Good to hear from you again. The reason is something else again. I really think Lem was right. At the moment I think we need to solve our own problems.
Now about our 1-d universe. First I had already discovered that 3.14159 was not right and that 3.383 was. The range of the size of the individual segments was much less See below for new table:
Circle Radians /Circle arc lengths/ Spiral arc Lengths
324-360 2.56 2.468
288-324 2.56 2,219
252-288 2.56 1.966
216-252 2.56 1.715
180-216 2.56 1.468
144-180 2.56 1.224
108-144 2.56 0.978
72-144 2.56 0.759
36-72 2.56 0.563
0-36 2.56 0.433
Totals –→ 25.6 13.8
You can compare it with the old one.
But that’s not what I have been thinking about. The 1-d whole universe idea you put forth is a great space-time exercise. It really describes time as if it were a geometric dimension making the one-dimensional universe a two-d universe. It occurs to me that what you could have drawn was a multi-universe. It seems it implies that there are an infinite number of universes. For instance if one takes a point on any place on the circle of greatest expansion, it has is own universe. See below:
If the two outer dots (suns) are two suns 90o removed from each other around the blue circle you can see how different each ones universe is. In this example they cross each other more than once and share a lobe of their spirals. That is where the two universes have the pieces of space in common. It is important to remember that a sun’s universe is only the one-dimensional line that can be seen or measured from that sun. Note also that someone (a one dimensional someone) can look in two directions down a 13.8 billion light-year line, their one-dimensional universe. If someone from the one sun wants to visit someone from the other sun, they must travel inwards along their universe line (the black line) to a time/distance of 5.67 billion light years.
I might point out that the spiral above is called an Archimedes spiral. It is compatible with a steadily expanding universe. If the rate of expansion were increasing one would expect a logarithmic spiral.
There of course is a Fermat spiral in which the arms get closer together. This would imply a universe that was expanding but at a slowing rate. That concept used to be popular, but I don’t think it is now. You would know about that at least as well as I. Obviously, the arc length would be different in both the latter two. I will stick with the Archimedes spiral for this letter.
The second thing I thought about was about the speed of light and about quantum entanglement. Lem, sitting off in his three dimensional world, can see all the present and past universes as well as the outer circle where all the suns exist at the same time. Our one –dimensional people, of course, cannot. Suppose though that light or some information was not constrained to the speed of light around the circle. Perhaps then an event on one sun could be instantly measured on another. At light speed around the outer circle, it would be a trip of 6.4 billion light years. Would that not look like what we call quantum entanglement? I really don’t know, but it’s fun to speculate. Another time problem is local time. Is that the same as time in the expanding universe?
I can visually see how these one-dimensional universes could be extended to two by simple rotation through three-space. I am sure the same is true of rotation through four-space, but I can’t visualize it. In the latter case Lem would have to live in five-space or more to visualize it. In the first book “Pygmalion Conspiracy” that is clearly the case because the gods are looking down on their creation which is supposed to be our universe.
For the moment I will stick to playing with the one-dimensional universe. I’ve gone one dimension smaller than “Flatlands”. Have you or either of your friends read that?
P.S. Best to Maglie and Artemy. Are they coming back to Earth