Last Update 11/ 09/ 2002
in English/ in Esperanto/ in French/ in Portuguese
The figure shows a set of dodecahedron edges represented mainly in a dark colour in perspective projection with convenient distortion, constituting an instructive application game of elements of symmetry. It works by clicking in the right sequence, the buttons with the respective element of symmetry in such orientation to turn the colour of the edges blue.
The application presents randomly a blue edge as starting point. Seven buttons at the right side of the figure may be used to operate the centre of inversion, 2 planes of symmetry and 4 axis of rotation oriented as shown in Table 1.
Table 1. The seven buttons of the interactive picture.
| Buttons | I | m | m | 2 | 2 | 3 | 3 |
| Orientation | at origin | contains
2 e 3 |
perpendicular
to m |
intersection
of m and m |
perpendicular
to 2 |
contained
in m |
contains
(100; 150) (243; 150) |
In Table 1, symbol I means centre of symmetry, m (any colour) indicates reflection plane and 3 and 2 are three fold and binary rotation axes respectively.
A scheme with the orientation of the axes and planes cited in table 1 is shown after a click on the grey background at the right side of the figure on the application, after the column of buttons. Symmetry elements coloured white in the scheme are coloured black in the buttons.
Each symmetry operation acts over all blue edges present at the time.
Answer with this game or with the aid of a three-dimensional model:
1) What is the minimum number of clicks on the buttons to solve the task?
2) The buttons that solved the task are always the same?
3) Commuting the symmetry operations gives always the same results?
Try to obtain:
1) Only 18 blue edges connected in a cycle.
2) Two independent cycles of 9 blue edges each.
The time spent can be checked at the end of the game, when the application shows the initial date (related to GMT) at the top of the figure and the final date at the bottom.
Click in the dodecahedron to start another game.
Please send your comments.
Table of subjects.