ICN5D wrote:Hmm, what are these 24 orientations you speak of? Are they different combinations of facing directions plus moving directions, plus inverted orientations?
Consider yourself standing upright somewhere in 3D space, with your arms stretched out sideways. There are a couple of vectors that determine your position: (1) which direction you're facing (the front vector), (2) which direction your feet are pointing (the down vector), (3) which direction your head is pointing (the up vector), (4) the direction your back faces (the back vector), (5) the direction your left arm is pointing, and (6) the direction your right arm is pointing. These 6 directions may be mapped to the 6 faces of a cube, or equivalently, the 6 vertices of an octahedron. You could, for example, label the vertices of the octahedron as F, D, U, B, L, R in the respective order of the above vectors. Now the question is, how many ways can you orient an octahedron in space? This is equivalent to the number of (primary) orientations you can have in space.
For example, if you spin the octahedron around the L-R axis, you'll find 4 orientations, where you face forward, up, back, or down, but your arms remain pointing in the same directions. You can also spin the octahedron around the U-D axis, where you'll find that you face forward, left, back, right, with your arms changing from L-R to B-F to R-L to F-B (then back to L-R), but your head and feet continue pointing the same way. Each of these represents a distinct orientation of your body in 3D space. There are many other such combinations, which, if you work it out, corresponds with the rotational symmetries of the octahedron (equivalently, the cube), which we know are 24 in number. Note that all of these are distinct orientations without changing your
location in 3D -- you're just spinning in-place.
So, to fully specify your configuration in 3D space, you require first a 3-vector that specifies your position, then you have to specify in which of these 24 orientations your body is. Of course, these 24 orientations represent the 24
principal orientations in 3D; it doesn't include combinations like rotating 45° to your left, then spinning 72° up, etc.. So you see, 3D isn't quite a simple as you may have thought.
Of course, in real-life we usually don't think of these 24 orientations, because gravity for the most part confines us to a fixed up/down direction, so that fixes the U/D vectors to 1 out of 6 possible directions, which effectively divides the 24 orientations by 6, resulting in the 4 familiar directions forward, backward, left, and right. These directions correspond to the 4 orientations of a square in 2D, which reflects how our floor is essentially a 2D surface.
Now in 4D, the floor becomes 3D, so there are 24 orientations you can have just by standing in one spot on the ground! We find this confusing, because in 2D, the 4 orientations are in a 1-to-1 relationship with the facing direction, so we are used to thinking that our orientation on the floor is equal to which direction we're facing. This is no longer true in 4D, because of the 3D floor! In fact, if we're floating in 3D space, then we'll realize that we can actually be in 4 different orientations without changing which way we're facing: to see this, consider looking straight forward, but your legs could be pointing down, up, left, or right. In 4D, therefore, while standing on the same spot on the ground, there are 24 possible orientations, divided into 6 groups of 4, each group of which represents one facing direction. So even while facing the same direction in 4D (
and keeping your head pointing up and feet on the floor), you still have 4 distinct orientations you can be in. Or, put another way, you can spin your body around in 4D while standing on the same spot, and facing the same direction, without changing the latter.
Now, this is just when you maintain an upright orientation in 4D... if you're floating through 4D space, then suddenly you discover that you have a
lot more distinct orientations, because now your up/down direction can point in any of the 8 directions of the tesseract, and each of these directions have 24 orientations as I described above. Which means there are a total of 24*8 = 192 possible orientations.
(Which means that if you're walking around in 5D, you'll have to keep track of 192 different orientations you could be in! And if you're floating around in 5D space, then it's a whopping 1920 possible orientations.
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