Topology question

[Brian-M]

Can anyone give me the official topological name for the surface you get when a two-dimensional plane with edges extending from zero to infinity is folded over so that the two edges are contiguous?

And joined seamlessly so that there are no edges anymore. If a position marker at (8,0) facing "up" or "north" on the plane were advanced one position forward, it would now be at (0,8) facing "right" or "east".

Would it just be a conical surface, or is there a special name for it?

(It's not really important, I'm just curious about how to describe it. I'm designing an esolang for my own amusement, and this is how I intend to have the memory space arranged. Probably pointless, as it's unlikely I'll be able to find someone to create an interpreter for me, and I wouldn't be able to do it myself. But I'm writing up specs for it anyway.)

 

[casebro]

Infundibula?

But I'd just call it a cone, since I eschew obfuscation.

Is that acute sentence, or am i being obtuse again?

It's plane to see I dropped out of geometry.

 

[Brian-M]

Just for fun, here's part of a specification of a popular well-known esolang called Funge that has an interesting topology for it's program code...

Befunge-93 handles the case of the IP travelling out of bounds (off the map of Funge-Space) by treating the space as a torus. If the IP leaves the west edge, it reappears on the east edge at the same row; if it leaves the south edge, it reappears at the north edge at the same column, and vice versa in both cases.

For various reasons, toroidal wrapping is problematic in Funge-98. Instead, we use a special wrapping technique that has more consistent results in this new, more flexible environment where Funge-Space can have an arbitrary size and the IP can fly. It is called same-line wrapping.

Same-line wrapping can be described in several ways, but the crucial idea it encompasses is this: unless the delta or position of the IP were to be changed by an intervening instruction, the IP will always wrap such that it would eventually return to the instruction it was on before it wrapped.

The mathematical description of same-line wrapping is known as Lahey-space wrapping, which defines a special topological space. It is generally of more interest to topologists and mathematicians than programmers. We won't cover it here, but it is included in the Appendix for completeness.

Full article: http://quadium.net/funge/spec98.html

 

[jojonete]

Can anyone give me the official topological name for the surface you get when a two-dimensional plane with edges extending from zero to infinity is folded over so that the two edges are contiguous?

And joined seamlessly so that there are no edges anymore. If a position marker at (8,0) facing "up" or "north" on the plane were advanced one position forward, it would now be at (0,8) facing "right" or "east".

Would it just be a conical surface, or is there a special name for it?

This all sounded somehow weird until I noticed the "programming language" part. So your coordinates are screen-pixel-style, that is, with (0,0) at the top left corner, x growing to the right and y growing to the bottom.

I'd say the figure is just a cone. As English is not my first language, I looked it up in wikipedia and it looks like there is actually something called "infinite cone" or "conical surface" which describer your figure pretty well.

To translate some (x,y) coordinates in the plane to their equivalent in the cone one would start by converting (x,y) to polar, which I'll denote as (r,a), with r going from zero to infinity and a going from zero to pi/2. Now, "r" would be distance from the cone apex and "a" would be a quarter of the angle around the cone.

If we were to draw a standard grid on the cone, it would look somehow weird, as travelling from (0,3) to (3,3) and then to (3,0) would start and end in the same point in the cone but it would follow some strange path to go around it (first going down then up again).

(It's not really important, I'm just curious about how to describe it. I'm designing an esolang for my own amusement, and this is how I intend to have the memory space arranged. Probably pointless, as it's unlikely I'll be able to find someone to create an interpreter for me, and I wouldn't be able to do it myself. But I'm writing up specs for it anyway.)

Hmmm... I might be willing to write that interpreter. But I promise nothing at all.
After all, I'm thinking about actually drawing the cone we're discussing

ETA: if you're interested, you may send me a PM with the specs, or with an e-mail address to discuss the matter.

 

[jojonete]

After all, I'm thinking about actually drawing the cone we're discussing

Okay, I just did it. How can I attach an image here? Here it says "attach files, valid file extensions... png", but I see no link or button. Possibly my paranoid-mode adblock... whatever, here it is: http://www.jojonete.com/00/20120605_Esolang/

The first image is a 3x3 grid turned into a cone. The second one is a 10x10 grid (the top part of it is the 3x3 grid). The red part is the x axis, which coincides exactly with the y axis. In the OP it was mentioned that these should be different, just "one position" apart. To model that, just convert the red "spinal cord" into a red ladder (i.e. two spinal cords with a horizontal link between each two nodes). I just didn't feel like modelling a finite distance between the two "spinal cords" and then adjusting all angles accordingly. Also, wouldn't it be a nicer esolang if (0,x) were the exact same memory position as (x,0)?

To "see" how the figures relate to a grid, draw a 3x3 grid and start only with node (0,0). Then add nodes (0,1)-(1,1)-(1,0). Then add (0,2)-(1,2)-(2,2)-(2,1)-(2,0). Go on adding layers and keeping the square shape.

Now, look at the "3x3 cone". The top node is (0,0). The next two-node layer is (0,1)-(1,1)-(1,0), however (0,1) and (1,0) are together in the red node, so there's just two nodes in that layer joined together, and the red node is linked to (0,0). The next layer adds (0,2)-(1,2)-(2,2)-(2,1)-(2,0), but again (2,0) is the same as (0,2) and it's painted red. Look at how the links in the cone match the links in the grid and add as many layers as you like.

ETA: I just had the idea of adding a third view from the top of the cone - and did it. Have to stop this right now before it gets worse

 

[The Man]

Can anyone give me the official topological name for the surface you get when a two-dimensional plane with edges extending from zero to infinity is folded over so that the two edges are contiguous?

And joined seamlessly so that there are no edges anymore. If a position marker at (8,0) facing "up" or "north" on the plane were advanced one position forward, it would now be at (0,8) facing "right" or "east".

Would it just be a conical surface, or is there a special name for it?

(It's not really important, I'm just curious about how to describe it. I'm designing an esolang for my own amusement, and this is how I intend to have the memory space arranged. Probably pointless, as it's unlikely I'll be able to find someone to create an interpreter for me, and I wouldn't be able to do it myself. But I'm writing up specs for it anyway.)

Yes it’s called a "nappe"


http://en.wikipedia.org/wiki/Conical_surface

Generally as the union of all lines that pass through a given point and some fixed curve, a conical surface consists of two. Each nappe is the union of all rays that start at a point (the apex) and pass through some fixed curve. This would be your curved plane that converges to a single point. As noted in the article often a “conical surface” just refers to one nappe. Not a bad sounding word either, so having a nappe or nappe like architecture for your language may be what you’re looking for.


ETA: Oh, by the way, excelent work jojonete.

 

[Brian-M]

Okay, I just did it. How can I attach an image here? Here it says "attach files, valid file extensions... png", but I see no link or button. Possibly my paranoid-mode adblock... whatever, here it is: http://www.jojonete.com/00/20120605_Esolang/

That's an awesome graphical representation, especially the last one. What software did you use to create it?

All I did was draw a grid on a sheet of paper, fold it diagonally, then tape the edges together. It's now a little Dunce's cap for my cat.

For attaching images, the white box directly below "Valid file extensions" containing the words "Manage Attachments" is the button you need to click on. (I know, it doesn't look like a button or a link, but it is.)

Yes it’s called a "nappe"


http://en.wikipedia.org/wiki/Conical_surface

Okay, so the phrase I'm looking for is "Single napped conical surface." Thanks!

 

[The Man]

Okay, so the phrase I'm looking for is "Single napped conical surface." Thanks!

No problem and not bad, a "Single napped conical surface." in reference to how your memory is, well, "mapped" has a nice lilt to it.


Q; "Wait a second, how do you map the memory in this structure you're proposing"?

A; "Well, it's a single nappe memory map"


Oh, I like it, such technical alliteration is rarely found.

Language and technology, a couple of my two favorite things.

 

[jojonete]

ETA: Oh, by the way, excelent work jojonete.

That's an awesome graphical representation, especially the last one. What software did you use to create it?

Thank you. I used POV-Ray.
Go to their home page and take a look at any of the pictures they have there. Then take a look at this other page. Zoom in and pay especial attention to the scratches on the road and the partially unpainted road lines. Also the clouds and the trees. Then look again at my cone single nappe memory map and see if you can still call it "excellent" or "awesome".

For attaching images, the white box directly below "Valid file extensions" containing the words "Manage Attachments" is the button you need to click on. (I know, it doesn't look like a button or a link, but it is.)

After a few tests, it's definitely only my ridiculously aggresively paranoid ad-block configuration.

 

[The Man]

Oh, I like it, such technical alliteration is rarely found.

Sorry should have been "Assonance" or just rhyme rather than "alliteration" (which is technically limited to the first sound or phoneme of the word). So still displaying my Ass-onance of the technical complexity of language.