What is the Omoglatron?
The Omoglatron is an amusement park ride in a 2 dimensional world whose purpose is to simulate being able to see through curved 2D space. It is was invented by Inventor. It is a follow up to his original amusement part ride to simulate what one would see moving through 3D space. (See 2D, 3D, 4D, 5D Thinking Made Simple Section 4.2) He contends that it is possible for his 2D universe to be curved around itself in the shape of what he calls a ball or a sphere (a 3D version of a circle for those who do not understand 3D concepts). Thus whatever direction one travels in, one could eventually return to the same spot. The Omoglatron shows how this would affect how one could see and understand reality in a spherically (or hyperspherically) curved Universe.

Why does the Omoglatron have a % symbol on it?
The % symbol is represents the basic concept of the Omoglatron. If a 2D Universe is curved, an object moving in a straight line would also be moving in two circles in two directions curving around two possible objects apart from each other at opposite ends of the Universe in an equal distance away from the point that trajectory is curving around. In 2D space this would be drawn as a line between two circles. In curved 3D space, it would look the same from one angle in line with that plane, as a 2D plane stretching between any two objects separating halfway between them, and also as as two balls surrounding each of them. As with the 2D version in which the line and both circles all represent the same line seen from different angles, in 3D the flat plane, and the two balls would represent the same plane curving in two opposite directions at once. In 4D space, one could image a 3D universe bending in two directions at once around two other centers in 4D space of an equal distance apart.

Ok, you said between any two objects, not ones a half a Universe apart, what is with that?
The curvature would cause any plane halfway between two objects to bend equally away from both at the same time. Though this is more pronounced the further the objects are away from each other, that curvature of the plane in two seemingly opposite directions at once in-between is there in the distance between any two objects no matter how close they are to each other or how minimal the curvature (how big the Universe is).

If there is a curvature around any straight line trajectory in curved space, why do you need two circles if the curvature extends away from every point equally, isn't that just unnecessarily confusing?
While one ball could explain how any point away from any object would lead back to the same spot of a ball or circle the entire lenght of the Universe away from that ball, using two balls shows aspects of dividing that space up into two sements understandable to a one dimensional less point of view. A 2D person could understand it as two circular regions of space curving away from every point, and another region of space curving back toward another point halfway across the universe away from there. To a 3D understanding person, they could see this as two views of a globe, if you were standing on the north pole, you could have a snapshot of a circle of 1/2 of the Earth as a big circle, taken above your head downwards with you in the center of the circle. You would need another separate picture of the other half of the earth taken from the point of opposite of the view from above your head, from below your feet. Similarly, if all you could understand was 3D, you could envision a 4D space, or curved 3D space into a 4th dimension as two balls away from every point. One would represent the half of the universe which curves away from wherever you are, and the other to represent space curving back upon that point opposite from where you are at the other end of the universe.

But if space is empty, how can it be curved?
It can't. To see or imagine the curvature we need to have or conceptually to put something there to mark it. The curvature only applies to objects within that region of space in their relation in distance and time to another region of space, how their shapes change in relation to objects in different directions and distances away from them look, and actually are, not just appear, bending in many directions at once relative to what else is around them, but only relative to the distance between them. At their surface, they know their shape as defined by themselves at that point in space and time.

Any examples?
This is from the Notes:
"Orange peel space is shown in the Omoglatron example. Where spinning and orbiting are aspects of the same thing, movement or speed collapses space ahead of it in a curved environment, slower speeds expand space like a circular orange peel. Thus full speed from a 3D point of view is breaking above the clouds so to speak of the 3D Universe and all points away are not only equally away but the same point."