-The \fIkleinbottle\fP program shows two different Klein bottles in 4d:
-the figure-8 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 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. 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 Klein bottle does not
-intersect itself 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 more,
-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.
+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 more, 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.