claustrophobic dudes

[Yoshi]

Did u realize that clausrtophobic people will have a hard life in the second dimension?

I was just thinking of that cause my friends is claustrophobic lol.

[Geosphere]

I'm sure tetronians wonder how we get along squished in our ridiculously limiting universe as well.

[Yoshi]

yah, they're probably like
"how could 3-d people live like that"
meanwhile we think 2-d people are squished

[Yoshi]

but dont u think that the gap between 2-d and 3-d is a lot bigger than the gap between 1-d and 2-d?

[Geosphere]

Geometrically.

And I mean that by every mathematical interpretation of the word.

[alkaline]

i think it's the reverse - the gap between 1-d and 2-d is bigger than 2-d to 3-d. That is because you get so much more out of moving up one dimension from the first, as opposed to moving up one dimension from the second. In the first case you're doubling the dimensions, in the second case you're increasing it by 50%.

[Geosphere]

I have to disagree, Alk.

Look at it this way.

Ever ride a Tricycle?

Then a Bicycle. Wow. That bike is a bit harder, with now controlling balance as well as motion.

What if you simply added 1 more dimension of freedom of movement to control?

Well, ever ride a unicycle?

The unicycle is insanely hard due to the combined directions that occur after simply freeing up forward/reverse motion.

[alkaline]

you're talking about balance. Of course, anything with only one contact with the ground will be the least stable, in any dimension. The higher the dimension, the more unstable something is if it has only one "leg".

I was talking about relative size, and relative freedom. After being stuck in a line, being able to move in a plane is a big step up. Getting an extra dimension beyond that isn't as big of a step up, because you already have two.

[Geosphere]

Its geometrically a step up.

Lets define a linear universe 2 units long. There are 3 points to travel to. The end, the start, and the middle, where the 2 units meet.

Planar, it is 2x2 lines, containing 9 points.

Realmic it is 2x2x2, containing 27 points.

3-9-27 is a geometric progression.

If it continues, there's be a whopping 81 points in the tetric version.

First you add 6, then another 18. Thats a big jump in complexity.

[Yoshi]

sure

[Jay]

No, Alklaline's right. It's like this: When you go from 1d to 2d, you go from 2 directions of motion to 4. That's a 100% increase, because you can now go twice as many ways. 4/2 = 2

But when you go from 2d to 3d, you go from 4 directions of motion to 6. That's only a 50% increase in directions, because you can now go 1.5 times as many ways. 6/4 = 1.5

The same thing hold true for 3d to 4d. You go from 6 directions of motion to 8. That's only a 33% increase in directions, because you can only go 1.33 times as many ways. 8/6 = 4/3 or approximately 1.33.

So to summarize:

1d-2d ---> +100%
2d-3d ---> +50%
3d-4d ---> +33%

The ratios between freedom of motions get smaller in the higher dimensions. It's less of a shock to go from 1d to 2d than it is to go from 2d to 3d. It would be the same to go from 1d to 2d, and from 2d to 4d, though.

[RQ]

Yeah, and I thought you said a line was at least 2 units long, not 3, and to go to the 2nd dimension you just add a dot so they are coplanar.

[Geosphere]

I'm not just adding to go coplanar. I'm adding to make equal lengths. Squaring the figure.

[Yoshi]

thats what i thought too

[RQ]

Well first of all, squaring the dimensions is unnecessary to get the next dimension, but even if we do follow your analogy, we would still have a ratio of 1 to 3 which would be the same for from 0d to 1D and from 1d to two, and from 2 to 3, and it seems random, so I don't really think it proves much.

[Geosphere]

Just replace it with 1 figure then. I was creating a fictional length to make it easier to visualize, discussing units rather than fractionals. Sorry if I confused you. Volumetrically, the results would be identical. And the result is geometrically, not additively larger, which is all the point is. Ratio, not addend.

[RQ]

It's still the same thing. The ratios are the same, and doesn't say that there is a bigger gap between 2D and 3D rather than 1d and 2d, since it's a ratio of 1/3. Let us suppose that we have ur linear universe of 1 unit, two points long, the length to area ratio is 1/2 (2 points length/4 points area). Now the volume would be 8 points, 2x2x2. so 4/8 is still 1/2.

The ratio of 3D to 2D as we would see it of a sphere/circle is 4/3 pi r^3 over pi r^2, so it would be 4/3r. The ratio of 2D to 1d is pi r^2/r (no polyshapes in the 1st dimension) which makes it pi r. we can see that pi is greater than 4/3, so the gap between 1d & 2d is greater than 2d and 3d.
Then the ratio of 3D and 1D is 4/3 pi r^2. So the gap between 1d and 2d is 3 pi/4 (about 2.33) times greater than that of the gap between 3D and 2D

[alkaline]

RQ, you're only speaking of the difference in round shapes between dimensions, not a more "general" difference. As far as the volume of a hypersphere in each dimension, I think it maxes out at dimension 7 and after that the volume actually gets smaller the higher the dimension you go.