The
ABCONWAY program is an interactive program for the exploration and
visualisation of Conway's group Co_{0}
(also denoted by .0 or mistakenly, by .O), which is the group of
congruence transformations of the Leech lattice which leave a specific
lattice point invariant. Co_{0}
is commonly denoted as the automorphism group
of the Leech lattice. Strictly speaking, one should say: the group of
automorphisms which leave a specific point invariant.
See [1], [2].
Both papers contain references to Leech's original papers.
The Leech lattice is a 24dimensional lattice with some special
properties. The group Co_{0}
is a finite subgroup of the 24D rotation group with equally remarkable
properties.
The matrix elements of any 24D rotation in Co_{0}
are integral multiples of 1/8. (By the way, the values 7/8 and −7/8 do
not appear.) Therefore an element of Co_{0}
is readily represented as a square
array of 24x24 tiles colourcoded according to the values of the
corresponding matrix elements.
Along with this picture a data panel is presented showing the order and
the trace (character) of the 24D rotation together with the rotation
angles in its twelve mutually perpendicular rotation planes.
MATHEMATICAL BACKGROUND
Following [1], the Leech lattice and its
automorphism group Co_{0}
are conveniently descrbed by means of the projective line
PL(23) = {0, 1, 2,
..., 20, 21, 22, ∞}
over the field of 23
elements. In the ABCONWAY program PL(23) is mapped onto the
coordinate numbers
1, 2, 3, ..., 21, 22,
23, 24.
more
coming in due time...
INSTALLATION AND GUIDED TOUR
THIS SOFTWARE WORKS UNDER
MSWINDOWS AS WELL AS UNDER MSDOS.
The file ABCONWAY.ZIP contains the ABCONWAY program and its
accompanying BAT, BGI and HLP files.
Unpack ABCONWAY.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 MSDOS batch file ABCO.BAT.
(B) The MSWindows way: open
Explorer, open directory D and doubleclick on ABCO (ABCO.BAT).
The opening screen shows some general program settings, a copyright notice (*) and a
component list with version numbers.
The last entry reads "ABCONWAY v2.092009". This is the
program itself.
Remark: When running an MSDOS program under MSWindows one
can switch between Windowed and Fullscreen modes by the keystroke
command AltENTER.

Figure 1
Opening screen: Component list with version numbers

Press
ENTER twice.

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


Figure 3
HELP and EXIT; Program run parameters

ABCO03.GIF:
this screen shows how to get out and how to get help, the command line
parameters and the actual parameter settings.
Parameter #1 is empty; most ARTIBODIESbased applications read an input
file; ABCONWAY builds the necessary data structures internally.
Parameter #2 = 9 specifies VGA graphics mode. Parameter #3 = 2
specifies 640x480 pixels in 16 colors.
Parameter #4 is empty; the program will find the required BGI (Borland
Graphics Interface) driver in the current directory.
Press
ENTER once.

Figure 4
No input file; data structure for matrix is generated internally.

ABCO04.GIF:
Free and occupied RAM memory: unfortunately a little bit above the
magical number 196560, the hypersphere kissing number in 24 dimensions.
Press
ENTER for a last time.

Figure 5
Initial picture: 24thorder identity matrix

The computer switches to graphics mode and shows the 24thorder
identity matrix. (ABCO05.GIF above)
Use
the ENTER key to switch between graphic mode and text mode.
(more and better to come at this place)
(PROVISIONAL EDIT)
The six generators of Co_{0}
according to [1] and the
mistaken sixth generator mentioned in
[7]

Figure G1
The values of the nonzero
matrix elements are given in units of 1/8.
The zero elements are
represented by grey squares.


Figure G2


Figure G3


Figure G4


Figure G5


Figure G6  A correct sixth generator


Figure G6M  Mistaken sixth generator

NUMERICAL
SHAPES OF CONWAY MATRICES
All entries of the matrices of all elements of the Conway group Co_{0}
are integral multiples of 1/8. The values 7/8 and −7/8 do not appear as
entries of Conway matrices. The numerical shape
of a Conway matrix is a specification of the number of times each of
the possible matrix element values actually appears. When
distinguishing absolute values only, there
are only six numerical
shapes:
(A) (8^{1}
. 0^{23})
in all 24 columns, meaning one +1 or −1 and 23 zeroes in units of 1/8;
(B) (4^{4}
. 0^{20})
in all 24 columns;
(C) (4^{1}
. 2^{12}
. 0^{11})
in all 24 columns;
(D) (4^{2}
. 2^{8}
. 0^{14})
in 16 columns and (2^{16}
. 0^{8})in
8 columns;
(E) (6^{1}
. 2^{7}
. 0^{16})
in 8 columns and (2^{16}
. 0^{8})
in 16 columns;
(F) (5^{1}
. 3^{2}
. 1^{21})
in 3 columns and (3^{5}
. 1^{19})
in 21 columns.
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 in order to keep the picture clean.
(*) COPYRIGHT NOTICE
ABCONWAY and the accompanying 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). They may be used
free of royalties.
LISTING OF THE FIRST SHELL OF THE LEECH LATTICE; ACKNOWLEDGEMENT
Was available for download at http://www.research.att.com/~njas/packings/dim24/leech196560.gz
This file contains a solution of the
hypersphere kissing problem in 24D space.
Lines 1 through 98304: the points of numerical
shape (3^{1}
. 1^{23});
98304 = 2^{12}.24
points;
Lines 98305
through 99408: shape (4^{2}
. 0^{22});
1104 = 2^{2}.276
points;
Lines 99409
through 196560: shape (2^{8}
. 0^{16});
97152 = 2^{7}.759
points;
Altogether 196560 = 2^{4}.3^{3}.5.7.13
points.
The author of this file is Warren
D.
Smith, currently (i.e. December 2014) on Occam's
Boxcutter, on blogspotdotcom
and on linkedindotcom,
previously (as of AD 2008) with Temple
University, Mathematics
Department ).
Warren D. Smith is a cofounder of The Center for Election
Science; see also Who We
Are.
This original file is because of its layout
not very suitable for processing by TurboPASCAL programs.
A link to a more convenient layout is at
the Downloads page.
The ZIP file LLABC.ZIP contains the
sources and executables of three TurboPASCAL programs. Technical
details at LLABC.htm
The GZ file as well as the ZIP file are readily expanded into TXT files
consisting of 196560 lines.
Each line contains the coordinates of the centre of one of the 196560
hyperspheres of radius sqrt(8) in 24D space which touch without
overlapping each other a central hypersphere, also of radius sqrt(8),
These points together form the first shell of the Leech lattice around
the centre of the central sphere.
A radius of sqrt(8) instead of a unit
radius (*)
was chosen because with this radius the Leech lattice can be oriented
in such a way that all coordinates of the centres of all kissing
spheres are integers.
(*)
The Leech lattice as defined by John H. Conway's characterisation
theorem ([3] (SPLAG),
Ch. 12) has 1 lattice point per unit volume (is unimodular) and has a
minimum distance between lattice points of 2 length units. The Leech
lattice in Warren D. Smith's file is Conway's Leech lattice scaled up
by a factor sqrt(8) and so contains one lattice point per (sqrt(8))^{24}
= 2^{36}
volume units.
CONJUGACY CLASSES OF THE CONWAY GROUP Co_{1}
AND ORDERS OF ELEMENTS OF THE GROUP Co_{0}
The group Co_{0}
has a centre Z_{2}
, consisting of the identity matrix I and the
central inversion matrix −I.
It is an almostsimple group in the sense that the factor group of Z_{2}
in Co_{0}
is a simple group. This factor
group is Conway's sporadic simple group Co_{1}.
Its elements are pairs of opposite elements of Co_{0},
sometimes denoted as diameters.
Co_{1}
consists of 101 conjugacy classes, in the ATLAS ([4])
denoted by
1A; 2ABC; 3ABCD; 4A...F; 5ABC;
6A...I;
7AB; 8A...F; 9ABC; 10A...F;
11A;
12A...M; 13A; 14AB;
15A...E; 16AB; 18ABC; 20ABC;
21ABC; 22A;
23AB; 24A...F; 26A; 28AB;
30A...E;
33A; 35A; 36A;
39AB; 40A; 42A; 60A.
Co_{0}
consists of 167 conjugacy classes; under the homomorphism of Co_{0}
onto Co_{1}
some of these classes are mapped in pairs onto single classes in Co_{1},
while the remaining classes of Co_{0}
are mapped onetoone onto the remaining classes of Co_{1}.
For instance, the conjugacy classes {I} and {−I}
in Co_{0}
are both mapped onto the identity conjugacy class of Co_{1}.
In general, whenever the two elements of a diameter
lie in different conjugacy classes of Co_{0},
these two classes are mapped onto a single conjugacy class of Co_{1}.
As a consequence of this situation there exist elements of Co_{0}
of orders unique to Co_{0}.
Exploration of Co_{0}
by means of the ABCONWAY program yields orders 46, 52, 56, 66, 70, 78
and 84. All cyclic subgroups of Co_{0}
of these orders contain the central inversion −I.
This is of course no proof that these orders are the only ones that are
unique to Co_{0}.
A speculation: does Co_{0}
contain a cyclic subgroup of order 120?  the largest cyclic subgroups
of the Monster group have order 119; therefore, if the Monster group
contains subgroups isomorphic to Co_{0}
then Co_{0}
cannot contain cyclic subgroups of order 120. The Monster group can be
realised as a finite simple subgroup of the rotation group SO(196883),
and as such it does not
contain the central inversion of its underlying 196883dimensional
vector space. This does not preclude the existence of
196883dimensional rotations that perform central inversions in
24dimensional subspaces.
Literature specifically on Conway's 24D rotation group Co_{0}: [5]
LITERATURE
[1]

J.H.Conway: A group of order
8,315,553,613,086,720,000.  Bull. London
Math. Soc.  1
(1969)  7988

[2] 
J.H.Conway: A perfect group of order
8315553613086720000 and the sporadic simple groups.
Proc.
Nat. Acad. Sci. USA.  1968
October; 61(2)  398400

[3] 
J.H.Conway and N.J.A.Sloane: Sphere Packings, Lattices And Groups.
(affectionately known as SPLAG)
Third edition  Springer, 1999, ISBN 0387985859

[4] 
J.H.Conway et al.: ATLAS of Finite Groups.
Oxford University Press, 1985,
ISBN 0198531990, ISBN13: 9780198531999

[5] 
Patterson, Nicholas James: On
Conway’s group .O and some subgroups : A dissertation submitted for the
degree of Doctor of Philosophy at the University of Cambridge.
Cambridge, 1973
Specifics: see page NickPatterson.txt.
Also see http://genealogy.math.ndsu.nodak.edu/id.php?id=26484
Remark: Apparently either the author or the Cambridge University
Library staff typed ".O" instead of the correct denotation ".0". Misspelled
symbols and keywords used as search terms often yield additional
information on the subject in question. For instance, http://www.croatianhistory.net/etf/janko/index.html
will not be found by searching for "conway group .0".

[6] 
Online ATLAS
of Finite Group Representations Version 3: Entry
for Co_{1}

[7] 
Thomas M. Thompson: From errorcorrecting codes through
sphere packings to simple groups.
Washington: Mathematical Association of America, 1983, ISBN
0883850230;
Uncorrected and unaltered 4^{th}
printing October 2004, ISBN 0883850370
(Carus mathematical monographs volume 21)
Page 152: mistaken sixth generator
of Conway's group Co_{0}:
Changing the signs of the 16 matrix elements a_{ij}
with i, j = 1, 4, 16, 24 yields a correct
sixth generator.

