Symmetry Operations JavaScript Simulation Applet HTML5
From young, most of us are taught that a + b = b + a, or a*b = b*a, and we take that for granted. But did you know that this is a property of real numbers, and may not actually be true for everything in mathematics?
This is called the commutative property, that the order of a binary operation (binary operation: an operation that takes in 2 inputs and returns 1 output) does not matter and is a property that comes into play when you add or multiply real numbers.
But let's think of situations where you don't see the commutative property apply. Trivially, subtraction and division are non-commutative, since a - b clearly does not equal b - a, and a/b likewise does not equal b/a (they are reciprocals).
A more interesting non-commutative set is the set of matrices. Matrix addition (for two matrices of the same size) is commutative, since you're just adding up the elements in each matrix. Matrix multiplication, however, is not. Let's say you have two 2x2 matrices A and B:
A*B ≠ B*A
Another example of a non-commutative operation is rotations and symmetry, which is what this simulation covers.
If you first apply a rotation to a object, then mirror that rotated object, you'll find that what you get is different than if you first applied the mirroring, then the rotation.
There are two panels in this simulation (link here): the left panel shows what happens when you first apply rotation, then the mirroring. the right panel shows what happens when the mirroring is first applied, then the rotation.
The rotation and base orientation of the triangle A can be adjusted using the sliders provided.
This is called the commutative property, that the order of a binary operation (binary operation: an operation that takes in 2 inputs and returns 1 output) does not matter and is a property that comes into play when you add or multiply real numbers.
But let's think of situations where you don't see the commutative property apply. Trivially, subtraction and division are non-commutative, since a - b clearly does not equal b - a, and a/b likewise does not equal b/a (they are reciprocals).
A more interesting non-commutative set is the set of matrices. Matrix addition (for two matrices of the same size) is commutative, since you're just adding up the elements in each matrix. Matrix multiplication, however, is not. Let's say you have two 2x2 matrices A and B:
A*B ≠ B*A
Another example of a non-commutative operation is rotations and symmetry, which is what this simulation covers.
If you first apply a rotation to a object, then mirror that rotated object, you'll find that what you get is different than if you first applied the mirroring, then the rotation.
There are two panels in this simulation (link here): the left panel shows what happens when you first apply rotation, then the mirroring. the right panel shows what happens when the mirroring is first applied, then the rotation.
The rotation and base orientation of the triangle A can be adjusted using the sliders provided.
