There are three ways for flipping them. That is, you can rotate the bed about the three axes (x, y and z).
First mark 1,2,3 and 4 in four corners of the bed. So that we can find out what happens after each flip.
1. H- horizontal flip: Rotate the bed about the lengthwise axis as shown in figure. The numbers shift to other side. Also, the mattress become upside down.
2. V- vertical flip: Spin the bed about the breadthwise axis. The numbers shift from top to bottom and bottom to top. Also bed turns upside down.
3. R- rotation: Just rotate bed about the vertical axis (not vertical flip). Top surface of the bed stays as it is. But the numbers move from top to bottom and vice versa.
4. I -Do not touch the bed. Bed stays as it is.
The elements H, V, R and I-- the binary operations--'rotations' form a 'group; They follow certain rules or axioms.
If you do horizontal flip two times, the bed goes back to initial state I. That is written as HH=I, similarly it can be proved.
RR=I; VV=I.
V=HR.
also, HR=RH, the operation obeys commutative law.
V=HR is the solution to our problem. That is, doing horizontal flip and rotation consecutively is equivalent to one vertical flip. So, H and R is enough. We can ignore V which is difficult to do. (Lifting the bed vertically up to the ceiling). Hence, for the mattress to wore out evenly, do H and R alternately and periodically. To remember it, the mattress companies have coined a sentence.
"Spin in the spring; flip in the fall'
Finally, we got a solution from group theory in mathematics. Also, the first algorithmic solution for Rubik cube was found using group theory.
Now, we are ready for the definition of group theory.
"Set of elements and a binary operations satisfying certain axioms".
SOME POINTS
1. Dice is also example for group theory.
2. Turning the two palms up and down is also a good example.
3. Chemical molecular rotations and vibrations are studied using group theory.
4. Symmetry is essence of group theory.