First edition 04/ 3/ 2002, last update 07/ 01/ 2014
in English/ in Esperanto/ in French/ in Portuguese
The figure shows a set of cube 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 yellow. The 09/ 01/ 2014 edition has more contrast.
The application presents randomly a yellow edge as starting point. Seven buttons at the right side of the figure may be used to operate the center 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 | 4 |
| Orientation | at origin | contains y,
intersects x at 45o |
perpendicular
to y, through 0,5 y |
intersection
of m and m |
contains
(95; 180) (204; 120) |
contained
in m |
contains
(100; 150) (243; 150) |
In table 1, symbol I means center of symmetry, m (any colour) indicates reflection plane, 4, 3 and 2 are four and three fold and binary rotation axes respectively.
A scheme with the orientation of axes and planes cited in table 1 is shown by clicking on the gray background at the right side of the figure on the application. Symmetry elements colored white in the scheme are colored black in the buttons.
Each symmetry operation acts over all yellow edges present at the moment and updates the game time in seconds displayed in blue on the lower left corner.
When the game is done the initial date appears at the top of the figure and the final date at the bottom. If the game is completed in up to 3 seconds 20 hexagonal colored stars, hexagrams, will be shown and the colours will change after any click on any button. A click on the center of the cube will start a new game in zero 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) What are the coordinates of the cube vertices intercepted by the triad axis in this game?
Try to obtain with this game yellow cycles with only two edges per vertex:
1) Only 4 yellow edges connected in a cycle.
2) Cycles with 6 and 8 edges.
How many different cycles of each kind were obtained?
Does the cube have a Hamiltonian path?
Please send your comments.
Table of subjects.