MATHEMATICAL BACKGROUND
In [1] Coxeter
considers the 24-cell 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)
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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 24-cell, a Petrie dodecagon.
The 24 vertices together describe two Petrie
dodecagons in this way. When the 24-cell
is projected along one of the plane-axes
of this rotation onto the other plane-axis,
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: (ωux , ωuy , ωuz , ωyz ,
ωzx , ωxy) =
60 ( -sqrt(1/3), -sqrt(1/3),
3 - sqrt(1/3); sqrt(1/3),
sqrt(1/3), 3/2 + sqrt(1/3) ) =
( -34.641016, -34.641016, 55.358984; 34.641016, 34.641016, 124.641016 ) degrees per time unit.
These components are
referred to the body-fixed coordinate system.
This rotational motion
has a period of 12 time units; each lap
of one time unit will rotate the 24-cell
to its next congruent orientation.
This rotational motion
is decomposed into the simple rotational
motions B and C given by
B: (ωux , ωuy , ωuz , ωyz , ωzx , ωxy) =
75 ( -sqrt(1/3), -sqrt(1/3), 1 - sqrt(1/3); sqrt(1/3), sqrt(1/3), 1 + sqrt(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: (ωux , ωuy , ωuz , ωyz , ωzx , ωxy) =
15 ( sqrt(1/3), sqrt(1/3), 1 + sqrt(1/3); -sqrt(1/3),
-sqrt(1/3), 1 - sqrt(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 24-cell is such that the space-fixed
XY plane is the axis-plane of B, the fast simple rotation, and the
space-fixed UZ plane is the axis-plane of
C, the slow simple rotation.
INSTALLATION AND
GUIDED TOUR
THIS
SOFTWARE WORKS UNDER MS-WINDOWS AS WELL
AS UNDER MS-DOS.
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 MS-DOS way: open an MS-DOS
box if you run an MS-Windows system, make directory D current and run
the batch command...
AA1
for the 24-cell with a pair of Petrie dodecagons,
AA2 for the 24-cell with a dodecagon and a dodecagram (star dodecagon),
AA3 for the 24-cell 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 MS-Windows way: open Explorer, open
directory D and double-click 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 v2.07-2006".
This is the program itself. (PIC-01.GIF
below).
Remark: When running
an MS-DOS program under MS-Windows one can
switch between Windowed and Full-screen
modes by the keystroke command Alt-ENTER.
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Fig. 1
Opening
screen: Component list
with version numbers
(The version numbers in this figure are partly outdated)
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Press ENTER twice.
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Fig. 2
Choosing
DEBUG yields text output, among
which a report of deleted data
structure elements when program
execution is ended.
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Fig. 3
HELP
and EXIT; Program run parameters
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PIC-03.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.
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Fig.
4
Input file as seen by the
parser
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PIC-04.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.
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Fig.
5
Initial picture: two identical
perpsective views of 4D scenery
defined by input file
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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. (PIC-05.GIF above)
The file COX-2.AS4
(invoked by AA2.BAT) specifies a scenery consisting of the 24
vertices of a 24-cell, a Petrie dodecagon
in yellow and a Petrie dodecagram in red.
The file COX-1.AS4 (invoked by AA1.BAT) shows the dodecagon of COX-2.AS4 and its complementary Petrie dodecagon in green.
The file COX-3.AS4 (invoked by AA3.BAT) shows the dodecagons of COX-1.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 windows-style menus.
Whereas
AB4SVDV is a general program, it is provided
with predefined settings aimed at the study
of the 24-cell 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 24-cell, and "a 2" to set the angular velocity bivector so as to produce Coxeter's double rotation (PIC-06.GIF below).
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Fig.
6
Dodecagonal aspects of
the 24-cell
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Press
the spacebar to enter the Running state. The figure starts rotating in steps of 0.10 time units. The left-hand panel shows the slow simple component rotation; the right-hand panel shows the fast simple component rotation.
The 24-cell was aligned in such
a way that the slow simple rotation takes
place in the XY plane (left-hand window)
and the fast one in the UZ plane (right-hand
window).
ALL
PRESETS OF GROUP A IN AB4SVDV SPECIFIC FOR THE 24-CELL
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a 1
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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.
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a 2
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Sets angular velocity
to (ωux , ωuy , ωuz , ωyz , ωzx , ωxy) =
60 ( -sqrt(1/3), -sqrt(1/3), 3 - sqrt(1/3); sqrt(1/3), sqrt(1/3), 3/2 + sqrt(1/3) ) degrees per time unit
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a 3
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L-isoclinic
component (speed 90 deg/[t], period 4 [t])
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a 4
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R-isoclinic component (speed 60 deg/[t], period 6 [t])
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a 5
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Fast simple component (speed 150 deg/[t], period 2.4 [t])
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a 6
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Slow simple component (speed 30 deg/[t], period 12 [t])
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a 7
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Sets standard orientation without affecting the angular velocity setting. Square aspects of the 24-cell.
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ALL PRESETS OF GROUP B IN AB4SVDV SPECIFIC FOR THE 24-CELL
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b 1
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Resets
orientation; sets angular velocity
to
ωux = ωyz = sqrt(1/2); ωuy = ωzx =
sqrt(1/2); ωuz = ωxy = 0 degrees per time unit
(left-isoclinic
rotation with period 360)
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b 2
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Resets
orientation; sets angular velocity
to
ωux = ωyz = sqrt(1/3); ωuy = ωzx =
sqrt(1/3); ωuz = ωxy = sqrt(1/3) degrees per time unit
(left-isoclinic
rotation with period 360)
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b 3
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Sets orientation
for dodecagonal aspect
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b 3 b 4
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Sets orientation
for hexagonal aspect
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b 5
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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.
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b 6
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L-isoclinic
component (period 180)
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b 7
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R-isoclinic
component (period 120)
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b 8
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Fast simple
component (period 72)
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b 9
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Slow simple
component (period 360)
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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 Ctrl-V.
By repeatedly pressing Ctrl-V the vertex
numbers are shown in turn in size 1, size
2, size 3, and made to disappear.
With
Alt-D a data panel is turned on showing
among other things the program's time and
time step, and the Parallel/Perspective
states of the 4D-to-3D and 3D-to-2D projections
(PIC-07.GIF below).
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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 (sqrt (1/2)) + 90 degrees.
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In Stopped state one
can single-step through the motion, and
see that only 0.20 time units are needed to
make a 30 degrees rotation of the right-hand
figure, whereas the left-hand figure takes
1.00 time units for a 30 degrees rotation.
(PIC-08.GIF and PIC-09.GIF below)
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Fig.
8
At time 0.20
The
star dodecagons in the right-hand
window have advanced
30 degrees.
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Fig.
9
At time 1.00
The dodecagons
in the left-hand window have
advanced 30 degrees.
The
star dodecagons in the right-hand
window
have advanced
5 x 30 degrees.
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With
parallel projections one sees nothing of
the fast rotation in the left-hand window
and of the slow rotation in the right-hand
window.
By repeatedly pressing
Alt-P one cycles through the four projection
modes 4D-Perspective/3D-Parallel, 4D-Perspective/3D-Perspective,
4D-Parallel/3D-Perspective and back to 4D-Parallel/3D-Parallel.
If one or both of
the projections are perspective then the
fast rotation component will manifest itself
in the left-hand picture, and the slow component
will show up in the right-hand picture (PIC-10.GIF
below).
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Fig.
10
Projection mode: 3D parallel
and 4D perspective
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If
one retains the afterimages of the motion
(Alt-W
to switch ON/OFF afterimage retention and
Del
key (*)
to refresh,
thus getting rid of unwanted afterimages)
then the left-hand picture will clearly
show the five-fold symmetry caused by the
5:1 ratio of the rotation speeds (PIC-11.GIF
below).
(*) Under some
systems the Del key in the numeric block
(with Num Lock OFF) does not work. The Delete
key in the six-keys-pad above the four arrow keys
does work.
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Fig.11
Slow
rotation in the XY plane; fast
rotation in the UZ plane
Left-hand
window: picture of slow rotation
perturbed by fast rotation
Right-hand
window: picture of fast rotation
perturbed by slow rotation
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With perspective projections
the right-hand 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 (PIC-12.GIF below).
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Fig.
12
After end of program run
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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
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[1]
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H.S.M. Coxeter:Two aspects of the
regular 24-cell 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 0-471-01003-0
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