4-D shoelaces?

[zero]

As it so happens, I still tie my shoelaces the way I leaned in kindergarten. But what would one learn about securing one;s shoes in a hypothetical 4-dimensional kindergarten?

Since the existence of knots appears to be unique to 3 dimensional manifolds (based on topological considerations), what alternatives are there for connecting things together in 4-D?

[Keiji]

In 4D, you can tie knots in flexible finite planes... and that's all I know about that

3D "hooks" can be securely fitted together in 4D, if one (but not both) of them is lathed in the fourth dimension.

[houserichichi]

4d staples? Scotch tape?

[Keiji]

Staples would be exactly the same, and they would penetrate a swock rather than paper, accounting for the added dimension. Scotch tape would be in a thin finite realm as opposed to a thin finite plane.

[zero]

Perhaps Velcro is transdimensional.

[wendy]

To create a link by using two sphere-surfaces, the sum of dimensions for which the sphere be solid is N+1. For example, a hollow circles is 2D, and two hollow circles link in 3D, since 3D+1D = 2D+2D.

You can generally fake a hollow line, by making the ends larger than the interior. In 3D, hollow lines and hollow spheres (1D+3D) add to 4D, which is greater than 3D.

In 4D, a simple link sums to 5D (eg 2D + 3D). Since 2D = line (hollow circle), we see that 2D+2D = 4D, does not link.

A lace can be made to go through holes, but in order to be made fast at the end, you need something that equates to a hollow 3-sphere. You could use something like the tassle one finds to hold hats on.

I am not really sure if a latrix (1D-cloth) is really the answer here. You must understand that the shoe needs a hedric (2d-manifold) opening, to separate the surface, and hedrices = 2D-cloth, being hollow spheres, would indeed knot (since 3D+3D > 4D)

So i should think that tieing shoes in 4D is pretty much the same as in 3D, except that one is using hedric threads (ie 2-cloth), and pretty much a prism of ordanary boot-lace tying.

In terms of velcrose, one notes that it is a paper with eye + hook ends, this can work in any dimension, since the thing is simply lots of little hooks (of any fabric), catching into complementry eyes (such that the hollow spheres add to a higher dimension than all-space).

Scotch tape works on the principle that the link is glued to both ends. It clearly works, but the link is not mechanical, but chemical.

[papernuke]

how would a 4D gun work then? the bullet i understand would have to be 4D to hit a four dimensional target. but how would the barrel work? and wouldnt it be harder to aim because there is one extra dimension to be covered?

if a 4D being shot a bullet into the third dimension, how would it be like to us? would we preceive it as a normal bullet? or would it be warped?

[pat]

A 4D gun would not be that tough. The barrel would be the hollowed out extrusion of a three-dimensional ball. There would be only one way to point the gun. It would be tougher to hit your target because there is one more degree of freedom for it to move or you to miss.

In 1-D, you pretty much can't miss your target. In 2-D, it's much harder. In 3-D, it's even harder. In 4-D, it's even harder. That said, I think the difference between 2-D and 1-D is much greater than the difference between 3-D and 2-D is much greater than the difference between 4-D and 3-D is much greater than the difference between n-D and (n-1)-D for n > 4.

[zero]

So i should think that tieing shoes in 4D is pretty much the same as in 3D, except that one is using hedric threads (ie 2-cloth), and pretty much a prism of ordanary boot-lace tying.

Hmm.

It's easy enough for me to see the generalization of an eye and hook (thank you), but the above will have me tripping all over my untied laces. Until I have time to learn the terminology better, I'm sticking with the velcro. It's kindergarten all over again! Except now I'm short a dimension.

[zero]

how would a 4D gun work then?

Or a 4D trigger lock. I'm concerned for your safety.

[kingmaz]

A 4D gun would not be that tough. The barrel would be the hollowed out extrusion of a three-dimensional ball. There would be only one way to point the gun. It would be tougher to hit your target because there is one more degree of freedom for it to move or you to miss.

In 1-D, you pretty much can't miss your target. In 2-D, it's much harder. In 3-D, it's even harder. In 4-D, it's even harder. That said, I think the difference between 2-D and 1-D is much greater than the difference between 3-D and 2-D is much greater than the difference between 4-D and 3-D is much greater than the difference between n-D and (n-1)-D for n > 4.

Perhaps, one could imagine like this. In 3d moving the gun up/down and left/right has the effect of causing a miss. The fourth dimension is equivalent to moving the gun backwards and forwards along the line of fire. This would have no effect whatsoever in 3d but would cause a miss in 4d*.

*assuming some coincident relationship between 2 of 4 axes, I guess.

[Keiji]

Perhaps, one could imagine like this. In 3d moving the gun up/down and left/right has the effect of causing a miss. The fourth dimension is equivalent to moving the gun backwards and forwards along the line of fire.

No, not really. The fourth dimension would just be another direction to cause a miss in; it isn't parallel to any of the existing three dimensions.

[kingmaz]

Removed full last post quote ~Keiji

Yes, you're right, they would all be mutually perpendicular. What I was saying was to try aid visualisation of a 4d effect in a realmic sense.
I would suspect that I was using an artefact of 4d projection into 3d to illustrate the point, which isn't really valid.

[quickfur]

Removed FLPQ ~Keiji

In projection from 4D to 3D, from a 1st person POV, when the gun is fired the bullet travels from the bottom of the projection volume towards the center, shrinking rapidly as it goes, with limiting point at the horizon (which in projection is a flat plane bisecting the projection volume) assuming you aim parallel to the ground, and that you hold the gun beneath eye-level. So, disregarding aiming up/down for a moment, this gives you already 2 dimensions to aim in (covering the horizon). If the object is also moving vertically, then you have to aim vertically as well, so you've to position your crosshairs in 3 dimensions.

An equivalent way of thinking about it is, in 3D, you're essentially trying to move your target crosshairs over a 2D plane (for simplicity, say it's a square) to the point where the target projects to. So there's left/right and up/down to consider. In 4D, you're trying to move your crosshairs to match a (possibly moving) point in a 3D cube, so there's left/right, front/back, as well as up/down to consider. So it's much harder to aim.

[PWrong]

The shoelace problem is really interesting. It's true that sheets can be tied into knots in 4D. Presumably if we lived there we could easily find one that could be made tight enough to keep the shoe on. But it's not immediately obvious that tetronians would need shoelaces at all. If they did, maybe they could do the same job with strings instead of sheets by restricting the strings to a realm.

Let's think about shoelaces really carefully. A 3D shoelace is the solution to the following problem.

A leg is roughly a solid cylinder ending in an odd shaped foot. It doesn't matter what shape the foot is, but it may not fit into the same size holes as the rest of the leg. A shoe is (topologically) a cylinder of material, roughly circular at the top and shoe shaped at the bottom. One end attached to the base of the shoe and the other end in the open air. Now the problem is you can't fit your foot in the hole, so you cut a long rectangular piece out. This allows you to bend the material and fit your foot inside. But now that your foot is inside, the shoe is too loose. How can you tighten the shoe around your ankle without taking out your foot?

The solution is to cut 2n holes in the shoe, with n holes along each side of the rectangle. You take a long string, and thread it through all the holes, crossing it over itself for each hole. Finally you have the two ends coming out of the top holes, which you pull on to tighten the shoe, and then tie in a knot.

Note that most of the "lacing", not including the knot, takes place in a plane with a bit of thickness to allow crossings. We should keep this fact in mind and use it in 4D.

We can probably think about 4D shoelaces by adding one to the dimension of every object. So you have a spherindrical leg and a spherindrical shoe with a square prism or a cylinder cut out. You need to tighten in two directions. I think we'll stick with a square prism and put n holes down each side. The holes have to be 1D to allow the 2D shoelace with its 1D ends to pass through it.

Important questions:
1. If you take it out of the 2D shoelace and straighten it out, what shape should the sheet be? A square? A long, thin rectangle? A disk? Maybe a square cross shape?

2. What lacing patterns can you get inside the prism?

3. Would it be better to cut out a cylinder?

4. Could they do it easier with 1D shoelaces by restricting them somehow?

[Keiji]

I can't really visualize this right now, but I do have an answer to the last of your questions. Restricting the shoelaces to a realm would not be an easy task. Even if there was some kind of lattice through which the laces were threaded, it would either be too spacey to stop laces from passing each other, or it would be too dense to let someone tie or untie the laces without deconstructing the lattice itself.

[PWrong]

I suspect there'd be a lot of complexity in lacing sheets.
http://www.fieggen.com/shoelace/lacingmethods.htm

We have a square prism with n rows, and 4 slits per row. I now think the sheet would have to be a long rectangle, with width slightly smaller than the length of each slit. The sheet can look like it passes through itself, since it's really crossing over itself. There'd be a huge number of unique lacing patterns available with this method.

The cylinder alternative is looking pretty good. We'd have a single slit per row (unless this would cause it to fall apart, I don't think it would), and a large square or circular shoelace. I think this might restrict the lacing patterns somewhat.

[Mrrl]

Okay, top of the boot is hollow spherinder. It horisontal section is the leather sphere in 3D. If I cut circular or square hole in it (i.e. cyliner or prism in the boot top), it will not help me to put the boot on the leg: sphere is rigit enough and one hole will not make it wider. One way I see is to make three cuts from the "front" pole to equator of the sphere (by meridians 0, 120W and 120E), so we get half-spherinder with three petals. When we unlace the boot, it will be easy to put it on. But then we need to stitch petals again.
I think that the best way is to make one or two rows of buttons and holes on the edges (3 or 6 rows in total). They will work in 4D as well as in 3D, and you don't have to tie anything in this model.
But if you still want to use laces, it's possible to use some clips on the ends. So you lace boots by 1D laces (using same 6 rows of holes), fix ends of laces by clips and you are ready to go...
Same construction works for 4D shirt. If tetronian has three arms, he may use shirt with three parts that meet at the front. And 6 (better 7) rows on buttons can hold these parts together.