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Projects archive / Animating symmetries of a cube

Code:cubesymWong2001
Title:Animating symmetries of a cube
Authors:Allan Wong, Meurig Beynon
Date:Jan-2001
Type:model
Funding:PhD - PhD thesis
Short description:S4 as a Cayley diagram and as symmetries of a cube
EM Technologies:tkeden, %scout, %sasami, %arca
Keywords:education, geometry
empublic references:
EMpress references:
Web site references:
Previous locations:
Tour:README.txt

This model is based on one of the original demonstration files developed to illustrate the use of the definitive notation ARCA. The original ARCA script to define a Cayley diagram for S4 was developed by Beynon, and has been translated into an eden file and displayed using the ARCA translator developed by Stuart Bird. (Note that the location of vertices of the diagram is determined by a projecting a cube onto the plane.) The vertices of the Cayley diagram correspond to group elements that can be variously interpreted as abstract symmetry operations, permutations and transformation matrices. The red, green and blue edges represent generators x, y and z respectively that generate S4 subject to the relations xx = yy = zz = xzxz = xyxyxy = yzyzyz = 1. The other relations between x, y and z correspond to cycles in the Cayley diagram. The model also includes a Sasami model of a cube and a button interface that was developed by Allan Wong.

More about the Cayley diagram

The Cayley diagram is associated with the symmetric group on 4 symbols, as generated by the three transpositions (12), (23) and (34). The presentation associated with these generators is defined by the following properties: all the transpositions have order 2, the (12)(34) is also of order 2, and (12)(23) and (23)(34) are of order 3. This group is also the group of symmetry operations on the cube, and the transpositions can be interpreted as rotations about diametrically opposite midpoints of the edges 12, 23 and 34 respectively of the cube. (There are of course several other triples of edges that could be chosen to establish this representation.) In the model, the indexing of the vertices of the cube is such that the midpoints actually used are (12), (15) and(56) respectively.

In the Cayley diagram, the blue, green and red edges respectively correspond to the transpositions (12), (23) and (34) and the rotations about the axes through the midpoints of (12), (15) and (56).

To adjust the size of the diagram, enter an integer value (e.g. between 5 and 15) for variable 'graphsize' in eden (or scout).

To operate the model in standard demo mode, press the reset button to relocate the cube in its 'initial' position, then inspect the effect of traversal of the Cayley diagram and the associated reconfiguration of the cube as the blue, green and red buttons are pressed.

cubesymWong2001 is available locally within DCS at /dcs/emp/empublic/projects/cubesymWong2001
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README.txt 3792 01:22 PM Sep 03 2002 01:19 PM Jul 27 2017
README.xml 2944 08:43 PM Nov 24 2010 01:19 PM Jul 27 2017
Run.e 173 07:34 PM Aug 30 2002 01:19 PM Jul 27 2017
S4screenshot.gif 8747 08:17 PM Aug 26 2002 01:19 PM Jul 27 2017
addwin2.s 2497 05:48 PM Aug 26 2002 01:19 PM Jul 27 2017
allansrubiks.s 5116 04:33 PM Jul 24 2002 01:19 PM Jul 27 2017
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colseq.e 330 04:33 PM Jul 24 2002 01:19 PM Jul 27 2017
ex_13 2768 09:56 PM Sep 26 2000 01:19 PM Jul 27 2017
front.png 3324 04:33 PM Jul 24 2002 01:19 PM Jul 27 2017
left.png 3724 04:33 PM Jul 24 2002 01:19 PM Jul 27 2017
link.e 1010 04:33 PM Jul 24 2002 01:19 PM Jul 27 2017
matmult.e 1890 04:33 PM Jul 24 2002 01:19 PM Jul 27 2017
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top.png 3448 04:33 PM Jul 24 2002 01:19 PM Jul 27 2017
tranex13.e 14398 04:33 PM Jul 24 2002 01:19 PM Jul 27 2017

empublic system initially created by Ashley Ward with assistance from Chris Roe and Meurig Beynon.