Noo's Spiral Space

I hope to expain clearly the geometry of space in my first article, also with the drawings posted. Noo's space is highly curved and non-Euclidean but I have ignored all relativistic efects for this article so we won't have time dilation, extreme gravity or black hole singularity stuff.

Higher dimensions are fun

The main concept of this article is what I called the Spiral Space until I come up with a better name, which in the 2-dimension case it's what we see in the sidebar top image. Imagine you're on a spiral staircase, there's a column in the middle and you can walk around it, and if you do it, you'll end in an upper or lower floor, but you won't return to the original point. That's the two-dimensional case of Spiral Space, and there is no mistery at all, you walked upside or downside so it makes sense to not ending where you started, but if you were a truly 2-D being, you wouldn't have the concept of upside or downside, and you would think you were walking in circles around the center column, so now it would have been very confusing, and we can show this confusion by upgrading to the 3-D case.

Noo's Spiral Space is the 3-D case, represented in this image of Peter's Infinite Room. Usually curious mathematical figures are named after their discoverer or developer (Sierpinski Triangle, Penrose Stairs, etc.) and probably Peter's Infinite Room has already a name but I haven't found it so I call like that because Peter lives in that room. Now, despite having the concept of three dimensions, Peter walks around the column and ends in a different place of the room without going up or down, because he is going "up" or "down" in a fourth dimension which he cannot visualize. The first problem I found it is that in order to make sense, the column in the middle has to be infinitely tall, which makes me uncomfortable, so I kept with a solution by adding a second column. To visualize the effects of a second column it's better to return to the two-dimensional space as the one I drawn.

Multiple column systems

Looking at the following image it's important to notice that while one column creates a clockwise spiral, the other one creates a counter-clockwise spiral, and the two combined make the effect for a flat being that travelling between the two columns and then turning around of one of them takes it to the space above the starting point, and walking around one column and then passing between both takes it to the space below the starting point. In the 3-D case, now we can suppose that analogously, Peter would have two infinite columns in their room but this is not required, instead we can have other shapes making the same effect and I chose a ring, which is finite and you can pass across to go to the "upper" space, or you can pass across from behind to go to the "lower" space. That is basically a portal, one type of portal, but this subject will be expanded in a future article. By now we can start visualizing Noo by imagining one of this rings in space. Noo looks like a spherical Earth-similar planet centered in the edge of the circle portal, but this is only the appearance, if you want to go around the world you'll not end where you started because all of what I wrote here. This Phileas Fogg's nightmare takes us to the second problem of the Spiral Space: Will I go travelling forever into the ocean without returning to my home? Has Noo an end or we can visit infinite amount of land?

I can't draw this part

As I don't like infinite things here, I came up with the solution by adding a new dimension, so now the 3-D case curves in five dimensions instead of four. This is the most difficult to see and explain because going back to the 2-D case makes a 4-D object, and also I'm not sure about how to write this, but the main idea is that going "up" a number of times will eventually take you to home by doing a loop. The fun part of the travel is that in order to return to the original space by going around the loop you will cross that original space one more time at the midpoint of the travel. This has an important consequence, that the ring where Noo is in has another part of Noo in the opposite side of the ring. If Earth was like that the effect would be interesting, imagine you are in China and you want to see the Moon, but instead of the Moon there's South America in the sky.

How Noo habitants understand it

There isn't a civilization who have explored all the world for now, so most of the people believes that the opposite part of Noo seen in the sky is a different unconnected world, with different myths explaining who lives there, often as a world for the dead people or for deities. Also, Noo's scientists and philosophers are struggling with the paradox that their planet is not spherical if you travel around it, but is spherical if you measure its curvature, and that mixed with muptiple myths and cosmovisions generates huge amounts of debate. May be one day some expedition, after years and years of travelling will arrive to the opposite part of Noo and look that the continents shown in the sky have the same shape that the ones drawn in their homeland map, and then probably think they just died or better than that will realize how their world is.

Wikipedia example of the Riemann surface of the complex logarithm, this particular shape will help to visualize the space where Noo is in, but in higher dimensions.

Final representation (not to scale) of what Noo looks like in the space with the spinning direction. Notice that the portal also has its own spin. It looks like a double twin planet but remember these are only two parts of the entire world, and you can travel by walking instead of using spaceships.