MATHEMATICAL BACKGROUND
In
[1] Coxeter considers the 24cell with the 24 vertices
(±2, 0, 0, 0), (0, ±2, 0, 0),
(0, 0, ±2, 0), (0, 0, 0,
±2) and (±1, ±1,
±1, ±1) ,
and applies the 4D rotation which sends (u, x, y, z), or in quaternion
representation: u + xi + yj + zk, into
the quaternion k * (u + xi + yj + zk) *
(1 − i − j − k) / 2.
This rotation has period 12, and a specific vertex forms together with
its eleven transforms a skew regular dodecagon consisting of 12
consecutive edges of the 24cell, a Petrie dodecagon. The 24 vertices
together describe two Petrie dodecagons in this way. When the 24cell
is projected along one of the planeaxes of this rotation onto the
other planeaxis, one of the Petrie dodecagons appears as a regular
dodecagon, and the other one as a regular dodecagram.
So far from Coxeter's article.
This rotational displacement was
turned into a uniform rotational motion by setting the angular velocity
components to
A:
(ω_{ux }, ω_{uy },
ω_{uz }, ω_{yz },
ω_{zx }, ω_{xy})
=
60 ( −√(1/3), −√(1/3), 3/2 − √(1/3);
√(1/3), √(1/3), 3/2 + √(1/3) ) =
( −34.641016, −34.641016, 55.358984;
34.641016, 34.641016, 124.641016 )
degrees per time unit.
These components are referred to the
bodyfixed coordinate system.
This rotational motion has a period of 12 time units; each lap of one
time unit will rotate the 24cell to its next congruent orientation.
This rotational motion is decomposed into the simple rotational motions
B and C given by
B:
(ω_{ux }, ω_{uy },
ω_{uz }, ω_{yz },
ω_{zx }, ω_{xy})
=
75 ( −√(1/3), −√(1/3), 1 − √(1/3);
√(1/3), √(1/3), 1 + √(1/3) ) =
( −43.301270, −43.301270, 31.698730;
43.301270, 43.301270, 118.301270 )
degrees per time unit;
angular speed 150 degrees per time unit, a complete revolution
in 2.40 time units;
C:
(ω_{ux }, ω_{uy },
ω_{uz }, ω_{yz },
ω_{zx }, ω_{xy})
=
15 ( √(1/3), √(1/3), 1 + √(1/3);
−√(1/3), −√(1/3), 1 − √(1/3) ) =
( 8.660254, 8.660254, 23.660254;
−8.660254, −8.660254, 6.339746 ) degrees
per time unit;
angular speed 30 degrees per time unit, a complete revolution in 12
time units.
The initial orientation of the 24cell is such that the spacefixed XY
plane is the axisplane of B, the fast simple rotation, and the
spacefixed UZ plane is the axisplane of C, the slow simple rotation.
INSTALLATION AND GUIDED TOUR
THIS SOFTWARE WORKS UNDER
MSWINDOWS AS WELL AS UNDER MSDOS.
The file AB4.ZIP contains the AB4SVDV program and its associated BGI,
AS4 and HLP files.
Unpack AB4.ZIP into an arbitrary directory D. This is the whole
installation. From this point one can proceed in two ways:
(A) The MSDOS way: open an
MSDOS box if you run an MSWindows system, make directory D current
and run the batch command...
AA1
for the
24cell with a pair of Petrie dodecagons,
AA2
for the 24cell with a dodecagon and a dodecagram (star
dodecagon),
AA3
for the 24cell with the pair of Petrie dodecagons of AA1 and
the associated dodecagrams....
or run the command
AB4SVDV
[filename with suffix included] 9 2 to
run the AB4SVDV program on AS4 files in general.
(B) The MSWindows way: open
Explorer, open directory D and doubleclick on AA1 (AA1.BAT), AA2
(AA2.BAT) or AA3 (AA3.BAT).
The opening screen shows some general program settings, a copyright notice (*) and
a component list with version numbers.
The last entry reads "AB4SVDV v2.072006". This is the
program itself. (PIC01.GIF below).
Remark: When running an
MSDOS program under MSWindows one can switch between Windowed and
Fullscreen modes by the keystroke command AltENTER.

Fig. 1
Opening screen: Component list with version numbers
(The version numbers in this
figure are partly outdated)

Press
ENTER twice.

Fig. 2
Choosing DEBUG yields text output, among which a report of deleted data
structure elements when program execution is ended.


Fig. 3
HELP and EXIT; Program run parameters

PIC03.GIF: this screen shows how to
get out and how to get help, the command line parameters and the actual
parameter settings. Parameter #4 is empty; the program will find the
required BGI (Borland Graphics Interface) driver in the current
directory.
Press
ENTER once.

Fig. 4
Input file as seen by the parser

PIC04.GIF: this screen shows the input file as seen by the parser.
Clauses specifying camera positions, vertex coordinates and other items
to specify the 4D scenery and its projections onto the 2D computer
screen are in white; comments are in blue. In this example the input
AS4 file is small enough to fit onto a single screen.
Press
ENTER for a last time.

Fig. 5
Initial picture: two identical perpsective views of 4D scenery
defined by input file

The computer switches to graphics mode. One gets two identical
perspective views of the 4D scenery defined in the input file. These
pictures look rather weird. This is typical for perspective projections
of 4D wireframe figures. (PIC05.GIF above)
The file COX2.AS4 (invoked by
AA2.BAT) specifies a scenery consisting of the 24 vertices of a
24cell, a Petrie dodecagon in yellow and a Petrie dodecagram in red.
The file COX1.AS4 (invoked by AA1.BAT) shows the dodecagon of
COX2.AS4 and its complementary Petrie dodecagon in green.
The file COX3.AS4 (invoked by AA3.BAT) shows the dodecagons of
COX1.AS4 together with their inscribed dodecagrams.
AB4SVDV is a general program for visualizing 4D wireframe figures. The
program is controlled by keystroke commands rather than by
windowsstyle menus.
Whereas AB4SVDV is a general program, it is provided with predefined
settings aimed at the study of the 24cell and its dodecagonal aspects.
These presets are put into action by pressing the "a"
key followed by "1"
... "7".
A second collection of presets is given by "b" followed
by "1"
... "9".
Press
the two keys "a 1" to obtain the two
dodecagonal aspects of the 24cell, and "a 2"
to set the angular velocity bivector so as to produce Coxeter's double
rotation (PIC06.GIF below).

Fig. 6
Dodecagonal aspects of the 24cell

Press
the spacebar to enter the Running state. The
figure starts rotating in steps of 0.10 time units. The lefthand panel
shows the slow simple component rotation; the righthand panel shows
the fast simple component rotation.
The 24cell was aligned in such a way that the slow simple rotation
takes place in the XY plane (lefthand window) and the fast one in the
UZ plane (righthand window).
ALL PRESETS OF GROUP A IN
AB4SVDV SPECIFIC FOR THE 24CELL
a 1

Sets orientation such that the
planes of the slow and fast simple component rotations are the XY plane
and the UZ plane, respectively. Sets time to zero and time step to 0.10.

a 2

Sets
angular velocity to (ω_{ux },
ω_{uy }, ω_{uz },
ω_{yz }, ω_{zx },
ω_{xy}) =
60 ( −√(1/3), −√(1/3), 3 − √(1/3);
√(1/3), √(1/3), 3/2 + √(1/3) )
degrees per time unit

a 3

Lisoclinic component (speed 90
deg/[t], period 4 [t])

a 4

Risoclinic component (speed 60
deg/[t], period 6 [t])

a 5

Fast
simple component (speed 150 deg/[t], period 2.4 [t])

a 6

Slow
simple component (speed 30 deg/[t], period 12 [t])

a 7

Sets standard orientation
without affecting the angular velocity setting. Square aspects of the
24cell.

ALL PRESETS OF GROUP B IN AB4SVDV SPECIFIC FOR THE 24CELL
b 1

Resets orientation; sets angular
velocity to
ω_{ux} = ω_{yz} = √(1/2); ω_{uy}
= ω_{zx} = √(1/2); ω_{uz} = ω_{xy}
= 0 degrees per time unit
(leftisoclinic rotation with period 360)

b 2

Resets orientation; sets angular
velocity to
ω_{ux} = ω_{yz} = √(1/3); ω_{uy}
= ω_{zx} = √(1/3); ω_{uz} = ω_{xy}
= √(1/3) degrees per time unit
(leftisoclinic rotation with period 360)

b 3

Sets orientation for dodecagonal
aspect

b 3 b 4

Sets orientation for hexagonal
aspect

b 5

Modified Coxeter's Double
Rotation (double rotation with a period 360)
Presets a6 and a7 refer to the decomposition into isoclinic components.
Presets a8 and a9 refer to the decomposition into simple rotations.

b 6

Lisoclinic component (period
180)

b 7

Risoclinic component (period
120)

b 8

Fast simple component (period 72)

b 9

Slow simple component (period
360)

The spacebar
switches between Running and Stopped.
In Stopped state one performs single steps forwards
by the "/"
keystroke and backwards by the "\"
keystroke.
Vertex numbers are made visible by the keystroke command CtrlV.
By repeatedly pressing CtrlV the vertex numbers are shown in turn in
size 1, size 2, size 3, and made to disappear.
With AltD
a data panel is turned on showing among other things the program's time
and time step, and the Parallel/Perspective states of the 4Dto3D and
3Dto2D projections (PIC07.GIF below).

Fig. 7
Time 0.00: initial situation
Data panel superimposed on picture
The two sets of Euler angles refer to Euler angle representations of
the
left and right quaternions of the 4D orientation.
The exact value of RTheta is atan (√(1/2)) + 90 degrees.

In Stopped state one can singlestep through the motion, and see that
only 0.20 time units are needed to make a 30 degrees rotation of the
righthand figure, whereas the lefthand figure takes 1.00 time units
for a 30 degrees rotation. (PIC08.GIF and PIC09.GIF below)

Fig. 8
At time 0.20
The star dodecagons in
the righthand window have advanced
30 degrees.


Fig. 9
At time 1.00
The dodecagons in the lefthand window have advanced 30 degrees.
The star dodecagons in the righthand window have advanced
5 x 30 degrees.

With parallel projections one sees nothing of the fast rotation in the
lefthand window and of the slow rotation in the righthand window.
By repeatedly
pressing AltP one cycles through the four
projection modes 4DPerspective/3DParallel,
4DPerspective/3DPerspective, 4DParallel/3DPerspective and back to
4DParallel/3DParallel.
If one or both of the projections are
perspective then the fast rotation component will manifest itself in
the lefthand picture, and the slow component will show up in the
righthand picture (PIC10.GIF below).

Fig. 10
Projection mode: 3D parallel and 4D perspective

If one retains the afterimages of the motion (AltW
to switch ON/OFF afterimage retention and Del key
(*) to refresh, thus getting
rid of unwanted afterimages) then the lefthand picture will
clearly show the fivefold symmetry caused by the 5:1 ratio of the
rotation speeds (PIC11.GIF below).
(*) Under some
systems the Del key in the numeric block (with Num Lock OFF) does not
work. The Delete key in the sixkeyspad above the four arrow keys does
work.

Fig.11
Slow rotation in the XY plane; fast rotation in the UZ plane
Lefthand window: picture of slow rotation perturbed by fast rotation
Righthand window: picture of fast rotation perturbed by slow rotation

With perspective projections the righthand picture will continue to
rotate fast, but in the meantime it will change slowly its shape.
With each five fast rotations the change pattern returns.
Remark: The solid colors in Fig.11 will appear by decreasing the time
step from 0.10 to 0.09 units by pressing F1 once, or
by.increasing to 0.11 units by pressing F2 once.
Press "e"
in Stopped state to end program execution (PIC12.GIF below).

Fig. 12
After end of program run

If the program is still in Running state then press the spacebar
to get in the Stopped state.
Switch between Graphics and Text modes by pressing ENTER.
In Text mode the blue status bar at the bottom shows among other things
the Stopped / Running state. In Graphics mode no status bar is
presented to keep the picture clean.
(*) COPYRIGHT NOTICE
AB4SVDV and the associated AS4 and HLP files are in the public domain.
The BGI files are in part property of Borland (Copyright 1987, 1992)
and in part of Jordan Hargraphix (Copyright 1991, 1994).
The Borland BGIs may be used free of royalties. The Hargraphix BGIs are
shareware.
LITERATURE
[1]

H.S.M. Coxeter:Two aspects of
the regular 24cell in four dimensions.
Paper nr 3 in F. Arthur Sherk  Peter McMullen  Anthony C.
Thompson  Asia Ivic Weiss: Kaleidoscopes  Selected Writings of H.S.M.
Coxeter.
John Wiley, 1995, ISBN 0471010030

