4d sponges

[alkaline]

An interesting property of sponges in the fourth dimension is that they can pass through each other. This is because two non-parallel lines can move past each other in the fourth dimension. Just imagine the sponge as a bunch of connected lines. Imagine one resting on top of the other, then imagine when it's pressed downwards, each of the lines of one sponge "roll through" the lines of the other, so that they pass through the same space. This property of lines has many far-reaching consequences in the fourth dimension.

[Polyhedron Dude]

I've considered this 4-D sponge property some time ago by thinking of 4-D crystals with "airspace" within the lattice, these solid 4-D crystals could be pushed into each other! Same thing with the 4-D equivalent to a wire screen (the screen looks 3-D, but with a slight thickness in the 4th dimension) - if you hold this screen - and dropped another on top - it will fall through it.

Polyhedron Dude

[Keiji]

two non-parallel lines

Don't you mean non-parallel PLANES? Non-parallel lines can move past each other in the THIRD dimension.

[alkaline]

two non-parallel lines

Don't you mean non-parallel PLANES? Non-parallel lines can move past each other in the THIRD dimension.

I guess that statement is too general to defend. If you take two infinite lines that are parallel when projected into one plane but non-parallel in another, when one line is brought towards the other and passes by it, then they will cross. i really need a picture to show what i'm thinking of here.

[Keiji]

Er... what?

And if you needed a picture, you can use the Attachment Mod

And while we're on the subject of mods, that's a very nice forum logo you have there

[alkaline]

oh i wouldn't have a problem putting up a picture, the problem is drawing it then getting it into the computer. Maybe sometime i could draw it in a paint program, but it's always more laborious to do it that way.

As far as the forum logo goes, that was included with the forum :-)

[Keiji]

I was joking, couldn't you tell that

[alkaline]

maybe you could create a picture for me (since you created the ones for the subject of getting lost in the dimensions). Make an infinite line (either a cylindrical or rectangular line) that crosses the image at an angle. Make a second line, a short measure upwards, that is rotated 90[sup]o[/sup] from the first line. Draw an arrow from the top line down to the bottom line, along the "line of travel" that the top line would move along in order to intersect the bottom line. Position the lines so that this "line of travel" is approximately in the center of the image. I hope that this wouldn't be too hard to create.

[Keiji]

What? :? :?

[alkaline]

I went ahead and created an image in povrary to illustrate what i was talking about:



There is no way to get those lines past each other without rotating them to make them parallel. In the fourth dimension, no rotation is necessary, only shifting.

[Aale de Winkel]

When two lines are moved in a direction which connects them they collide in any dimension.
the two shown lines can freely move when the movement is parrallel the plane that is spanned
by the lines directional vectors ie:
l[sub]1[/sub] = x1[sub]k[/sub] + λ v1[sub]k[/sub]
l[sub]2[/sub] = x2[sub]k[/sub] + λ v2[sub]k[/sub]
the free move plane:
p[sub]1[/sub] = x1[sub]k[/sub] + λ v1[sub]k[/sub] + μ v2[sub]k[/sub]
as long as there is no λ and μ such that x2 lies in this plane.
the lines are free to move in any direction given by:
λ v1[sub]k[/sub] + μ v2[sub]k[/sub]
for any λ and μ

Boys this is most basic vector algebra!

[alkaline]

well not all of us have had vector algebra yet. As for me, it's been a lot time since i've vector math. In any case, what i was trying to say is that in realmspace it's not possible to get the blue line on the "other side" (bottom side) of the red line without intersecting it. In tetraspace, the blue line can move towards it, shift into ana or kata, move down more, then shift back into the original realm, and now it is below the red line but back on the original path. It never had to cross it.

[Aale de Winkel]

Yes crossing the lines would take a move non-parallelel to the given plane.

[alkaline]

back to the original subject of sponges, the property of lines being able to pass each other means that sparse structures can move through each other without there being a hole anywhere big enough for either. Linked networks can have whole subsections migrate internally. In the brain for example, you could have a module of interconnected neurons move from one side of the brain to the other. In our 3d brains, networks couldn't do this - they would have to disconnect themselves first.