ABCONWAY  -  DOWNLOAD THE SOFTWARE  -  INSTALLATION AND GUIDED TOUR  -  A 24D JOKE   OCTOBER 23RD 2015
     


The ABCONWAY program is an interactive program for the exploration and visualisation of Conway's group Co0 (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. Co0 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 24-dimensional lattice with some special properties. The group Co0 is a finite subgroup of the 24D rotation group with equally remarkable properties.
The matrix elements of any 24D rotation in Co0 are integral multiples of 1/8. (By the way, the values 7/8 and −7/8 do not appear.) Therefore an element of Co0 is readily represented as a square array of 24x24 tiles colour-coded 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 Co0 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 MS-WINDOWS AS WELL AS UNDER MS-DOS.


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 MS-DOS way: open an MS-DOS box if you run an MS-Windows system, make directory D current and run the MS-DOS batch file ABCO.BAT.

(B) The MS-Windows way: open Explorer, open directory D and double-click 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  v2.09-2009". This is the program itself.


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.

ABCO-01.GIF: Opening screen


Figure 1
Opening  screen: Component list with version numbers


Press ENTER twice.

ABCO-02.GIF: Debug? Yes or No?


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

 

ABCO-03.GIF: Help; Exit; Program run parameters


Figure 3
HELP and EXIT; Program run parameters

ABCO-03.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 ARTIBODIES-based 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.

ABCO-04.GIF: Input file as seen by parser


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


ABCO-04.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.

ABCO-05.GIF: Initial picture


Figure 5
Initial picture: 24th-order identity matrix


The computer switches to graphics mode and shows the 24th-order identity matrix. (ABCO-05.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 Co0 according to [1] and the mistaken sixth generator mentioned in
[7]

ABCO-Generator-1.gif


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



 

ABCO-Generator-2.gif


Figure G2



 

ABCO-Generator-3.gif


Figure G3



 

ABCO-Generator-4.gif


Figure G4



 

ABCO-Generator-5.gif


Figure G5



 

ABCO-Generator-6.gif


Figure G6 - A correct sixth generator



 

ABCO-Generator-6-Mistaken.gif


Figure G6M - Mistaken sixth generator



NUMERICAL SHAPES OF CONWAY MATRICES

 
All entries of the matrices of all elements of the Conway group Co0 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)  (81 . 023) in all 24 columns, meaning one +1 or −1 and 23 zeroes in units of 1/8;

(B)  (44 . 020) in all 24 columns;

(C)  (41 . 212 . 011) in all 24 columns;

(D)  (42 . 28 . 014) in 16 columns and (216 . 08)in 8 columns;

(E)  (61 . 27 . 016) in 8 columns and (216 . 08) in 16 columns;

(F)  (51 . 32 . 121) in 3 columns and (35 . 119) 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 Not Found - 404.png 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 (31 . 123); 98304 = 212.24 points;

Lines 98305 through 99408: shape (42 . 022); 1104 = 22.276 points;
Lines 99409 through 196560: shape (28 . 016); 97152 = 27.759 points;
Altogether 196560 = 24.33.5.7.13 points.

The author of this file is Warren D. Smith, currently (i.e. December 2014) on Occam's Boxcutter, on blogspot-dot-com and on linkedin-dot-com,
previously (as of AD 2008) with Temple University, Mathematics Department ).
Warren D. Smith is a co-founder 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 Turbo-PASCAL 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 Turbo-PASCAL 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 = 236 volume units.


CONJUGACY CLASSES OF THE CONWAY GROUP Co1 AND ORDERS OF ELEMENTS OF THE GROUP Co0

The group Co0 has a centre Z2 , consisting of the identity matrix I and the central inversion matrix −I.
It is an almost-simple group in the sense that the factor group of Z2 in Co0 is a simple group. This factor group is Conway's sporadic simple group Co1. Its elements are pairs of opposite elements of Co0, sometimes denoted as diameters.

Co1 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.

Co0 consists of 167 conjugacy classes; under the homomorphism of Co0 onto Co1 some of these classes are mapped in pairs onto single classes in Co1, while the remaining classes of Co0 are mapped one-to-one onto the remaining classes of Co1. For instance, the conjugacy classes {I} and {−I} in Co0 are both mapped onto the identity conjugacy class of Co1. In general, whenever the two elements of a diameter lie in different conjugacy classes of Co0, these two classes are mapped onto a single conjugacy class of Co1.
As a consequence of this situation there exist elements of Co0 of orders unique to Co0.
Exploration of Co0 by means of the ABCONWAY program yields orders 46, 52, 56, 66, 70, 78 and 84. All cyclic subgroups of Co0 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 Co0.

A speculation: does Co0 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 Co0 then Co0 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 196883-dimensional vector space. This does not preclude the existence of 196883-dimensional rotations that perform central inversions in 24-dimensional subspaces.

Literature specifically on Conway's 24D rotation group Co0: [5]


LITERATURE
   

[1]

J.H.Conway: A group of order 8,315,553,613,086,720,000. - Bull. London Math. Soc. - 1 (1969) - 79-88 
 

[2]

J.H.Conway: A perfect group of order 8315553613086720000 and the sporadic simple groups.  
Proc. Nat. Acad. Sci. USA. - 1968 October; 61(2) - 398-400    
 

[3]

J.H.Conway and N.J.A.Sloane: Sphere Packings, Lattices And Groups. (affectionately known as SPLAG)
Third edition - Springer, 1999, ISBN 0-387-98585-9

[4]

J.H.Conway et al.: ATLAS of Finite Groups. Oxford University Press, 1985,
ISBN 0198531990, ISBN-13: 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 Co1

[7]

Thomas M. Thompson: From error-correcting codes through sphere packings to simple groups.
Washington: Mathematical Association of America, 1983,  ISBN 0-88385-023-0;
Uncorrected and unaltered 4th printing October 2004,  ISBN 0-88385-037-0
(Carus mathematical monographs volume 21)
Page 152: mistaken sixth generator of Conway's group Co0:
Changing the signs of the 16 matrix elements aij with i, j = 1, 4, 16, 24 yields a correct sixth generator.
 

 

SEE ALSO

Conway groups

http://en.wikipedia.org/wiki/Conway_group

http://mathworld.wolfram.com/ConwayGroups.html

Leech lattice

http://en.wikipedia.org/wiki/Leech_lattice

http://mathworld.wolfram.com/LeechLattice.html

Kissing number

http://en.wikipedia.org/wiki/Kissing_number

http://mathworld.wolfram.com/KissingNumber.html

Golay code

http://en.wikipedia.org/wiki/Binary_Golay_code

http://mathworld.wolfram.com/GolayCode.html

Mathieu groups

http://en.wikipedia.org/wiki/Mathieu_group

http://mathworld.wolfram.com/MathieuGroups.html

Steiner systems

http://en.wikipedia.org/wiki/Steiner_system

http://mathworld.wolfram.com/SteinerSystem.html
http://mathworld.wolfram.com/SteinerTripleSystem.html
http://mathworld.wolfram.com/SteinerQuadrupleSystem.html

Sporadic groups

http://en.wikipedia.org/wiki/Sporadic_group

http://mathworld.wolfram.com/SporadicGroup.html