Updated: Oct 25
Apollinaria woke up early in the morning, it was cool, and she decided to lie in bed longer. Her thoughts and gaze moved from one object in the room to another. She seemed to be half asleep, and in this state unexpected thoughts always came to her. Now she was looking at the closet, and saw that the closet had a left and a right door. It had a front side and a back side. The closet also had an inner side that contained shelves with things and an outer side, part of which was wood and part of which was glass.
It could be told that the closet was symmetrical from several points of view. First, its left door was symmetrical to the right door. The uppermost part was symmetrical to the lowermost part. The side parts were also symmetrical. Even the angles could be called symmetrical. But the shelves in the closet could not be called symmetrical. Why? Because there were things at the top of each shelf, and there were no things at the opposite side of each shelf. Things were stacked in one direction, from bottom to top. Things weren't symmetrical.
“What does symmetry depend on?” thought Apollinaria, “in symmetry there is a center, it can look like a point, or it can look like a line that separates left and right, for example. If left and right reflect each other as in a mirror, then this is symmetry. But if the clothes just lie in a row, even if there is a left and right side of the shelf, there is no more symmetry. As there is no mirror effect, and there is no center. To have symmetrical shelves in the closet, I will have to put two of the same things on the left and on the right. Symmetry doubles the space!"
“The position of each object can only be stated in relation to another object,” the teacher explained in class. "For example, the pen on your desk may be to the right of the notebook. And the textbook may be to the left of the notebook. But if your friend turns to face you and looks at your desk, he will say that your pen is on the left and the textbook is lying on the right of the notebook. If you turned to face each other, then you are in symmetry to each other. And the notebook in the center of the desk between you will be the center of symmetry".
"You can take each other's hands. Your right hand will connect with your friend's left hand, and vice versa. And if you walk around in a circle and change places, and the objects on the desk remain lying without moving, then your position in relation to the objects will change. What you saw on the right, you will see on the left, and vice versa".
"Therefore, when you speak about the position of any object in space, you need to find the fixed parts of that space and say on which side of these parts the object is located. We cannot tell that some part of our room is left. It depends on the position of the one who is telling. But we can say that this is the eastern wall, and that wall is the western one, because the parts of the world can be called the fixed parts of our world, and that’s what they are called.”
“So if I stand in the center of the room, I can turn in all directions, I will be the center of symmetry for the east, west, north and south?” - thought Apollinaria. "They are opposites. But people who constantly move between north, east, west and south break the symmetry if they do not turn to face each other. People also do not move one by one in line or in parallel, they move in different directions, rather chaotically."