1klein(6x) XScreenSaver manual klein(6x)
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6 kleinbottle - Draws a 4d Klein bottle.
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9 kleinbottle [-display host:display.screen] [-install] [-visual visual]
10 [-window] [-root] [-delay usecs] [-fps] [-klein-bottle bottle-name]
11 [-figure-8] [-lawson] [-mode display-mode] [-wireframe] [-surface]
12 [-transparent] [-appearance appearance] [-solid] [-bands] [-colors
13 color-scheme] [-twosided] [-rainbow] [-depth] [-view-mode view-mode]
14 [-walk] [-turn] [-walk-turn] [-orientation-marks] [-projection-3d mode]
15 [-perspective-3d] [-orthographic-3d] [-projection-4d mode] [-perspec‐
16 tive-4d] [-orthographic-4d] [-speed-wx float] [-speed-wy float]
17 [-speed-wz float] [-speed-xy float] [-speed-xz float] [-speed-yz float]
18 [-walk-direction float] [-walk-speed float]
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21 The kleinbottle program shows two different Klein bottles in 4d: the
22 figure-8 Klein bottle or the Lawson Klein bottle. You can walk on the
23 Klein bottle, see it turn in 4d, or walk on it while it turns in 4d.
24 The figure-8 Klein bottle is well known in its 3d form. The 4d form
25 used in this program is an extension of the 3d form to 4d that does not
26 intersect itself in 4d (which can be seen in the depth colors mode).
27 The Lawson Klein bottle, on the other hand, does intersect itself in
28 4d. Its primary use is that it has a nice appearance for walking and
29 for turning in 3d. The Klein bottle is a non-orientable surface. To
30 make this apparent, the two-sided color mode can be used. Alterna‐
31 tively, orientation markers (curling arrows) can be drawn as a texture
32 map on the surface of the Klein bottle. While walking on the Klein
33 bottle, you will notice that the orientation of the curling arrows
34 changes (which it must because the Klein bottle is non-orientable).
35 The program projects the 4d Klein bottle to 3d using either a perspec‐
36 tive or an orthographic projection. Which of the two alternatives
37 looks more appealing depends on the viewing mode and the Klein bottle.
38 For example, the Lawson Klein bottle looks nicest when projected per‐
39 spectively. The figure-8 Klein bottle, on the other hand, looks nicer
40 while walking when projected orthographically from 4d. The projected
41 Klein bottle can then be projected to the screen either perspectively
42 or orthographically. When using the walking modes, perspective projec‐
43 tion to the screen should be used. There are three display modes for
44 the Klein bottle: mesh (wireframe), solid, or transparent. Further‐
45 more, the appearance of the Klein bottle can be as a solid object or as
46 a set of see-through bands. Finally, the colors with with the Klein
47 bottle is drawn can be set to two-sided, rainbow, or depth. In the
48 first case, the Klein bottle is drawn with red on one "side" and green
49 on the "other side". Of course, the Klein bottle only has one side, so
50 the color jumps from red to green along a curve on the surface of the
51 Klein bottle. This mode enables you to see that the Klein bottle is
52 non-orientable. The second mode draws the Klein bottle with fully sat‐
53 urated rainbow colors. This gives a very nice effect when combined
54 with the see-through bands mode or with the orientation markers drawn.
55 The third mode draws the Klein bottle with colors that are chosen
56 according to the 4d "depth" of the points. This mode enables you to
57 see that the figure-8 Klein bottle does not intersect itself in 4d,
58 while the Lawson Klein bottle does intersect itself. The rotation
59 speed for each of the six planes around which the Klein bottle rotates
60 can be chosen. For the walk-and-turn more, only the rotation speeds
61 around the true 4d planes are used (the xy, xz, and yz planes). Fur‐
62 thermore, in the walking modes the walking direction in the 2d base
63 square of the Klein bottle and the walking speed can be chosen. This
64 program is somewhat inspired by Thomas Banchoff's book "Beyond the
65 Third Dimension: Geometry, Computer Graphics, and Higher Dimensions",
66 Scientific American Library, 1990.
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69 kleinbottle accepts the following options:
70 -window Draw on a newly-created window. This is the default.
72 -root Draw on the root window.
74 -install
76 Install a private colormap for the window.
77 -visual visual
79 Specify which visual to use. Legal values are the name of a
80 visual class, or the id number (decimal or hex) of a specific
81 visual.
82 -delay microseconds
84 How much of a delay should be introduced between steps of the
85 animation. Default 10000, or 1/100th second.
86 -fps Display the current frame rate, CPU load, and polygon count.
88 The following three options are mutually exclusive. They determine
90 which Klein bottle is displayed.
91 -klein-bottle random
93 Display a random Klein bottle (default).
94 -klein-bottle figure-8 (Shortcut: -figure-8)
96 Display the figure-8 Klein bottle.
97 -klein-bottle lawson (Shortcut: -lawson)
99 Display the Lawson Klein bottle.
100 The following four options are mutually exclusive. They determine how
102 the Klein bottle is displayed.
103 -mode random
105 Display the Klein bottle in a random display mode (default).
106 -mode wireframe (Shortcut: -wireframe)
108 Display the Klein bottle as a wireframe mesh.
109 -mode surface (Shortcut: -surface)
111 Display the Klein bottle as a solid surface.
112 -mode transparent (Shortcut: -transparent)
114 Display the Klein bottle as a transparent surface.
115 The following three options are mutually exclusive. They determine the
117 appearance of the Klein bottle.
118 -appearance random
120 Display the Klein bottle with a random appearance (default).
121 -appearance solid (Shortcut: -solid)
123 Display the Klein bottle as a solid object.
124 -appearance bands (Shortcut: -bands)
126 Display the Klein bottle as see-through bands.
127 The following four options are mutually exclusive. They determine how
129 to color the Klein bottle.
130 -colors random
132 Display the Klein bottle with a random color scheme (default).
133 -colors twosided (Shortcut: -twosided)
135 Display the Klein bottle with two colors: red on one "side" and
136 green on the "other side".
137 -colors rainbow (Shortcut: -rainbow)
139 Display the Klein bottle with fully saturated rainbow colors.
140 If the Klein bottle is displayed as see-through bands, each
141 band will be displayed with a different color.
142 -colors depth (Shortcut: -depth)
144 Display the Klein bottle with colors chosen depending on the 4d
145 "depth" of the points.
146 The following four options are mutually exclusive. They determine how
148 to view the Klein bottle.
149 -view-mode random
151 View the Klein bottle in a random view mode (default).
152 -view-mode walk (Shortcut: -walk)
154 View the Klein bottle as if walking on its surface.
155 -view-mode turn (Shortcut: -turn)
157 View the Klein bottle while it turns in 4d.
158 -view-mode walk-turn (Shortcut: -walk-turn)
160 View the Klein bottle as if walking on its surface. Addition‐
161 ally, the Klein bottle turns around the true 4d planes (the xy,
162 xz, and yz planes).
163 The following options determine whether orientation marks are shown on
165 the Klein bottle.
166 -orientation-marks
168 Display orientation marks on the Klein bottle.
169 -no-orientation-marks
171 Don't display orientation marks on the Klein bottle (default).
172 The following three options are mutually exclusive. They determine how
174 the Klein bottle is projected from 3d to 2d (i.e., to the screen).
175 -projection-3d random
177 Project the Klein bottle from 3d to 2d using a random projec‐
178 tion mode (default).
179 -projection-3d perspective (Shortcut: -perspective-3d)
181 Project the Klein bottle from 3d to 2d using a perspective pro‐
182 jection.
183 -projection-3d orthographic (Shortcut: -orthographic-3d)
185 Project the Klein bottle from 3d to 2d using an orthographic
186 projection.
187 The following three options are mutually exclusive. They determine how
189 the Klein bottle is projected from 4d to 3d.
190 -projection-4d random
192 Project the Klein bottle from 4d to 3d using a random projec‐
193 tion mode (default).
194 -projection-4d perspective (Shortcut: -perspective-4d)
196 Project the Klein bottle from 4d to 3d using a perspective pro‐
197 jection.
198 -projection-4d orthographic (Shortcut: -orthographic-4d)
200 Project the Klein bottle from 4d to 3d using an orthographic
201 projection.
202 The following six options determine the rotation speed of the Klein
204 bottle around the six possible hyperplanes. The rotation speed is mea‐
205 sured in degrees per frame. The speeds should be set to relatively
206 small values, e.g., less than 4 in magnitude. In walk mode, all speeds
207 are ignored. In walk-and-turn mode, the 3d rotation speeds are ignored
208 (i.e., the wx, wy, and wz speeds). In walk-and-turn mode, smaller
209 speeds must be used than in the turn mode to achieve a nice visualiza‐
210 tion. Therefore, in walk-and-turn mode the speeds you have selected
211 are divided by 5 internally.
212 -speed-wx float
214 Rotation speed around the wx plane (default: 1.1).
215 -speed-wy float
217 Rotation speed around the wy plane (default: 1.3).
218 -speed-wz float
220 Rotation speed around the wz plane (default: 1.5).
221 -speed-xy float
223 Rotation speed around the xy plane (default: 1.7).
224 -speed-xz float
226 Rotation speed around the xz plane (default: 1.9).
227 -speed-yz float
229 Rotation speed around the yz plane (default: 2.1).
230 The following two options determine the walking speed and direction.
232 -walk-direction float
234 The walking direction is measured as an angle in degrees in the
235 2d square that forms the coordinate system of the surface of
236 the Klein bottle (default: 7.0).
237 -walk-speed float
239 The walking speed is measured in percent of some sensible maxi‐
240 mum speed (default: 20.0).
241
243 If you run this program in standalone mode in its turn mode, you can
244 rotate the Klein bottle by dragging the mouse while pressing the left
245 mouse button. This rotates the Klein bottle in 3D, i.e., around the
246 wx, wy, and wz planes. If you press the shift key while dragging the
247 mouse with the left button pressed the Klein bottle is rotated in 4D,
248 i.e., around the xy, xz, and yz planes. To examine the Klein bottle at
249 your leisure, it is best to set all speeds to 0. Otherwise, the Klein
250 bottle will rotate while the left mouse button is not pressed. This
251 kind of interaction is not available in the two walk modes.
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254 DISPLAY to get the default host and display number.
255 XENVIRONMENT
257 to get the name of a resource file that overrides the global
258 resources stored in the RESOURCE_MANAGER property.
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261 X(1), xscreensaver(1)
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264 Copyright © 2005-2009 by Carsten Steger. Permission to use, copy, mod‐
265 ify, distribute, and sell this software and its documentation for any
266 purpose is hereby granted without fee, provided that the above copy‐
267 right notice appear in all copies and that both that copyright notice
268 and this permission notice appear in supporting documentation. No rep‐
269 resentations are made about the suitability of this software for any
270 purpose. It is provided "as is" without express or implied warranty.
271
273 Carsten Steger <carsten@mirsanmir.org>, 03-aug-2009.
274X Version 11 5.15-3.fc14 (18-Oct-2011) klein(6x)