In this first lesson, we look at some basic properties of spatial orientations and rotations. If you're already quite familiar with 3-D rotations, or if you're eager to get to the movement-based puzzles, you can safely skip ahead to Lesson Two: exploring orientation space through movement.
When exploring the set of all spatial orientations, one of the most basic questions we can ask is, how big is this set?
Let's try to answer this question:
With three degrees of freedom, the set of spatial orientations is rather large! How can we navigate this vast set?
Write down your procedure, then proceed to the next video, which gives a possible solution:
Did you find a way to achieve a 90° z-rotation, using only x-rotations and y-rotations? Here's one method, which you can compare to your own:
Although they have many quirks, 3-D rotations are still the most natural way to travel through the set of all spatial orientations.
Let's bring our attention back to the set of all spatial orientations -- which is actually much more than just a set. It is natural to think of it as a space, namely, orientation space !
