Date: 2014-12-18
To: Curvy Flatland!
Miles today: 0.0
Weather: few clouds

Since Flatlanders can¢t perceive the 3rd dimension, they can¢t tell if their universe is perfectly flat in three dimensional space or if it¢s curved in some way or another. It might be wavy like the surface of the ocean. It might be folded up like a blanket on the closet shelf. Or, it might be curved back onto itself, like the surface of a doughnut or the surface of a sphere.The blanket analogy, by the way, is where the idea of wormholes is played out in the movie ¢Interstellar.¢ Where one part of the blanket rests on another part, there might be a way for the Flatlanders to jump directly from one layer to the other using a wormhole to quickly transit in the 3rd dimension between the two layers. Otherwise, the Flatlanders would have to travel many, many light years staying in the blanket and going around the fold. (We would need a 4D wormhole to shortcut our 3D universe.)Let¢s have some fun with the idea of Flatland being on the surface of a sphere. Keep in mind that Flatlanders can¢t detect the curvature of their universe. When they see the light coming from something in the distance, the light has traveled within Flatland along the surface of the sphere. They perceive it as coming in a straight line without any perception of the curvature in the 3rd dimension.One fun aspect of this to consider is that if you¢re in this Flatland and look straight into the distance far enough, you¢d look all the way around the surface of the sphere and see the back of your head! No matter what direction you looked!Of course, the light from the back of your head takes time to go around the surface of the sphere. If the sphere were big enough, it might take millions or billions of years for the light to travel that far. So what you¢d see is whatever was standing where you are so many years ago.Let¢s throw in another thought. Our scientists have observed that our universe is expanding - all the galaxies are moving further apart from each other. Big Bang theories hypothesize that this expansion has been going on since the beginning of our universe. But expanding into what?I used to picture an infinite-in-all-directions 3D space with all our galaxies in one particular area of that space, expanding out into the rest of the space that¢s already there. The corresponding analogy in 2D Flatland would be a flat plane extending infinitely in both directions off the table top, and all the Flatland galaxies contained in one area that is spreading out like liquid in a paper towel. In this case there would be a certain spot in the universe from where everything started moving out, and therefore one direction in which I should look back through the galaxies to see that spot. If I looked in the opposite direction, I would see the space into which we were expanding.Flatland on a sphere gives another option to consider. What if Flatland began in 3D space as the surface a teeny tiny sphere so small it was a point, and then blew up to be the surface of a bigger sphere that continued to get bigger and bigger, like someone blowing up a balloon? As the balloon continues to get bigger, all the points on the surface of the balloon get further apart from each other. The galaxies in Flatland would be moving away from each other because the existing 2D space is stretching, not because they are moving out into some 2D space that already existed!In this scenario, when you look into the distance on Flatland trying to see the back of your head, the further into the distance you look, the smaller Flatland was when the light left that object you¢re seeing. When observed from 3D land, the light would be following a spiral rather than a circle. (In either case, as a Flatlander, you would see the light as going in a straight line.)If you looked far enough, the inward spiral of your sight would wind around until you saw the very beginning of Flatland as the surface of a teeny tiny sphere or a point. And you would see this no matter in which direction you looked! Depending on the relative speeds of the light you¢re seeing and the expansion rate of the sphere, you might see back to the beginning point before the light had traveled around the sphere once. Otherwise you might see your own galaxy in its earlier stages in between you and the beginning!Let¢s take this analogy to our 3D universe! As telescopes become more and more capable, astronomers are able to detect light coming from objects further and further away, and hence, further and further back in time. And they see these objects no matter in which direction they look! Perhaps our 3D universe is the surface of a 4D sphere - whatever that is!