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My daughter asked why this motion starts with palm up but ends with palm down? HOW DOES THIS WORK? Why does it turn your hand over? And how can your arm move like this? 😅 There’s actually an answer for this in MATHEMATICS of all places. (Short thread) /1
2/ Math is amazing and can be used for just about anything. In this case we are asking about the rotation of solid objects. Each part of your arm has a location in three dimensions (x, y, z). In math, a list of numbers like this called a vector. (Credit: en.m.wikipedia.org/wiki/Euclidean…) 
3/ When you rotate your arm, the position of the top part of your hand ends up different than it was at first. The same for almost every part of your arm. So the (x,y,z) values change for almost every part of your arm. We can mathematically change them all with 1 simple equation:
4/ We multiply the initial vector by a matrix and that gives us the new vector. You can plug in the value of (x,y,z) for any part of your arm, and multiplying by that matrix will give the new (x,y,z) for that part of your arm. This amazing matrix is called a Rotation Matrix.
5/ We use rotation matrixes all the time in physics & engineering. I was writing software this week to predict where dust goes when it’s ejected off the Moon. I used rotation matrices to simplify the calculations and make it run faster, for example. (More on that later...)
6/ When you take a math object like a number or a vector or anything, and then change it by multiplying with a number or a vector or doing anything else, we call that an “operation”. The Rotation Matrix is an “operator”. It is interesting to study the properties of operators.
7/ Like for example you can study the Multiplication operator. It turns out you can do those operations in any order and you get the same result. Take 4 and multiply it by 2 and then by 3. You get 24. Now reverse the operators. Multiply by 3 and THEN by 2. You still get 24.
8/ We say that the multiplication operator “commutes”. We can make either operator cut ahead of the other (commute past it) and it makes no difference. The same is true of Addition. 3+4 is the same as 4+3. Operators that commute are in the same Group, according to Group Theory.
9/ There’s actually a lot of deep philosophical physics in this simple observation. The fact you can find the square footage of a house by measuring Length x Width or equally Width x Length tells us the spacetime of the universe is described by operators in the commuting group.🤯
10/ But not all operators commute like this! In particular, there are operators in Quantum Mechanics that don’t commute, which like everything else in QM tells us the universe is more freaky than our everyday experience suggests! But there are other non-commuting operators, too.
11/ And there is a noncommuting operator that we can see in our everyday experience: this Rotation Matrix! If you multiply the vector by Matrix A and then by Matrix B (two different rotations one after the other) you don’t get the same result as multiplying by B and then by A.
12/ To describe rotations you need three numbers, because you can rotate on any axis in x, y, or z dimensions. In engineering we often use the Euler Angles as a system to describe all three rotations. The order you do these rotations is vital.(credit: en.m.wikipedia.org/wiki/Euler_ang…)
13/ If your arm does a rotation in the Alpha direction, then the axis of its Beta rotation is pointing in a different direction than it was before (see the picture), so the Beta rotation becomes a different matrix. I.e., the operators act on each other. They don’t commute.
14/ So the motions my daughter did were a set of three rotations. The elbow rotates in Pitch. Then the shoulder causes your arm to execute a Roll. Then your elbow causes it to execute a Yaw. Pitch, Roll, and Yaw. The result is the same as just turning your palm over, a Roll. How?
15/ Because the operators don’t commute (based on symmetries of Spacetime in the overall universe!), doing the Roll causes the elbow motion to change from a Pitch into a Yaw—the operators change each other!—so the 3rd matrix cancels the 1st one even though it was different.🤯
16/16 I will be honest. I never liked Rotation operators when I was in grad school. They are one of the more complicated parts of everyday physics. No quantum mechanics required. But I‘ve had to use them a lot, especially calculating orbital dynamics of dust blown off the Moon.
