Widely used thought device when trying to free the mind from the constraints of our 3D thinking or something like that.

It is basically a two-dimensional universe, with 2D beings, who can't perceive the third dimension.

You can then suggest allsorts of experiments to prove that this third dimension does in fact exist.

Yep it's just like living as a pixel in the middle of this computer screen :)

Onwards, to more crazy mathematics

An inhabitant in Flatland would perceive a sphere, an angelic being from a higher dimension, passing through its plane as a circle growing larger and smaller.

Somewhat abbreviated title of Flatland: A Romance of Many Dimensions, a 19th century text about living in a world of two dimensions. It was allegedly written by Flatland inhabitant A. Square and edited for our perusal by English scholar Edwin A. Abbott, but it doesn't take the Chief Circle to determine the true story behind the book's authorship.

In addition to describing the two-dimensional realm, the book introduces us to various zero, one, and three-dimensional creatures, and dares us to dream about the possibilities of the Fourth Dimension and higher. Of course, it's also a very thinly-veiled satire of Victorian society. :) Take from it what you wish...

Go ahead and read it yourself.

Flatland is a beautiful story about how the Universe would appear from the point of view of 2 dimensional creatures. The narrator of the story is a square and the world he lives in is called Flatland. All inhabitants of this world are polygons. Low caste beings are irregular and the social status of a being is denoted by the number of sides he has. For example the square, being 4 sided, is a member of the middle class.

Edwin Abbot, who wrote the story, explores many interesting aspects of how the world would appear from such a perspective. For example, all objects would seem like lines. Though this may seem counter intuitive it is true and can easily be verified by keeping a coin on a table and bringing one's eye down to the level of the table. When the line of sight is exactly along the surface of the table the coin will appear to be a line. In the story he also explores how the world would appear from the point of view of one dimensional creatures (lines) and talks about the Universe from the point of view of a point! This point of course cannot perceive anything else and thinks the entire universe just consists of him and nothing else. Abbot ends by daring us to dream of dimensions higher than 3. Truly an interesting book...

Also a very technical skateboarding style. Rodney Mullen is the undefeated king of flatland. Flatland tricks are quite often done totally still. Though mobility is often employed in tricks, high speed is an extreme rarity.

To be succesful at flatland it is a must to be able to blance Kaspers and manuals impeccibly. Long strings of flips, Kaspers and manuals are common in flatland. Grinds are less common as they are usually held on truly flat, barren concrete areas.

Flatland is a little-known, ever-growing way to ride a BMX bike. It is where one rides a bike in a parking lot or other flat,paved area and attempts to balance his or her self in different positions on a bike. This can be achieved in many ways, while moving forwards, backwards, standing still, or even upside down. While it is an activity enjoyed by many people of every age, race, and social status, it is still seen by the man as a hazardous activity and results in many cases of being removed from a riding area.

Flatland is normally done on BMX bikes much smaller than normal, however the bikes are smaller because it makes it much easier to manuever around the bike.

Despite the fact that most people have yet to hear of flatland, it has been around for years. It originated in the early 70s, a few years after BMX racing began. Kids who raced became bored with just racing and decided to look for other things to do on their bikes. And with that, flatland was born.

This writeup assumes you know what flatland is. If not, read the above writeups, or rather, the book.

There's a problem with flatland. It's forgetting a few things. Small things really, not that important. Just the basic laws of physics. Sure, there's squares, triangles and other things (which happen to be living), but how do they move? What are they standing on? Which brings me to the next point, what happened to gravity, anyway?

Poof. Add gravity to flatland, and observe the results. Things are clustering together, while empty space will exist in other places. After millions of years, you will have great balls of matter. Perhaps they'll even circle around each other, you know, in orbit. Congratulations, you've just created your own universe in 2D, complete with gravity and solar systems!

Now this is the point at which I usually put my flatland up straight. Up straight as in, when you have a cube, your flatland will be parallel to the plane on the cube which has width and height. Too vague? Just put a piece of paper on its edge on the table. There's your flatland.

Now, wait another million years. Look at that? What's that? It looks like a being1.. With eyes and arms and legs... But it's not like a human. Its eyes are at the top of its head, so he can look both ways by bending his head. It has one arm in the front, and one in the back. It's walking to the left like it's walking on stalks. Look! There's another one! But what are they doing? They can't go past each other, but one simply goes over the other. This is done quite often, even among strangers, since it's the only way to get 'past' someone else.

The being continues its walk. It's walking to what looks like a house... except that the door opens upward, like those fancy sports cars. But what's that? Why is the left wall so much wider than the right one? Well, because the right one isn't bearing any weight. If it was, it would collapse as soon as you opened the door. This house also needs a rope from the top to the left, to keep it in balance. People occasionally walk over the rope to get to the other side of the house, since there's no other way. Walking over the rope isn't hard, since you can't fall off to the sides, you can only slide down. If people want to go over from the other side, they have to use the notches to climb up the house.

So far so good. So, what's not possible in this world? Well, pillars holding the roof up (You couldn't get past them), subways (You couldn't hold the roof up), fast transportation would be very hard since you'd need a system to make every person in front of you jump up the very moment the train is coming, so the train can drive underneath them, but actually people themselves are impossible. You see, we have blood vessels, but imagine our 2D people having blood vessels. Their skin would instantly fall off since there's nothing holding it up anymore. The blood vessels cut the link between the skin and the muscles.

So what makes this straight up flatland better than the original? I feel it's more natural. It sticks to more laws of physics, and you can imagine a community to develop there, with its own rules and etiquette. Also, it is required for my future writeup about width, height and a timeline. Just the existence of gravity makes for a lot more thought experiments you can do, for example the next one.

Imagine a formula for the strength of the muscles of a 2D being, The width of the muscle times the height of the muscle is the maximum force exerted by the muscle. Now place this 2D being in our 3D world. We have different laws for the muscle power here, since we have 3 dimensions. Our formula would be length•width•height=force. Which means our being is in trouble, since its length is zero. Thus, in our 3D world, its maximum force is also zero. In the same way, if a 3D object happened to intersect the 2D world of the being, he would not be able to lift it.

Now imagine a 3D object falling down in the 2D world. It hits the floor. But the floor is 2D, and is therefore unable to exert any force on the 3D object, which means the object totally destroys everything 2D it goes through.

Now relating these things to our dimensions and the fourth one (which is what flatland is most useful for), We can state that a 4D object intersecting our dimension would be immovable by us, since we have no length on the 4th dimension axis, and thus are unable to excert any force on the 4D object. This object would, to us, classify as an immovable object. Then we have a 4D being, waving his hand through our dimension. Everything it comes into contact with would be moved or destroyed, since it's unable to exert any force on the hand. This would classify, to us, as an unstoppable force. Ringing a bell yet?

So what happens when they meet? Well, that depends on how big our unstoppable force is in comparison to our immovable object. In general, the object would take a part of the kinetic energy of the force. There would not be an endless transfer of energy, time would not stop2, no blubberjugs, no alternate dimensions (well, one of course ^^)

Another use of this land is this one: In my writeup How to draw a 4-dimensional object, when you're limited to 3 dimensions I wrote in the notes:

You can't actually see your tesseract. That's because you drew it in a solid cube. But a 4-dimensional being could see it, since they can observe the whole 3D world at once. Don't bother wearing clothes. You're just a mere 3D being. They can see right through your clothes (or rather, into your clothes). And 4D clothes are way too expensive.
This can be seen in flatland. You can see inside the 2D being, and inside a house when the door is closed, even though the 2D beings can't.

As you can see, the 2D world can often help you understand the fourth dimension better. I'll leave you now to ponder your self-made world (With gravity!). Keep an eye out for that writeup about 2 dimensions and a timeline. It's going to be interesting. The writeup has been noded under the name of: Spaceland: A Romance of Two Dimensions and a Timeline.

1. We'll get to the impossibility of that later, no worries.
2. Since the transfer of energy would be instant, and endless, one might think time would stop.

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