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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 |
| Download cubesymWong2001.tar.gz Uncompressed, cubesymWong2001 is 63.6 Kbytes (in 17 files). Your download will be smaller as it will be compressed. Help on extracting the files from the download. |
README.txt |
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README.xml |
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empublic system initially created by Ashley Ward with assistance from Chris Roe and Meurig Beynon.
