There are two hands, an hour hand and a minute hand. The set of all possible positions that the hour hand can take describes a circle. The same is true of the minute hand: its set of all possible positions describes a circle. Consequently, the set of all ordered pairs of possible positions (h,m), where h is the position of the hour hand and m is the position of the minute hand, is the Cartesian product of the two individual sets, and thus the Cartesian product of two circles. This is a two-dimensional torus.
Did you take into account that the positions of the two hands are not independent? When the hour hand of a given clock is at its 12:00:00 position, there’s only one possible location of the minute hand for that clock, and this is true for any position of the hour hand.
If you read elsewhere in the thread, you’ll see that johnclark draws a distinction between all possible clock faces and “valid clock faces”, i.e., those that obey the constraint you describe. Constant is addressing the former, not the latter.
You’re talking about what I’ve been calling valid clock faces, the faces of a working clock. That set forms a circle. Right here I’m talking about what we’ve been calling the set of possible clock faces, where we no longer assume the gears of the clock are constraining the positions of the hands. This set forms a two-dimensional torus, the surface of a donut.
Yes.
Show me.
John K Clark
There are two hands, an hour hand and a minute hand. The set of all possible positions that the hour hand can take describes a circle. The same is true of the minute hand: its set of all possible positions describes a circle. Consequently, the set of all ordered pairs of possible positions (h,m), where h is the position of the hour hand and m is the position of the minute hand, is the Cartesian product of the two individual sets, and thus the Cartesian product of two circles. This is a two-dimensional torus.
Did you take into account that the positions of the two hands are not independent? When the hour hand of a given clock is at its 12:00:00 position, there’s only one possible location of the minute hand for that clock, and this is true for any position of the hour hand.
If you read elsewhere in the thread, you’ll see that johnclark draws a distinction between all possible clock faces and “valid clock faces”, i.e., those that obey the constraint you describe. Constant is addressing the former, not the latter.
Yes. Thanks.
Ah. Okay.
Good point. The minute hand is entirely redundant.
You’re talking about what I’ve been calling valid clock faces, the faces of a working clock. That set forms a circle. Right here I’m talking about what we’ve been calling the set of possible clock faces, where we no longer assume the gears of the clock are constraining the positions of the hands. This set forms a two-dimensional torus, the surface of a donut.