2D Universes Readings and Videos
Experts think that the universe has many more physical
dimensions than we directly experience. In order to try and understand
the idea of more physical dimensions, we will step back and try to
address an easier, but related question: How could a 2D creature
understand the 3rd dimension?
In the process, we will see some ideas that at first appear counterintuitive.
We will look at 3D objects in new ways in an attempt to develop
visualization skills that we will need in order to understand the 4th
physical dimension and the shape of the universe.
We previously explored Homer's transition.
For now, assume that the Simpson's really were 2D creatures living
in an x-y plane of some blackboard, as Dr. Frink suggests,
and that Homer and Bart had made the transformation to 3D creatures.
While a 2D Marge can't
really understand the 3rd dimension and would feel like there isn't any
room for another dimension, she could
see weird behavior occurring that suggests
that the 3rd dimension exists (for example the "wall" that Homer
disappeared into could not be explained using only 2 dimensions).
2D Marge wouldn't be able to comprehend the concept of
depth or an entire 3D Homer, since only 2D pieces would make sense to her. In fact, 2D Marge can only pass
others in her space by jumping over them, like a leap frog, since there is no depth to pass in front of or behind them.
Taken from Hyperspace & A
Theory of Everything, by Dr. Michio Kaku
Taken from David Henderson's Experiencing Geometry in the
Euclidean, Spherical, and Hyperbolic Spaces
Watch this video:
The Shape of Space
This animated video produced by The
Geometry Center introduces the two-dimensional space of flatland, looks at possible shapes for flatland from the perspective of three dimensions, and represents those shapes of space in two dimensions. Then the animation uses the same kind of representation to look at possible shapes for three-dimensional space. Viewers are taken on a ride across the boundless three-dimensional surface of a three-torus and a four-dimensional Klein bottle. As viewers see these
imaginary universes from inside the spaceship, they experience the illusion
of seeing copies of the universes.
Read p. 349 in Heart of Mathematics on the Klein Bottle
surfaces with no edge [https://youtu.be/T0rZ41jV0rI]
This video I created explores tic-tac-toe on surfaces with no edges
Flatland: The Movie -
Official Trailer [http://www.youtube.com/watch?v=C8oiwnNlyE4]
Flatland: The Movie is an animated film inspired by
Edwin A. Abbott's classic novel, Flatland.
Set in a world of only two dimensions inhabited by sentient geometrical
shapes, the story follows Arthur Square and his ever-curious granddaughter
Hex. When a mysterious visitor arrives from Spaceland, Arthur and Hex must
come to terms with the truth of the third dimension, risking dire
consequences from the evil Circles that have ruled Flatland for a thousand
Mathematics... reason... imagination... will help reveal the truth. -Arthur Square
Think about what a 2D creature like Arthur Square or a 2D Marge
Simpsons would see as spherius (a sphere) passes through their
2D plane of existence. They can't see above or below the plane because
they are limited to their views of being inside.
We would see (in her plane) a point turning into
a circle which gets larger and larger
then smaller and smaller until it turns into a point
and finally disappears.
However, when we look at a building from the front, we see just one side or
face of it. In order to see the entire building, we must walk all
the way around it.
Just as we cannot see an entire building all at once,
Arthur Square cannot
see an entire circle or 2D curve all at once.
As a sphere passes through his plane of existence
Arthur Square (standing to our right of the sphere)
might see a point and then a curve (the part of the circle we see that
would be visible to him) that gets closer to her, then farther from her,
then turns into a point that finally disappears.
This would seem very strange. In fact, this behavior does not make sense
in 2 dimensions - objects don't just appear and disappear. A 3rd
dimension would be needed to explain the behavior.
Click on the link below.
Think about Arthur Square standing on the right and think about what
she would see.
Use the Play Button
Davide Cervone's Spheres Sliced in 2D.
Notice that you can use the other buttons to play it slide by slide
or to rewind or replay the movie.
Next, go through the following movie by
using the play button.
explanations to further help you develop
visualization skills that we will need in order to
understand the shape of the universe.
Davide Cervone's Rotating Cube
From Davide Cervone's Orthographic and Stereographic Projections
"This movie shows how projections of a cube can be made more understandable by looking at a sequence of images as the cube rotates above the plane of the projection. The various shadows make it clearer which parts are in front and which are behind.
We begin with a view of the 3D cube above the projection plane, with the light source above. The relation between the shadow and the rotating object is quite clear. After seeing the cube rotate, we move to a new viewpoint where all we can see is the shadow; we have to imagine the rotating cube from these 2D images. "
2D Marge would think that the shadow behaves in
ways that are impossible since the intersection movements are
unlike anything that she would have ever experienced.