Leading up to the Omoglatron...

6.4:  Mapping 4D objects on spinning spheres 

         When Inventor walked into the lab, he thought he knew 
the kind of greeting he would get from Assistwo, but as always, 
Assistwo surprised him yet again. 
         "Hey Inventor, I’m not dead!" Assistwo said. 
         "I thought you going to say "Hey Inventor, I thought 
you were dead!" That has been the reaction I have been getting 
everywhere I go today. But you are right, from your point of 
view, your own continued existence is more important or 
noteworthy to you. Either that or you are smart enough not to 
remind me that I am still here, but it is nice to see you too," 
Inventor said while settling back into work mode. 
         "I thought we were done, or done for," Assistwo said. 
         "Nothing lasts forever, not even time," Inventor 
began, "all I really said was now is not the time to be 
discussing too heavily the ramifications of space and time 
being completely interchangeable, and I still believe that to 
be the case. It is still not time to go there yet. However, my 
favorite topic has come around again, and I couldn’t sit this 
one out." 
         "Oh no," Assistwo said bracing himself, "not again! 
Please don’t tell me." 
         "Yes, that is right, ball spinning!" Inventor said 
smiling. "What, you thought I was going to say something else?" 
         "I am not touching that one," Assistwo said, "too easy 
and too many ways to go, I will let you continue without 
comments for awhile." 
         "Ok, I will take the ball from here now, so to speak," 
Inventor began. "Stick spinning, and my new favorite, ball 
spinning, illustrates spinning on one, two, and three axises. 
You can spin a stick on two axises but it is not as fun as 
spinning it in only one direction, plus you can’t paint spots 
on it." 
         Assistwo was beginning to turn purple from restraining 
himself from making a comment on this. It was still just too 
easy to say something smart, but he stayed true to his word and 
remained silent. 
         "Ball spinning," Inventor continued, "will represent 
curved 3D space, which as we have already mentioned can be 
analogous to 4D space. We also said 3D space can be thought to 
us 2D people as being similar to 2D space curving around back 
into itself, and 4D space can be thought of as 3D space curving 
around itself as well. But that will come later, for now we 
will first deal with the spinning." 
         Enough time had passed for Assistwo to control himself 
from making a smart, wise-ass, and possibly very rude joke. It 
was not easy for Assistwo to let it go, but he simply said, 
"Paint spots on it?" 
         "Yes, with a ball or sphere we can paint spots on it 
relative to the positions of the sides of our cube. Since the 
ball is curved outwards, we can see the spot at the top and 
the spot at the bottom at the same time, something not even a 
3D person could not see in regular 3D space which is not 
curved, or two opposite sides of a cube at once. Since from a 
4D or 3 lines of sight plus away perspective this is possible, 
it appears from a 3D 2 lines of sight plus away perspective 
that 3D space can be curved, which as we already said 
repeatedly only appears to be the case, and only from a limited 
3D perspective." 
         "Right," said Assistwo, "before getting all cryptic 
and mysterious about space and time being interchangeable, we 
did, ok, you did, a pretty good job of explaining all points of 
a mostly 3D cube appearing as flat from 2 directions 
perpendicularly and equally "away" from its 8 corners at once. 
From those two angles outside of normal 3D perceived space, a 
mostly 3D cube would be indistinguishable from a fully 4D cube 
or Tesseract until it gets turned or you try to move around it. 
Are the 8 corners of the cube analogous to the 8 cube sides of 
a Tesseract?" 
        "I said we will first deal with the spinning before 
getting into curved space or 4D," Inventor reminded. "Ok, since 
stick spinning is my forte, I get to explain stick spinning in 
3D terms, only now we are using balls because of the spots we 
can paint on them. Each of the six spots, best represented by 
different colors, represent 3 sticks, or lines of sight, all 
moving through the center of the ball or sphere at 90 degree 
right angles to each other. Spinning on one axis only can be 
viewed in more than one way. We in the 2D world can only view 
spinning on one axis one way, what we will call edgewise. In 
stick terms, it would seem to be getting thinner and wider as 
it moved with one end further and one end closer, then both 
equal distance, then the other end closer and its opposite end 
further away. There is no top down view of a spinning stick 
from our perspective, alas, what a sight that would be, as the 
spinning requires both of our 2 dimensions to occur. As it has 
already been pointed out, from a 3D 2 line of sight plus away 
point of view, spinning on 2 lines of sight perpendicular to 
each other and the away line of sight would be similar to each 
other, and to the way we view a spinning stick but the 3rd way 
where all points of a plane or 2D world are equal points away, 
that is quite different." 
         "Yes," said Assistwo, "where the tips go around each 
other but never get closer or further away to the observer, 
where the spinning axis is completely in line with the away 
line of sight." 
         "Exactly Assistwo," said Inventor. "With our ball 
with spots on it, the equivalent of that event would be having 
one spot always in the center and four other spots chasing each 
other around the edge. That is one way to view spinning on one 
axis. The other way to view it is 90 degrees away from that 
point where both axises spin points are equally in the "away" 
direction and both spots at the "poles" around which it spins 
are visible at once. The you have a very different view of the 
same spin. Now instead of making a circular motion around the 
visible edge, we see stick spinning the way we see it in the 
2D world, as the spots representing the axises or sticks come 
closer to the viewer and then move away. When both poles 
around which it spins are equal distance away, the other 2 
axises move in a straight line across the middle in-between 
them." 
         "Don’t they always run in a straight line between 
them?" Assistwo asked. 
         "Yes, but it does not always look that way. Depending 
upon the position of the observer, each spot can seem to travel 
around the edge, stay right in front of you, or seem to 
continually move in a straight line in one direction across, or 
a combination of these depending on the angle of your "away" 
position relative to the angle of the spin. If we spin it on 2 
axises at once, it gets even more complicated." 
         "So if both of the "non-away" axises are both at 90 
degree angles to the away line of sight, they automatically 
line up with each other and follow each other in a straight 
line from the away line of sight or away from whichever other 
axis is turning’s perspective. The old "Once an away direction 
is determined, all other points of the other axises are on a 
plane away from it." What about triplanes?" Assistwo asked. 
         "Not yet Assistwo, you keep running ahead of me," 
Inventor admonished. "Yes, whichever one of the 3 axises of a 
sphere is turning, it serves the same principle as an away line 
of sight, it makes the other two seem to line up in a 
particular way, relative to that motion. In this case they seem 
to follow each other in a line like a big letter X spinning 
when viewed from above. That is something we can see clearly 
when the line of sight is directly over one of the poles around 
which it spins. When both poles are visible or equal distance 
away, both parts of the X converge into what we know of as 2 
line of sight convergence." 
         "And spinning on two axises at once", Assistwo quipped.
         "That would seem way if you were directly over one of the poles of spin, the other pole of spin, were it to be spinning on a 90 degree angle to the first, that extra point of spin would be around one of the spots moving around the edge. This means that if you were directly over the first spinning axis and turned exactly in sync at the same speed with it, it would appear normally to you as if spinning on only one axis," Inventor said.
         "Double spinning, I don't think I like where this is potentially going," Assistwo said wondering if he should have had eaten first on the way to work.
         "Right you are, to the game room, Assistwo."
 

6.5:  The Centrifuge and the Omoglatron

         Ever since the success of the 3D amusement park ride, Inventor's experiments were privately funded by the proceeds from them, leaving Inventor free to invent all kinds of weird things with seemingly no purpose. Many of these could also double for new amusement park rides since, as I said, pretty much were lacking and basically consisted of spinning you around since they have no verticle axis in their 2D world. One of the older rides already invented was called the Centrifuge, which Inventor bought anyway since he was now rich, rides were the reason they had all the funding, and said it was necessary for his research. He also thought it was fun.
         The Centrifuge was basically a big spinning circle with walls. If you stood in the center and opened the doors you would see the room outside of it spinning around like most other Flatland rides. However, if you moved away from the center spot, you would spin more erratically and faster and eventually if you moved further away or could not hold a point further away from the center, you would get thrown up against the wall. Inventor speculated this force seemed similar to one he called gravity, which of course no one took seriously as their is no gravity in Flatland or their Universe would seem 1D more than 2D if they did have such a force. (The lack of gravity also would make the Centrifuge not work, but never mind that for now, that will come later.)
          On Inventor's instructions, he and Assistwo got into the Centrifuge ride. In the middle he set up a video camera. He instructed Assistwo to hold on to one wall while he stood on the opposite side and turned the ride up to full speed for 5 minutes. Assistwo was right, it was not a good idea to have had lunch first on the way to work.
         After they were done, Inventor showed the videotape.
         "You don't look so well, Assistwo," he noticed.
         "Was it really necessary for us to be in the videotape? Couldn't you have used dummies or poles or anything else?" Assistwo wondered aloud.
         "Yes, but then it wouldn't have been as fun. You really should try to enjoy your work more, Assistwo," Inventor added.
         "I just had lunch," Assistwo said.
         "Well, you should have said something. As we can see on the tape, though we are clearly spinning around the camera, it looks identical to this tape we made yesterday where we stood in the same spots and spun the camera around, except for the part where you throw up. Next time, say something," he added.