane (default: 1.7).
+.TP 8
+.B \-speed-xz \fIfloat\fP
+Rotation speed around the xz plane (default: 1.9).
+.TP 8
+.B \-speed-yz \fIfloat\fP
+Rotation speed around the yz plane (default: 2.1).
+.PP
+The following two options determine the walking speed and direction.
+.TP 8
+.B \-walk-direction \fIfloat\fP
+The walking direction is measured as an angle in degrees in the 2d
+square that forms the coordinate system of the surface of the Klein
+bottle (default: 7.0).
+.TP 8
+.B \-walk-speed \fIfloat\fP
+The walking speed is measured in percent of some sensible maximum
+speed (default: 20.0).
+.SH INTERACTION
+If you run this program in standalone mode in its turn mode, you can
+rotate the Klein bottle by dragging the mouse while pressing the left
+mouse button.  This rotates the Klein bottle in 3D, i.e., around the
+wx, wy, and wz planes.  If you press the shift key while dragging the
+mouse with the left button pressed the Klein bottle is rotated in 4D,
+i.e., around the xy, xz, and yz planes.  To examine the Klein bottle
+at your leisure, it is best to set all speeds to 0.  Otherwise, the
+Klein bottle will rotate while the left mouse button is not pressed.
+This kind of interaction is not available in the two walk modes.
+.SH ENVIRONMENT
+.PP
+.TP 8
+.B DISPLAY
+to get the default host and display number.
+.TP 8
+.B XENVIRONMENT
+to get the name of a resource file that overrides the global resources
+stored in the RESOURCE_MANAGER property.
+.SH SEE ALSO
+.BR X (1),
+.BR xscreensaver (1)
+.SH COPYRIGHT
+Copyright \(co 2005-2014 by Carsten Steger.  Permission to use, copy,
+modify, distribute, and sell this software and its documentation for
+any purpose is hereby granted without fee, provided that the above
+copyright notice appear in all copies and that both that copyright
+notice and this permission notice appear in supporting documentation.
+No representations are made about the suitability of this software for
+any purpose.  It is provided "as is" without express or implied
+warranty.
+.SH AUTHOR
+Carsten Steger <carsten@mirsanmir.org>, 03-oct-2014.