Dimensional Views [ 01.10 ] Looking Inside 2D - Flatland

Published: 22.02.2006
Updated: 22.05.2008
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2D World - Inside

Todays visit is adressing the centre of our 5 dimensions,
the
2nd dimension.

As we live in 3 dimensions, we have to be aware of the limitations of having only 2 dimensions:

  • There is no space, only plane land.
  • 2D creatures are flat bodies in a flat land.
  • Their bodies are bordered by a line around them.
  • They have only 2 degrees of freedom:
    1. forward - backwards
    2. right - left



Look Inside 2D, which is called: Flatland

3D View:

Let us get introduced to Mr.Star.


and to Mrs.Circle, a friend of Mr.Star.



This is how we 3D creatures see a 2D creature. We look at them from a 3D space position, from top.




2D View:

In 2D flatland both have a different view from each other:

Mrs.Circle recognizes Mr.Star:
and Mr. Star recognizes Mrs. Circle:
  • Both face only the front line [1D] of each other.

When they walk tight together they are seen by another 2D creature like this:




Notice:

  • A 2D creature in 2D land sees all 2D objects in 1D quality (as a "line of...")
  • A 3D creature in 3D land sees all 3D objects in 2D quality (as a "picture of...")

If you like you can go on with 1D creatures
and/or 4D creatures...

Help:

  • To draw a line [1D] you need a paper [2D]
  • To hang up a picture [2D] you need a wall [3D]
  • To realize space [3D] you have to be in [4D]



Back to Flatland

Mrs.Circle is alarmed. Mr.Star has disappeared.
She asked the neighbours and they told her the police came the day before und took Mr.Star to jail.
He was accused of treason.

Here we can watch Mr.Star behind jail lines in a really small square cell.

Indeed, Mr.Star had had a visit from a 3D creature and he could not keep his mouth shut afterwards and told everybody everything he had heard about the real world.

Now let's help Mr.Star. We peel him off the 2D world, and redeposit him outside jail.

  • This feat, which is quite ordinary in 3D, appears fantastic in 2D.
  • No one in the 2D world understands what the up direction means.

Mr. Star is very thankful, but deeply desperate of what had happened to him.
We tell him to keep cool and meditate regulary.




We [3D] Do Look Inside [2D]

The internal organs of a 2D creature would be visible to us. It would be trivial to reach inside a 2D creature and perform surgery without cutting the skin.

  • Viewing this 2D flatland, notice that we [from 3D] are omnipotent.
  • The 2D creature cannot hide from us.
  • He would see us as having magical powers.



The Limits

After his adventures at jail Mr.Star found back into normal life and became a very famous scientist.
His wish for looking inside had grown so strong - he had started to cut the bounderies of 2D objects to get inside.

  • Unfortunately he never succeeded in getting inside of anything.
  • His efforts always resulted in producing devided parts of his objects, which he tried again and again, getting smaller and smaller parts.....



"I CALL our world Flatland, not because we call it so, but to make its nature clearer to you, my happy readers, who are privileged to live in Space.

Imagine a vast sheet of paper on which straight Lines, Triangles, Squares, Pentagons, Hexagons, and other figures, instead of remaining fixed in their places, move freely about, on or in the surface, but without the power of rising above or sinking below it, very much like shadows--only hard with luminous edges--and you will then have a pretty correct notion of my country and countrymen. Alas, a few years ago, I should have said "my universe": but now my mind has been opened to higher views of things..... "

This is the introduction of "Flatland: A Romance of Many Dimensions", a book written 1884 by Edwin A. Abbott. (an early and copyright free edition of the book is online available and we hope you enjoy it very much)

Flatland [inside 2D] characterises the adventures of A Square in Lineland [1D] and Spaceland [3D] and it offers a great view on what it means to live in 2 dimensions and to leave their limits.

Sources

© Christian & Carla Geerdes, Editors HereNow4U online magazine

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