1klein(6x) XScreenSaver manual klein(6x)
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6 klein - Draws a 4d Klein bottle.
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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
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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73 klein accepts the following options:
74 -window Draw on a newly-created window. This is the default.
76 -root Draw on the root window.
78 -install
80 Install a private colormap for the window.
81 -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 -delay microseconds
88 How much of a delay should be introduced between steps of the
89 animation. Default 10000, or 1/100th second.
90 -fps Display the current frame rate, CPU load, and polygon count.
92 The following three options are mutually exclusive. They determine
94 which Klein bottle is displayed.
95 -klein-bottle random
97 Display a random Klein bottle (default).
98 -klein-bottle figure-8 (Shortcut: -figure-8)
100 Display the figure-8 Klein bottle.
101 -klein-bottle squeezed-torus (Shortcut: -squeezed-torus)
103 Display the squeezed torus Klein bottle.
104 -klein-bottle lawson (Shortcut: -lawson)
106 Display the Lawson Klein bottle.
107 The following four options are mutually exclusive. They determine how
109 the Klein bottle is displayed.
110 -mode random
112 Display the Klein bottle in a random display mode (default).
113 -mode wireframe (Shortcut: -wireframe)
115 Display the Klein bottle as a wireframe mesh.
116 -mode surface (Shortcut: -surface)
118 Display the Klein bottle as a solid surface.
119 -mode transparent (Shortcut: -transparent)
121 Display the Klein bottle as a transparent surface.
122 The following three options are mutually exclusive. They determine the
124 appearance of the Klein bottle.
125 -appearance random
127 Display the Klein bottle with a random appearance (default).
128 -appearance solid (Shortcut: -solid)
130 Display the Klein bottle as a solid object.
131 -appearance bands (Shortcut: -bands)
133 Display the Klein bottle as see-through bands.
134 The following four options are mutually exclusive. They determine how
136 to color the Klein bottle.
137 -colors random
139 Display the Klein bottle with a random color scheme (default).
140 -colors twosided (Shortcut: -twosided)
142 Display the Klein bottle with two colors: red on one "side" and
143 green on the "other side".
144 -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 -colors depth (Shortcut: -depth)
151 Display the Klein bottle with colors chosen depending on the 4d
152 "depth" of the points.
153 The following four options are mutually exclusive. They determine how
155 to view the Klein bottle.
156 -view-mode random
158 View the Klein bottle in a random view mode (default).
159 -view-mode walk (Shortcut: -walk)
161 View the Klein bottle as if walking on its surface.
162 -view-mode turn (Shortcut: -turn)
164 View the Klein bottle while it turns in 4d.
165 -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 The following options determine whether orientation marks are shown on
172 the Klein bottle.
173 -orientation-marks
175 Display orientation marks on the Klein bottle.
176 -no-orientation-marks
178 Don't display orientation marks on the Klein bottle (default).
179 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 -projection-3d random
184 Project the Klein bottle from 3d to 2d using a random projec‐
185 tion mode (default).
186 -projection-3d perspective (Shortcut: -perspective-3d)
188 Project the Klein bottle from 3d to 2d using a perspective pro‐
189 jection.
190 -projection-3d orthographic (Shortcut: -orthographic-3d)
192 Project the Klein bottle from 3d to 2d using an orthographic
193 projection.
194 The following three options are mutually exclusive. They determine how
196 the Klein bottle is projected from 4d to 3d.
197 -projection-4d random
199 Project the Klein bottle from 4d to 3d using a random projec‐
200 tion mode (default).
201 -projection-4d perspective (Shortcut: -perspective-4d)
203 Project the Klein bottle from 4d to 3d using a perspective pro‐
204 jection.
205 -projection-4d orthographic (Shortcut: -orthographic-4d)
207 Project the Klein bottle from 4d to 3d using an orthographic
208 projection.
209 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 -speed-wx float
221 Rotation speed around the wx plane (default: 1.1).
222 -speed-wy float
224 Rotation speed around the wy plane (default: 1.3).
225 -speed-wz float
227 Rotation speed around the wz plane (default: 1.5).
228 -speed-xy float
230 Rotation speed around the xy plane (default: 1.7).
231 -speed-xz float
233 Rotation speed around the xz plane (default: 1.9).
234 -speed-yz float
236 Rotation speed around the yz plane (default: 2.1).
237 The following two options determine the walking speed and direction.
239 -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 -walk-speed float
246 The walking speed is measured in percent of some sensible maxi‐
247 mum speed (default: 20.0).
248
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
261 DISPLAY to get the default host and display number.
262 XENVIRONMENT
264 to get the name of a resource file that overrides the global
265 resources stored in the RESOURCE_MANAGER property.
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268 X(1), xscreensaver(1)
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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
280 Carsten Steger <carsten@mirsanmir.org>, 03-oct-2014.
281X Version 11 5.43-2.fc31 (27-Aug-2019) klein(6x)