1klein(6x)                     XScreenSaver manual                    klein(6x)
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NAME

6       klein - Draws a 4d Klein bottle.
7

SYNOPSIS

9       klein [-display host:display.screen] [-install] [-visual visual] [-win‐
10       dow] [-root] [-delay usecs] [-fps] [-klein-bottle  bottle-name]  [-fig‐
11       ure-8]  [-squeezed-torus]  [-lawson]  [-mode display-mode] [-wireframe]
12       [-surface] [-transparent] [-appearance  appearance]  [-solid]  [-bands]
13       [-colors  color-scheme]  [-twosided]  [-rainbow]  [-depth]  [-view-mode
14       view-mode] [-walk] [-turn] [-walk-turn] [-orientation-marks]  [-projec‐
15       tion-3d   mode]  [-perspective-3d]  [-orthographic-3d]  [-projection-4d
16       mode] [-perspective-4d] [-orthographic-4d] [-speed-wx float] [-speed-wy
17       float] [-speed-wz float] [-speed-xy float] [-speed-xz float] [-speed-yz
18       float] [-walk-direction float] [-walk-speed float]
19

DESCRIPTION

21       The klein program shows three different Klein bottles in 4d:  the  fig‐
22       ure-8  Klein  bottle,  the  squeezed  torus Klein bottle, or the Lawson
23       Klein bottle.  You can walk on the Klein bottle, see it turn in 4d,  or
24       walk  on  it  while  it turns in 4d.  The figure-8 Klein bottle is well
25       known in its 3d form.  The 4d form used in this program is an extension
26       of the 3d form to 4d that does not intersect itself in 4d (which can be
27       seen in the depth colors mode).  The squeezed torus Klein  bottle  also
28       does  not intersect itself in 4d (which can be seen in the depth colors
29       mode).  The Lawson Klein bottle, on  the  other  hand,  does  intersect
30       itself  in  4d.   Its  primary use is that it has a nice appearance for
31       walking and for turning in 3d.  The Klein bottle  is  a  non-orientable
32       surface.   To make this apparent, the two-sided color mode can be used.
33       Alternatively, orientation markers (curling arrows) can be drawn  as  a
34       texture  map  on the surface of the Klein bottle.  While walking on the
35       Klein bottle, you will notice  that  the  orientation  of  the  curling
36       arrows  changes  (which  it  must  because the Klein bottle is non-ori‐
37       entable).  The program projects the 4d Klein bottle to 3d using  either
38       a perspective or an orthographic projection.  Which of the two alterna‐
39       tives looks more appealing depends on the viewing mode  and  the  Klein
40       bottle.   For  example,  the Lawson Klein bottle looks nicest when pro‐
41       jected perspectively.  The figure-8 Klein bottle, on  the  other  hand,
42       looks nicer while walking when projected orthographically from 4d.  For
43       the squeezed torus Klein bottle, both  projection  modes  give  equally
44       acceptable  projections.   The  projected Klein bottle can then be pro‐
45       jected to the screen either perspectively  or  orthographically.   When
46       using the walking modes, perspective projection to the screen should be
47       used.  There are three display modes for the Klein bottle: mesh  (wire‐
48       frame),  solid,  or  transparent.   Furthermore,  the appearance of the
49       Klein bottle can be as a solid object or as a set of see-through bands.
50       Finally,  the  colors with with the Klein bottle is drawn can be set to
51       two-sided, rainbow, or depth.  In the first case, the Klein  bottle  is
52       drawn with red on one "side" and green on the "other side".  Of course,
53       the Klein bottle only has one side, so the  color  jumps  from  red  to
54       green  along  a  curve  on  the surface of the Klein bottle.  This mode
55       enables you to see that the Klein bottle is non-orientable.  The second
56       mode  draws the Klein bottle with fully saturated rainbow colors.  This
57       gives a very nice effect when combined with the see-through bands  mode
58       or  with the orientation markers drawn.  The third mode draws the Klein
59       bottle with colors that are chosen according to the 4d "depth"  of  the
60       points.   This  mode  enables you to see that the figure-8 and squeezed
61       torus Klein bottles do not intersect themselves in 4d, while the Lawson
62       Klein bottle does intersect itself.  The rotation speed for each of the
63       six planes around which the Klein bottle rotates can  be  chosen.   For
64       the  walk-and-turn  mode,  only  the rotation speeds around the true 4d
65       planes are used (the xy, xz, and yz planes).  Furthermore, in the walk‐
66       ing modes the walking direction in the 2d base square of the Klein bot‐
67       tle and the walking speed can be  chosen.   This  program  is  somewhat
68       inspired  by Thomas Banchoff's book "Beyond the Third Dimension: Geome‐
69       try, Computer Graphics, and  Higher  Dimensions",  Scientific  American
70       Library, 1990.
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OPTIONS

73       klein accepts the following options:
74
75       -window Draw on a newly-created window.  This is the default.
76
77       -root   Draw on the root window.
78
79       -install
80               Install a private colormap for the window.
81
82       -visual visual
83               Specify  which  visual  to use.  Legal values are the name of a
84               visual class, or the id number (decimal or hex) of  a  specific
85               visual.
86
87       -delay microseconds
88               How  much  of a delay should be introduced between steps of the
89               animation.  Default 10000, or 1/100th second.
90
91       -fps    Display the current frame rate, CPU load, and polygon count.
92
93       The following three options are  mutually  exclusive.   They  determine
94       which Klein bottle is displayed.
95
96       -klein-bottle random
97               Display a random Klein bottle (default).
98
99       -klein-bottle figure-8 (Shortcut: -figure-8)
100               Display the figure-8 Klein bottle.
101
102       -klein-bottle squeezed-torus (Shortcut: -squeezed-torus)
103               Display the squeezed torus Klein bottle.
104
105       -klein-bottle lawson (Shortcut: -lawson)
106               Display the Lawson Klein bottle.
107
108       The  following four options are mutually exclusive.  They determine how
109       the Klein bottle is displayed.
110
111       -mode random
112               Display the Klein bottle in a random display mode (default).
113
114       -mode wireframe (Shortcut: -wireframe)
115               Display the Klein bottle as a wireframe mesh.
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117       -mode surface (Shortcut: -surface)
118               Display the Klein bottle as a solid surface.
119
120       -mode transparent (Shortcut: -transparent)
121               Display the Klein bottle as a transparent surface.
122
123       The following three options are mutually exclusive.  They determine the
124       appearance of the Klein bottle.
125
126       -appearance random
127               Display the Klein bottle with a random appearance (default).
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129       -appearance solid (Shortcut: -solid)
130               Display the Klein bottle as a solid object.
131
132       -appearance bands (Shortcut: -bands)
133               Display the Klein bottle as see-through bands.
134
135       The  following four options are mutually exclusive.  They determine how
136       to color the Klein bottle.
137
138       -colors random
139               Display the Klein bottle with a random color scheme (default).
140
141       -colors twosided (Shortcut: -twosided)
142               Display the Klein bottle with two colors: red on one "side" and
143               green on the "other side".
144
145       -colors rainbow (Shortcut: -rainbow)
146               Display  the  Klein bottle with fully saturated rainbow colors.
147               If the Klein bottle is displayed  as  see-through  bands,  each
148               band will be displayed with a different color.
149
150       -colors depth (Shortcut: -depth)
151               Display the Klein bottle with colors chosen depending on the 4d
152               "depth" of the points.
153
154       The following four options are mutually exclusive.  They determine  how
155       to view the Klein bottle.
156
157       -view-mode random
158               View the Klein bottle in a random view mode (default).
159
160       -view-mode walk (Shortcut: -walk)
161               View the Klein bottle as if walking on its surface.
162
163       -view-mode turn (Shortcut: -turn)
164               View the Klein bottle while it turns in 4d.
165
166       -view-mode walk-turn (Shortcut: -walk-turn)
167               View  the Klein bottle as if walking on its surface.  Addition‐
168               ally, the Klein bottle turns around the true 4d planes (the xy,
169               xz, and yz planes).
170
171       The  following options determine whether orientation marks are shown on
172       the Klein bottle.
173
174       -orientation-marks
175               Display orientation marks on the Klein bottle.
176
177       -no-orientation-marks
178               Don't display orientation marks on the Klein bottle (default).
179
180       The following three options are mutually exclusive.  They determine how
181       the Klein bottle is projected from 3d to 2d (i.e., to the screen).
182
183       -projection-3d random
184               Project  the  Klein bottle from 3d to 2d using a random projec‐
185               tion mode (default).
186
187       -projection-3d perspective (Shortcut: -perspective-3d)
188               Project the Klein bottle from 3d to 2d using a perspective pro‐
189               jection.
190
191       -projection-3d orthographic (Shortcut: -orthographic-3d)
192               Project  the  Klein  bottle from 3d to 2d using an orthographic
193               projection.
194
195       The following three options are mutually exclusive.  They determine how
196       the Klein bottle is projected from 4d to 3d.
197
198       -projection-4d random
199               Project  the  Klein bottle from 4d to 3d using a random projec‐
200               tion mode (default).
201
202       -projection-4d perspective (Shortcut: -perspective-4d)
203               Project the Klein bottle from 4d to 3d using a perspective pro‐
204               jection.
205
206       -projection-4d orthographic (Shortcut: -orthographic-4d)
207               Project  the  Klein  bottle from 4d to 3d using an orthographic
208               projection.
209
210       The following six options determine the rotation  speed  of  the  Klein
211       bottle around the six possible hyperplanes.  The rotation speed is mea‐
212       sured in degrees per frame.  The speeds should  be  set  to  relatively
213       small values, e.g., less than 4 in magnitude.  In walk mode, all speeds
214       are ignored.  In walk-and-turn mode, the 3d rotation speeds are ignored
215       (i.e.,  the  wx,  wy,  and  wz speeds).  In walk-and-turn mode, smaller
216       speeds must be used than in the turn mode to achieve a nice  visualiza‐
217       tion.   Therefore,  in  walk-and-turn mode the speeds you have selected
218       are divided by 5 internally.
219
220       -speed-wx float
221               Rotation speed around the wx plane (default: 1.1).
222
223       -speed-wy float
224               Rotation speed around the wy plane (default: 1.3).
225
226       -speed-wz float
227               Rotation speed around the wz plane (default: 1.5).
228
229       -speed-xy float
230               Rotation speed around the xy plane (default: 1.7).
231
232       -speed-xz float
233               Rotation speed around the xz plane (default: 1.9).
234
235       -speed-yz float
236               Rotation speed around the yz plane (default: 2.1).
237
238       The following two options determine the walking speed and direction.
239
240       -walk-direction float
241               The walking direction is measured as an angle in degrees in the
242               2d  square  that  forms the coordinate system of the surface of
243               the Klein bottle (default: 7.0).
244
245       -walk-speed float
246               The walking speed is measured in percent of some sensible maxi‐
247               mum speed (default: 20.0).
248

INTERACTION

250       If  you  run  this program in standalone mode in its turn mode, you can
251       rotate the Klein bottle by dragging the mouse while pressing  the  left
252       mouse  button.   This  rotates the Klein bottle in 3D, i.e., around the
253       wx, wy, and wz planes.  If you press the shift key while  dragging  the
254       mouse  with  the left button pressed the Klein bottle is rotated in 4D,
255       i.e., around the xy, xz, and yz planes.  To examine the Klein bottle at
256       your  leisure, it is best to set all speeds to 0.  Otherwise, the Klein
257       bottle will rotate while the left mouse button is  not  pressed.   This
258       kind of interaction is not available in the two walk modes.
259

ENVIRONMENT

261       DISPLAY to get the default host and display number.
262
263       XENVIRONMENT
264               to  get  the  name of a resource file that overrides the global
265               resources stored in the RESOURCE_MANAGER property.
266

SEE ALSO

268       X(1), xscreensaver(1)
269
271       Copyright © 2005-2014 by Carsten Steger.  Permission to use, copy, mod‐
272       ify,  distribute,  and sell this software and its documentation for any
273       purpose is hereby granted without fee, provided that  the  above  copy‐
274       right  notice  appear in all copies and that both that copyright notice
275       and this permission notice appear in supporting documentation.  No rep‐
276       resentations  are  made  about the suitability of this software for any
277       purpose.  It is provided "as is" without express or implied warranty.
278

AUTHOR

280       Carsten Steger <carsten@mirsanmir.org>, 03-oct-2014.
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284X Version 11              5.42-1.fc30.2 (03-Feb-2019)                klein(6x)
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