Directions on the Surface of a Klein Bottle

I am exploring the potential for a work that takes place on the surface of a Klein bottle that is not self-intersecting, which is to say a Klein bottle with a twist in four-dimensional space rather than three-dimensional. (Let’s set aside the matter of physical impossibility. I am only interested in the math.)

In lieu of cardinal directions, what terms could I use to refer to the directions in which I could travel?

Also, has someone already made an IF work that takes place entirely on the surface of a Klein bottle? (I have done some web searches, and the closest thing I can find is Erehwon, but reviews seem to indicate that it is not set entirely on a Klein bottle.)


The best-known Klein bottle is in Trinity. Not the entire game – just a few rooms – but it gets the idea across well enough to support a good puzzle.

It’s in the “Let’s Play Trinity” thread somewhere… Let's Play: Trinity by Brian Moriarty


You need 4 direction letters if you want to handle it mathematically. Like plus/minus x y z u (Or replace u).

Or you take the 6 existing 3d directions (U, D, E, W, N, S) and add two:

  • in and out?
  • or forward and back?

You could do it all with cardinal directions. A square grid of rooms with NSEW movement and “wrapping” is a torus. If you also reverse east-west when wrapping north-to-south, I believe that’s a Klein bottle. You have to use variable room descriptions that change “east” to “west”… Trinity demonstrates this.


From the perspective of someone on the surface, though, there won’t be a fourth (or even really third) dimension, right?

An ant on a Möbius strip doesn’t know that it’s twisted through three-dimensional space, it just knows that it’s a two-dimensional space that connects weirdly.


A picture to illustrate a Klein bottle:


If you appreciate the quad-face modelling, you’ll notice that any part of the surface can be traversed in cardinal directions. Each square face can be a grid cell. Eventually, the top and bottom of the grid will connect with each other as well as the left and right (imagine unwrapping this surface as a flat grid). Like in the game Asteroids, if you go off the left side of the screen, you will eventually appear on the right side.

If you think of the spine of the shape being Y axis (takes the longest to return back to where you started) and the X going left or right (shortest path to return to where you are - is also a circular shape - 18 grid cells across/circumference in the picture), you’ll notice that the going 180 degrees (9 cells distance) in X will end up on the opposite side of the Y as well (half the total length of possible travel on Y). Neat!

And no, I won’t count the number of Y axis rows in the picture. :wink:

Exactly. Space time is relative.


I assumed the player character can move freely in all 4 dimensions. I get a knot in my brain if I try to understand a 4D Klein bottle. Don’t even get the 3D one. But I can easily understand a Möbius band where the ant moves only on the surface.


Oh wait, due to the folding… I mentioned being half way on the Y axis as you move half way on the X axis (and vice versa).

I think the Y axis wrapping needs to be programmed a bit differently. Now I appreciate the conundrum. I’ll make a diagram to illustrate how it needs to wrap along the spine. It’s actually kind of tricky to imagine, but I think it only needs a simple rule to maintain. I’m not sure, but I think the Y axis wrapping needs to work like this?

In this example, the X axis is a 6 grid distance/circumference. The Y axis wrapping needs to occur at whatever half the X axis total distance is. (Keep in mind the the outer most edges of the flat map are the same/shared lines - left most / right most lines and top most / bottom most lines.)

Someone smarter than me needs to confirm this though because I think I just gave myself an ice cream headache.

Edit: Damn it. I think the diagram needs to be altered.


I think the diagram needs to be altered.

Yeah, it doesn’t need an offset, it needs a reversal. Pardon the ascii art:

  V W X Y Z
A +-+-+-+-+ A
  | | | | |
B +-+-+-+-+ B
  | | | | |
C +-+-+-+-+ C
  | | | | |
D +-+-+-+-+ D
  | | | | |
E +-+-+-+-+ E
  Z Y X W V

(You can use any square or rectangular dimensions for this.)


Three jolly sailors from Blaydon-on-Tyne
They went to sea in a bottle by Klein.
Since the sea was entirely inside the hull
The scenery seen was exceedingly dull.
(Frederick Winsor)


Actually, you might have been onto something that the rest of us didn’t see.

Travelling long the surface takes you to the inside surface too. If you moved along the Y axis, stopping at half the distance required before you end up where you began, you’d end up on the exact opposite side of the surface/floor you started at. Okay, I think I’m seeing the “other side” of this problem. :wink:

I think the diagram I made showing the wrapping of the Y axis is right (Edit: wrote this before Andrew posted), but now I know that there is an opposite side that you cross. Z axis movement could be (flipping to the opposite side of the surface you’re standing on. Plus, if you jumped directly up (relative to you), you could hit another surface in some cases. You could flip + jump + flip and really travel around like a pro.

Personally, now that I see how a Klein bottle works (I think), the flipping/jumping to the opposite side seems like great way to melt minds even more.

I’m going to create a low-poly/quad model of the Klein bottle and lay out the grid map and explain it all. OCD kicking in… can’t stop over analyzing… brain melting… must stop thinking… rebooting operating system.


Thank you, all who have replied!

I have a question related to the conversation that has been unfolding between zarf and HAL9000. (I welcome answers from anyone and everyone.)

First, a review of my understanding of Klein bottles (please correct me if I get anything wrong): Let’s call a point on the Klein bottle my “home”. If I decide to walk in a straight line “north” (as the term has been used above), then eventually I will return home. But my 3-D perception will be astounded to see that the world is now flipped from my perspective—for example, if the doorknob of my front door was on the left side, it will appear to be on the right—although it would be more accurate to say that I have become rotated in a way that is not possible in 3-D space.

With that fascinating weirdness out of the way, here is my question: What would I see if at the halfway point of my journey I decided to stop and walk the rest of the way home going “east” or “west”? If my understanding of the Klein bottle is correct, my perpective would be analogous to what a flatlander would see when a rectangle is entirely perpendicular to the plane the flatlander lives in. In other words, I would not be able to see anything at all because my three-dimensional eyes would not be able to see my home from the necessary angle.

How am I doing?


Okay, I literally got my PhD in this field (topology).

A surface like this has two kinds of coordinates, intrinsic and extrinsic.

Extrinsic coordinates are the coordinates of the bigger space you’ve put the surface in. An example would be a ghost, unaffected by gravity, floating around and through the earth. They have 3 natural directions of movement, even though the face of the earth is only 2-dimensional.

Intrinsic coordinates are the coordinates of the space itself viewed from ‘inside’. For example, on our earth some intrinsic coordinates are latitude and longitude.

Generally, from a math perspective, intrinsic coordinates are strongly preferred. People living in a manifold don’t even know they’re in a manifold. The secret is there may not be any outside space. Space can be curved, yes, but it doesn’t have to be curved in something larger.

With intrinsic coordinates on a ‘flat’ klein bottle (a square where going off one edge teleports you to another, with a ‘reflection’ when teleporting from north to south but not from east to west), all you would see around you is an infinite repeating chessboard-like pattern, where everything on one ‘color’ of squares (say, white) would be oriented the same way as you and the other color would be flipped. But you wouldn’t actually see any hard boundaries between the two. There would be an infinite grid of yourself, half of them flipped, half not.

The website Geometry Games has little simulators allowing you to see what living in a curved space would be like.

If my understanding of the Klein bottle is correct, my perpective would be analogous to what a flatlander would see when a rectangle is entirely perpendicular to the plane the flatlander lives in. In other words, I would not be able to see anything at all because my three-dimensional eyes would not be able to see my home from the necessary angle.

This is true if using extrinsic coordinates. It is not true when using intrinsic coordinates.

It would look a bit like this, but only one ‘plain’. This is more of a 3d version with layers of klein bottles stacked on each other or where going up loops with going down. This is from that game I linked:


Also, if you want to be a 3d person walking on a klein bottle, you could just take a 3d space called ‘the klein bottle crossed with an interval’. This just ‘thickens’ the klein bottle to make it 3d, much like the atmosphere on our earth’s crust makes it a ‘sphere cross an interval’ rather than a truly 2-dimensional sphere.

Editeditedit: Also I highly recommend this film to people who want to understand what walking around in these spaces is like:


If the entity is living in a 2D only world, then yes, they would only see what’s in cardinal directions. In fact, they would not see shapes, but only dots and horizontal lines (perfectly horizontal - remember, no Z dimension). Up and down are an extra dimension. However, I find it would be fascinating to be a three dimensional being living on the surface of a Klein bottle (like it provided gravity or kept you on it with magnetism)… then you could see all the weirdness happening and use your extra perception to create new strategies for navigating. Like, literally opening a hole in the floor would take you half way across the world in the Y axis, but in a mirror-like X axis. Crazy!

Mind blown… defragmentation of brain commencing. :upside_down_face:


I’m so confused. Isn’t it just another one of those shapes like the mobius strip?



You can’t get home that way! This isn’t a sphere.

On my diagram above, say your home is at the center (CX). If you walk halfway north, you’re at AX (or EX – A and E are really the same meridian, which I could have made clearer, oh well).

If you now start walking east, you’ll move along the A meridian until you’re back at AX. You can keep doing that forever, but you’ll never get to CX going east or west.

(The answer would be the same on a torus, and it’s easier to visualize.)

Note: I am always using what Brian called intrinsic coordinates.


Both do have something in common: They are non-orientable.


True! In fact, if you cut a hole out of Klein bottle it becomes a moebius strip!


I knew you (and a number of other people here) were math deities, but I had no idea this was your specialty. So cool!

This seems like an important distinction. Thank you!


Right. I am still confused.

Perhaps this will help explain where my confusion lies:

Chess in a Klein Bottle

It’s basically FIDE chess played on an eight by fourteen board with wrapping. The literal twist is that the pieces move as though a11 has been glued to h12, b11 has been glued to g12, and so on. (That bit is unhelpfully explained on a separate page: Chess in a Moebius Strip.)

Now imagine that there are no pieces preventing the rook from moving an infinite number of spaces north or south. In order to move to the square it starts from it must first pass through a11, which is immediately followed by h12, and three spaces beyond that is h1, right? At that point it is halfway to home, and it is directly east or west of its point of origin.

Is the maker of Chess in a Klein Bottle using extrinsic coordinates? If not, what is going on here?