! Where did you go? Did you router get fried too? Just as I got you ready and warm for the kill, you disappear on me. It's not fair
4Dspace wrote:No. The fact that only one side of a plane is visible to a POV in a Euclidean space is a basic fact that you can check yourself.
No, it's not a self-evident fact, it's a wrong fact for me and the other persons in this discussion, and I've spent a lot of time checking the basics of Euclidean geometry and couldn't find a single reference to this. Planes don't have sides with different things on them.
So you've been denying that planes have directions or chirality. Here is the ref from google books, it's Geometry of four dimensions
By Henry Parker Manning, 1914, includes Schlafli's work that was published 3 years prior. Here is the relevant quotes from it:
p 162 - 163
We cannot speak of the right and left sides of a line except as we associate the line with some plane in which it lies; but direction along a line is a property of the line itself and is independent of any plane of space that contains it.
Likewise, we cannot speak of one and the other side of a plane except as we associate the plane with some hyperplane in which it lies; but "order of the plane", or direction of rotation, is independent of any hyperplane.
So, since you cannot deal with a concept of direction of a sheet of paper shown in different colors, because for you it immediately implies a "thin cuboid", I came up with the solution to this problem: Instead of a sheet of paper we have a plane figure proper
. And, to show its direction or "order", which is relevant to the discussion at hand, we have a bunch of 2Ders living on this plane (finally! mythical creatures are good for something -- you have to agree that this is in line with the rest of the discussions on this board). Like a line of ants, the 2Drs keep walking in a circle non-stop, in such a way that you can clearly say in which direction (from your POV) they are walking, clockwise or counterclockwise. So, where the sheet of paper had "yes" written, we have 2Ders walk clockwise and where the sheet had "no", the 2Ders walk counterclockwise.
Here. Now you have no excuses. Which way do you see them walk from a POV in a positive quadrant of the axes?