klein(6) XScreenSaver manual klein(6)

NAME

klein - Draws a 4d Klein bottle.

SYNOPSIS

klein [--display host:display.screen] [--install] [--visual visual]
[--window] [--root] [--window-id number] [--delay usecs] [--fps]
[--klein-bottle bottle-name] [-figure-8] [-pinched-torus] [-lawson]
[--mode display-mode] [--wireframe] [--surface] [--transparent]
[--appearance appearance] [--solid] [--bands] [--colors color-scheme]
[--onesided] [--twosided] [--rainbow] [--depth] [--change-colors]
[--view-mode view-mode] [--walk] [--turn] [--walk-turn]
[--orientation-marks] [--projection-3d mode] [--perspective-3d]
[--orthographic-3d] [--projection-4d mode] [--perspective-4d]
[--orthographic-4d] [--speed-wx float] [--speed-wy float] [--speed-wz
float] [--speed-xy float] [--speed-xz float] [--speed-yz float]
[--walk-direction float] [--walk-speed float]

DESCRIPTION

The klein program shows three different Klein bottles in 4d: the
figure-8 Klein bottle, the pinched 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 when using static
colors). The pinched torus Klein bottle also does not intersect
itself in 4d (which can be seen in the depth colors mode when using
static colors). 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 pinched 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
one-sided, two-sided, rainbow, or depth. In one-sided mode, the
Klein bottle is drawn with the same color on both "sides." In two-
sided mode (using static colors), 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. If changing colors are
used in two-sided mode, changing complementary colors are used on the
respective "sides." The rainbow color mode (using static colors)
draws the Klein bottle with a color wheel of fully saturated rainbow
colors. If changing colors are used, the color wheel's colors change
dynamically. The rainbow color mode gives a very nice effect when
combined with the see-through bands mode or with the orientation
markers drawn. The depth color mode draws the Klein bottle with
colors that are chosen according to the 4d "depth" of the points. If
static colors are used, this mode enables you to see that the
figure-8 and pinched 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.

OPTIONS

klein accepts the following options:

--window
Draw on a newly-created window. This is the default.

--root Draw on the root window.

--window-id number
Draw on the specified window.

--install
Install a private colormap for the window.

--visual visual
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.

--delay microseconds
How much of a delay should be introduced between steps of the
animation. Default 10000, or 1/100th second.

--fps Display the current frame rate, CPU load, and polygon count.

The following three options are mutually exclusive. They determine
which Klein bottle is displayed.

--klein-bottle random
Display a random Klein bottle (default).

--klein-bottle figure-8 (Shortcut: --figure-8)
Display the figure-8 Klein bottle.

--klein-bottle pinched-torus (Shortcut: --pinched-torus)
Display the pinched torus Klein bottle.

--klein-bottle lawson (Shortcut: --lawson)
Display the Lawson Klein bottle.

The following four options are mutually exclusive. They determine
how the Klein bottle is displayed.

--mode random
Display the Klein bottle in a random display mode (default).

--mode wireframe (Shortcut: --wireframe)
Display the Klein bottle as a wireframe mesh.

--mode surface (Shortcut: --surface)
Display the Klein bottle as a solid surface.

--mode transparent (Shortcut: --transparent)
Display the Klein bottle as a transparent surface.

The following three options are mutually exclusive. They determine
the appearance of the Klein bottle.

--appearance random
Display the Klein bottle with a random appearance (default).

--appearance solid (Shortcut: --solid)
Display the Klein bottle as a solid object.

--appearance bands (Shortcut: --bands)
Display the Klein bottle as see-through bands.

The following five options are mutually exclusive. They determine
how to color the Klein bottle.

--colors random
Display the Klein bottle with a random color scheme
(default).

--colors one-sided (Shortcut: --onesided)
Display the Klein bottle with a single color.

--colors two-sided (Shortcut: --twosided)
Display the Klein bottle with two colors: one color one
"side" and the complementary color on the "other side." For
static colors, the colors are red and green.

--colors rainbow (Shortcut: --rainbow)
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.

--colors depth (Shortcut: --depth)
Display the Klein bottle with colors chosen depending on the
4d "depth" of the points.

The following options determine whether the colors with which the
Klein bottle is displayed are static or are changing dynamically.

--change-colors
Change the colors with which the Klein bottle is displayed
dynamically.

--no-change-colors
Use static colors to display the Klein bottle (default).

The following four options are mutually exclusive. They determine
how to view the Klein bottle.

--view-mode random
View the Klein bottle in a random view mode (default).

--view-mode walk (Shortcut: --walk)
View the Klein bottle as if walking on its surface.

--view-mode turn (Shortcut: --turn)
View the Klein bottle while it turns in 4d.

--view-mode walk-turn (Shortcut: --walk-turn)
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).

The following options determine whether orientation marks are shown
on the Klein bottle.

--orientation-marks
Display orientation marks on the Klein bottle.

--no-orientation-marks
Don't display orientation marks on the Klein bottle
(default).

The following three options are mutually exclusive. They determine
how the Klein bottle is projected from 3d to 2d (i.e., to the
screen).

--projection-3d random
Project the Klein bottle from 3d to 2d using a random
projection mode (default).

--projection-3d perspective (Shortcut: --perspective-3d)
Project the Klein bottle from 3d to 2d using a perspective
projection.

--projection-3d orthographic (Shortcut: --orthographic-3d)
Project the Klein bottle from 3d to 2d using an orthographic
projection.

The following three options are mutually exclusive. They determine
how the Klein bottle is projected from 4d to 3d.

--projection-4d random
Project the Klein bottle from 4d to 3d using a random
projection mode (default).

--projection-4d perspective (Shortcut: --perspective-4d)
Project the Klein bottle from 4d to 3d using a perspective
projection.

--projection-4d orthographic (Shortcut: --orthographic-4d)
Project the Klein bottle from 4d to 3d using an orthographic
projection.

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.

--speed-wx float
Rotation speed around the wx plane (default: 1.1).

--speed-wy float
Rotation speed around the wy plane (default: 1.3).

--speed-wz float
Rotation speed around the wz plane (default: 1.5).

--speed-xy float
Rotation speed around the xy plane (default: 1.7).

--speed-xz float
Rotation speed around the xz plane (default: 1.9).

--speed-yz float
Rotation speed around the yz plane (default: 2.1).

The following two options determine the walking speed and direction.

--walk-direction float
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).

--walk-speed float
The walking speed is measured in percent of some sensible
maximum speed (default: 20.0).

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.

ENVIRONMENT

DISPLAY to get the default host and display number.

XENVIRONMENT
to get the name of a resource file that overrides the global
resources stored in the RESOURCE_MANAGER property.

XSCREENSAVER_WINDOW
The window ID to use with --root.

SEE ALSO

X(1), xscreensaver(1)

COPYRIGHT

Copyright (C) 2005-2020 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.

AUTHOR

Carsten Steger <carsten@mirsanmir.org>, 11-jan-2020.

X Version 11 6.09 (07-Jun-2024) klein(6)