cubocteversion(6) XScreenSaver manual cubocteversion(6)

NAME

cubocteversion - Displays a cuboctahedron eversion.

SYNOPSIS

cubocteversion [--display host:display.screen] [--install] [--visual
visual] [--window] [--root] [--window-id number] [--delay usecs]
[--fps] [--eversion-method method] [--morin-denner] [--apery] [--mode
display-mode] [--surface] [--transparent] [--edges edge-mode]
[--self-intersections self-intersection-mode] [--colors color-scheme]
[--twosided-colors] [--face-colors] [--earth-colors] [--deformation-
speed float] [--projection projection-mode] [--perspective]
[--orthographic] [--transparency transparency-method] [--correct-
transparency] [--approximate-transparency] [--standard-transparency]
[--speed-x float] [--speed-y float] [--speed-z float]

DESCRIPTION

The cubocteversion program shows a cuboctahedron eversion, i.e., a
smooth deformation (homotopy) that turns a cuboctahedron inside out.
During the eversion, the deformed cuboctahedron is allowed to
intersect itself transversally. However, no fold edges or non-
injective neighborhoods of vertices are allowed to occur.

The cuboctahedron can be deformed with two eversion methods: Morin-
Denner or Ap'ery. The Morin-Denner cuboctahedron eversion method is
described in the following two papers: Richard Denner: "Versions
poly'edriques du retournement de la sph`ere", L'Ouvert 94:32-45, March
1999; Richard Denner: "Versions poly'edriques du retournement de la
sph`ere, retournement du cubocta`edre", L'Ouvert 95:15-36, June 1999.
The Ap'ery cuboctahedron eversion method is described in the following
paper: Fran,cois Ap'ery: "Le retournement du cubocta`edre",
Pr'epublication de l'institut de recherche math'ematique avanc'ee,
Universit'e Louis Pasteur et C.N.R.S., Strasbourg, 1994.

The deformed cuboctahedron can be projected to the screen either
perspectively or orthographically.

There are three display modes for the cuboctahedron: solid,
transparent, or random. If random mode is selected, the mode is
changed each time an eversion has been completed.

The edges of the faces of the cuboctahedron can be visualized in
three modes: without edge tubes, with edge tubes, or random. If edge
tubes are selected, solid gray tubes are displayed around the edges
of the cuboctahedron. This makes them more prominent. If random
mode is selected, the mode is changed each time an eversion has been
completed.

During the eversion, the cuboctahedron must intersect itself. It can
be selected how these self-intersections are displayed: without self-
intersection tubes, with self-intersection tubes, or random. If
self-intersection tubes are selected, solid orange tubes are
displayed around the self-intersections of the cuboctahedron. This
makes them more prominent. If random mode is selected, the mode is
changed each time an eversion has been completed.

The colors with with the cuboctahedron is drawn can be set to two-
sided, face, earth, or random. In two-sided mode, the cuboctahedron
is drawn with magenta on one side and cyan on the other side. In
face mode, the cuboctahedron is displayed with different colors for
each face. The colors of the faces are identical on the inside and
outside of the cuboctahedron. Colors on the northern hemi-
cuboctahedron are brighter than those on the southern hemi-
cuboctahedron. In earth mode, the cuboctahedron is drawn with a
texture of earth by day on one side and with a texture of earth by
night on the other side. Initially, the earth by day is on the
outside and the earth by night on the inside. After the first
eversion, the earth by night will be on the outside. All points of
the earth on the inside and outside are at the same positions on the
cuboctahedron. Since an eversion transforms the cuboctahedron into
its inverse, the earth by night will appear with all continents
mirror reversed. If random mode is selected, the color scheme is
changed each time an eversion has been completed.

It is possible to rotate the cuboctahedron while it is deforming.
The rotation speed for each of the three coordinate axes around which
the cuboctahedron rotates can be chosen arbitrarily.

BRIEF DESCRIPTION OF THE CUBOCTAHEDRON EVERSION BASICS

A sphere eversion turns the standard embedding of the unit sphere
inside-out in a smooth manner. Creases, pinch points, holes, etc.
may not occur during the eversion. However, the sphere may intersect
itself during the eversion. In mathematical terms, the eversion is a
regular homotopy between the sphere and the sphere point reflected at
its center. A convex bounded polyhedron of Euler characteristic 2 is
homeomorphic to a sphere. Since a polyhedron does not have a
continuous tangent bundle, it cannot be everted by a regular
homotopy, which requires the tangent bundle induced by the homotopy
to be continuous. Instead, it is required that polyhedron does not
develop fold edges during the eversion and that a neighborhood of
each vertex is injective throughout the eversion. Fold edges occur
whenever two faces that share an edge become coplanar and all
vertices of the two faces lie on the same side of the edge in the
plane in which they are coplanar. Furthermore, it is required that
all self-intersections between edges that occur during the eversion
are transversal, which means that they must not occur at the vertices
of the edges.

Any eversion of the sphere (smooth or polyhedral) must contain a
quadruple point. This is a point in which four different parts of
the deformed sphere intersect transversally. For a polyhedron, this
means that four different faces must intersect transversally. Four
faces are defined by four planes, each of which, in turn, is defined
by three vertices. By the above requirements, none of the twelve
vertices that define the four planes may coincide. Therefore, the
minimum number of vertices of a polyhedron that allows it to be
everted is twelve. The cuboctahedron has twelve vertices and the
papers cited above show that a cuboctahedron can indeed be everted.

A cuboctahedron has 14 faces: six squares and eight equilateral
triangles. To perform the eversion, the cuboctahedron is oriented
such that two opposite squares are horizontal. One of these squares
corresponds to the north polar region and one to the south polar
region if the cubctahedron is identified with the round sphere. The
four remaining squares are vertical and lie in the tropical region
around the equator. Each square is then divided into two isosceles
right triangles. The four tropical squares are divided along the
equator and the north and south pole squares are divided in
orthogonal directions: the edge introduced in the north pole square
is orthogonal to that introduced in the south pole square. This
results in a triangulated version of the cuboctahedron with 12
vertices, 30 edges, and 20 triangular faces. This is the version of
the cuboctahedron that can be everted.

BRIEF DESCRIPTION OF THE MORIN-DENNER CUBOCTAHEDRON EVERSION METHOD
The approach of Morin and Denner is to evert the cuboctahedron in 44
steps, resulting in 45 different polyhedra that occur as models. The
eversion is symmetric in time, so the 44 steps can be visualized by
time running from -22 to 22. Of the 45 models, 44 possess a twofold
rotational symmetry. The halfway model at time 0 possesses a
fourfold rotational symmetry. The halfway model is the model at
which the cuboctahedron is turned halfway inside-out. In each of the
44 steps, two vertices of the cuboctahedron are moved along two
respective straight lines, each of which is an edge or an extension
of an edge of the cuboctahedron. After the eversion has been
completed, the inside of the cuboctahedron lies on the outside.
Furthermore, all points of the everted cuboctahedron lie at the
antipodal points of the original cuboctahedron.

The following description assumes that the cuboctahedron is
visualized in two-sided color mode. In the first 16 steps, the
magenta cuboctahedron is deformed into a magenta polyhedron that
Morin and Denner call the bicorne. During this phase, no self-
intersections occur. Topologically, the bicorne is still an embedded
sphere. The next twelve steps, from time -6 to 6, are the most
interesting steps of the eversion: the cuboctahedron intersects
itself. It no longer is an embedding but an immersion. In this
phase, progressively more of the cyan inside becomes visible. These
steps are shown at a two times slower speed compared to the rest of
the steps. At time 6, the eversion has produced a cyan bicorne. At
this step, the cuboctahetron has been everted: it is an embedding of
the everted sphere. In the remaining 16 steps, the cyan bicorne is
deformed to the everted cuboctahedron.

BRIEF DESCRIPTION OF THE AP'ERY CUBOCTAHEDRON EVERSION METHOD
The original approach of Ap'ery is to evert the cuboctahedron in four
steps, resulting in five different polyhedra that occur as models.
The eversion is symmetric in time, so the four steps can be
visualized by time running from -2 to 2. Of the five models, four
possess a twofold rotational symmetry. The halfway model at time 0
possesses a fourfold rotational symmetry. The halfway model is the
model at which the cuboctahedron is turned halfway inside-out. In
addition to the start and end models at times -2 and 2, which both
are cuboctahedra, and the halfway model at time 0, the two
intermediate models at times -1 and 1 are embeddings of the
cuboctahedron. Ap'ery calls them gastrula because they correspond to
a cuboctahedron in which the northern hemi-cuboctahedron has been
pushed downwards so that it lies inside the southern hemi-
cuboctahedron. In each of the four steps, the cuboctahedron is
deformed by linearly interpolating the corresponding vertices between
two successive models. After the eversion has been completed, the
inside of the cuboctahedron lies on the outside. Furthermore, all
points of the everted cuboctahedron lie at the antipodal points of
the original cuboctahedron.

During the development of this program, it was discovered that the
linear interpolation between the cuboctahedron and the gastrula
causes the deformed cuboctahedron to intersect itself for a brief
period of time shortly before the gastrula is reached. Therefore, an
additional model, devised by Fran,cois Ap'ery and called pre-gastrula
by him, was inserted at times -1.25 and 1.25. This additional model
avoids the self-intersections before the gastrula is reached. The
rest of Ap'ery's approach remains unaffected: the vertices are
interpolated linearly between successive models.

The following description assumes that the cuboctahedron is
visualized in two-sided color mode. In the first two steps, the
magenta cuboctahedron is deformed into a magenta gastrula. During
this phase, no self-intersections occur. Topologically, the gastrula
is still an embedded sphere. The next two steps, from time -1 to 1,
are the most interesting steps of the eversion: the cuboctahedron
intersects itself. It no longer is an embedding but an immersion.
In this phase, progressively more of the cyan inside becomes visible.
At time 1, the eversion has produced a cyan gastrula. At this step,
the cuboctahetron has been everted: it is an embedding of the everted
sphere. In the remaining two steps, the cyan gastrula is deformed to
the everted cuboctahedron.

OPTIONS

cubocteversion 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 20000, or 1/50th second.

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

The following three options are mutually exclusive. They determine
which cuboctahedron eversion method is used.

--eversion-method random
Use a random cuboctahedron eversion method (default).

--eversion-method morin-denner (Shortcut: --morin-denner)
Use the Morin-Denner cuboctahedron eversion method.

--eversion-method apery (Shortcut: --apery)
Use the Ap'ery cuboctahedron eversion method.

The following three options are mutually exclusive. They determine
how the deformed cuboctahedron is displayed.

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

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

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

The following three options are mutually exclusive. They determine
whether the edges of the cuboctahedron are displayed as solid gray
tubes.

--edges random
Randomly choose whether to display edge tubes (default).

--edges on
Display the cuboctahedron with edge tubes.

--edges off
Display the cuboctahedron without edge tubes.

The following three options are mutually exclusive. They determine
whether the self-intersections of the deformed cuboctahedron are
displayed as solid orange tubes.

--self-intersections random
Randomly choose whether to display self-intersection tubes
(default).

--self-intersections on
Display the cuboctahedron with self-intersection tubes.

--self-intersections off
Display the cuboctahedron without self-intersection tubes.

The following four options are mutually exclusive. They determine
how to color the deformed cuboctahedron.

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

--colors twosided (Shortcut: --twosided-colors)
Display the cuboctahedron with two colors: magenta on one
side and cyan on the other side.

--colors face (Shortcut: --face-colors)
Display the cuboctahedron with different colors for each
face. The colors of the faces are identical on the inside and
outside of the cuboctahedron. Colors on the northern hemi-
cuboctahedron are brighter than those on the southern hemi-
cuboctahedron.

--colors earth (Shortcut: --earth-colors)
Display the cuboctahedron with a texture of earth by day on
one side and with a texture of earth by night on the other
side. Initially, the earth by day is on the outside and the
earth by night on the inside. After the first eversion, the
earth by night will be on the outside. All points of the
earth on the inside and outside are at the same positions on
the cuboctahedron. Since an eversion transforms the
cuboctahedron into its inverse, the earth by night will
appear with all continents mirror reversed.

The following option determines the deformation speed.

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

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

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

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

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

The following three options are mutually exclusive. They determine
which transparency algorithm is used to display the transparent faces
of the cuboctahedron. If correct transparency is selected, a correct
but slower algorithm is used to render the transparent faces. If the
frame rate of this algorithm is too slow and results in a jerky
animation, it can be set to one of the other two modes. If
approximate transparency is selected, an transparency algorithm that
provides an approximation to the correct transparency is used.
Finally, if standard transparency is selected, a transparency
algorithm that only uses standard OpenGL transparency rendering
features is used. It results in a lower-quality rendering of the
transparent faces in which the appearance depends on the order in
which the faces are drawn. The approximate and standard transparency
algorithms are equally fast and, depending on the GPU, can be
significantly faster than the correct transparency algorithm. The
correct and approximate transparency algorithms are automatically
switched off if the OpenGL version supported by the operating system
does not support them (for example, on iOS and iPadOS).

--transparency correct (Shortcut: --correct-transparency)
Use a transparency algorithm that results in a correct
rendering of transparent surfaces (default).

--transparency approximate (Shortcut: --approximate-transparency)
Use a transparency algorithm that results in an approximately
correct rendering of transparent surfaces.

--transparency standard (Shortcut: --standard-transparency)
Use a transparency algorithm that uses only standard OpenGL
features for the rendering of transparent surfaces.

The following three options determine the rotation speed of the
deformed cuboctahedron around the three possible axes. 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.

--speed-x float
Rotation speed around the x axis (default: 0.0).

--speed-y float
Rotation speed around the y axis (default: 0.0).

--speed-z float
Rotation speed around the z axis (default: 0.0).

INTERACTION

If you run this program in standalone mode, you can rotate the
deformed cuboctahedron by dragging the mouse while pressing the left
mouse button. This rotates the cuboctahedron in 3d. To examine the
deformed cuboctahedron at your leisure, it is best to set all speeds
to 0. Otherwise, the deformed cuboctahedron will rotate while the
left mouse button is not pressed.

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) 2023 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>, 06-mar-2023.

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