The Shape of a 2D Universe
A 2D creature could live on many differently shaped universes,
such as the surface of a sphere (a spherical universe) or
a blackboard (a Euclidean universe). Here are some additional possibilities:
Torus - A Euclidean Universe
A torus is a mathematician's name for a donut.
The surface of a torus is the topological space found in
old-style video games such as Pac Man,
where a spaceship goes off the right-hand side of the screen only to
reappear on the left, or off the top to reappear on the bottom.
If you take a sheet and try to glue the left edge to the right edge
(the single arrows tell you to do this)
and the top edge to the bottom edge (the double arrows tell you to do this),
the paper will crumple up and you'll get a big mess.
2-Holed Donut - A Hyperbolic Universe
But, we can do this identification of a square
with a stretchy piece of rubber. First we identify the left and right hand
side in order to form a cylinder.
We use these arrows and the points that correspond to
glue the top and the bottom together and this forms a donut or torus.
Search the web to find the measure of one
interior angle of a flat octagon ______________
||Here we glue the side with a number on it with the
side that has the same number on it.
It is an exercise in visualization skills to see that the resulting figure
is a 2-holed donut. Here are directions for
2-holed donut that may be helpful for visual purposes.
To understand why the laws of Euclidean geometry that you learned in
high school do not apply to the 2-holed donut, we can look to see whether
octagons will tile the plane in the same way that Escher created his works
of art. So we would like to know whether we can take a certain number of
octagons (instead of birds like Escher took in the Euclidean
Sun and Moon work,
or angles and devils like Escher used in the
distorted hyperbolic Heaven and
and put them together around a vertex in order
to form 360 degrees.
How much of
360 degrees is left over when two octagons are placed side by side like
What happens if you try and place three octagons together at a vertex?
Do they fit into 360 degrees? __________________ If so, we could
create a flat tiling work of art (like Escher)
without distorting them. If not, then we must
distort them in order to create a flat work of art.
We can create a hyperbolic 2-holed donut by using a hyperbolic
octagon with 45 degree interior angles. Eight of these glue together in
hyperbolic space to form 360 degrees at a vertex and so they tile the space.
The laws of hyperbolic geometry hold for this work of art and we now
understand some of the issues that Escher faced.
Klein Bottle - A Euclidean Universe
Notice that this identification
labeling of a square looks similar to the one that resulted
in a torus, but the top and bottom edges are glued with a twist -
a reflection in the line between the midpoints of the sides.
Just as a 2D Flatlander could not imagine how to construct a cylinder out
of a piece of paper, we have problems
figuring out how to put together this square, because when we label
corresponding points (such as the green squares, which are the same
in this space because they are identified via the reflection in the
there seems to be no way to glue them together.
However, an inhabitant of 4-space would have no trouble because he would
have enough space to glue the edges together.
The space that we have represented here has a nasty intersection when
the slinky passes through itself.
Our gluing instructions give no hint of this and in fact,
the intersections only arose when we tried to glue the space in 3 dimensions.
In order to properly see an actual model of the glued square,
we would really need a 4th physical dimension.
In the 4th dimension there is enough room to glue the edges together
without creating intersections, and it is here where the glued square
Try forming the Klein bottle slinky.
There is only one way to for us to put this space together
when we identify like-colored points using a slinky
by having the slinky pass through itself.
Where you able to form the Klein bottle slinky? ____________
Both the Klein bottle and torus satisfy the laws of Euclidean geometry from
high school because we can tile the plane with a square - four 90 degree
angles come together at a vertex to form 360 degrees. If
we sketch what a creature living in the place would see in all directions,
this is called a tiling view.
The top left is really just below the bottom right
in this Klein Bottle universe because they are identified by the mirror
reflection. However, if we look off to the right or left, we see
exactly the same image instead of the reflected one.
Play on one computer with a partner:
||Experience what it is like to live on a Klein Bottle by playing
Klein Bottle Tic-Tac-Toe.
* Download the Torus Games for Mac OS 10.3.9 or later (2.5 MB) and then
double click on the file titled Torus Games.
* Using the Game menu, select Tic-Tac-Toe.
* Using the Options menu, change to Human vs. Human.
* Using the Topology menu, change to Klein Bottle.
* You are allowed to scroll the board
(once a square has been labeled X or O, you can click on it, hold down,
and move the board around to see the identifications)
in order to help develop your intuition. Try this. Notice that
the top left square is really just below the bottom right square in this
Klein Bottle universe.
* Use Esc to go to the options, and click Erase
Tic-Tac-Toe Board under the Game menu in order to bring up a new game.
Winner - best out of 3______________________
Next play without scrolling the board. It may help if you sketch a
tiling view on a piece of paper as you make your moves:
|No Scrolling Winner - best out of 3______________________