say I have a dice, with following numbering arrangements:
top front right bottom back left
1 2 3 6 5 4the dice can be rolled freely of course
I think it is enough to give the setout of the dice by only give the top number and front number say a and b respectively ( assume they are valid status )
how do I deduce other faces from there: say if telling the top and front is 5 and 1 respetively
the setting should be 5 1 3 2 6 4
I know this might be related with group theory about the rotation groups just don't know how to figure this out
$\endgroup$ 12 Answers
$\begingroup$The full group of rotations has $24$ elements and is in fact isomorphic to the symmetric group $S_4$ (this can be seen from the way it acts on the four space diagonals). The group acts transitively on the faces, so each of the $6$ faces has the possibilty to become top face. The fix group of a single face is cyclic of order $4$ (these are the rotations around the axis orthogonal to the fixed face); it operates transitively on the four adjacent faces (but also keeps the diametrically opposite face fixed). Thus all four adjacent faaces can indeed be brought to front. But this also implies that specifying the top face and the front face already determines completely which of the $6\cdot 4=24$ orientations the die has.
$\endgroup$ $\begingroup$Yes, specifying the top and front faces of a die are enough to define the position. You can determine the other faces by inspection. Because opposite faces add to $7$, you can find the bottom and back faces easily, in your case $2$ and $6$. It isn't easy to describe how to figure out which way the other two faces are positioned. There are two chiralities or handednesses for dice. Yours is counterclockwise. The easiest way to get the sides is to see that you roll the die so the back face becomes top, so the right and left faces stay where they are. $4$ is on the left and $3$ on the right.
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