two dimentional chains

[fatrat97]

Hey, i'm new here, so sorry if you already knew this:

I was just reading the tetraspace explanation (http://tetraspace.alkaline.org/page4.htm) and noticed it said 2-dimentional chains aren't possible. Of course they are possible. Each link would be like the letter "S" and would hook onto the ends of each other.
S
S
S
(except those S's aren't hooked together) :wink:

[wendy]

Topologically connected chains (ie where rubbery links will not fall apart), are possible in 3d and higher.

The most common implementation of a chain in 2d is the rod+cavity chain, formed by rods with knobs at each end + spheres with cavities. One sees the same sort of chain holding bathroom plugs, pens at banks etc, since they are not as rough as chains with links, although not as strong.

Chains formed out of double-end hooks are not useful where the hooks tend to drift apart. When one can stop that, they are quite efficient, even in three dimensions (most train couplings involve some kind of open hook, eg buckeye couplings, or chain and hook couplings).

W

[thigle]

maybe what alkaline meant was similar to the point that you can untie any knot in 4d. similarily, a chain would not be stable because any 2 circles could be freed without tearing them ? so it is possible, but might be not practical, because it could "untie" itself by wiggling motion when used ?

[Keiji]

I was just reading the tetraspace explanation (http://tetraspace.alkaline.org/page4.htm) and noticed it said 2-dimentional chains aren't possible. Of course they are possible. Each link would be like the letter "S" and would hook onto the ends of each other.

That's certainly possible in 2D but relies on gravity to hold it together. It could be broken simply by lifting one of the hooks. Chains on the other hand are only possible in 3D because they consist of several interlocking rings. In 4D or above, the chain would simply fall apart.

The necklace-type chain is also only possible in 3D, as it consists of a string threaded through several rings. Again, in 4D or above, it would fall apart.

[Neues Kinder]

I think 4D chains are possible. Of course 3D chain links would fall apart in tetraspace. But that doesn't mean that you couldn't make a chain in tetraspace. It's just like saying it's impossible to make a 3D counterpart of a 2D chain just because you can't hang S-shaped pieces of paper edge-to-edge. Instead of using 3D chain links in tetraspace, you should use 4D chain links.

[wendy]

A four-dimensional chain made of solid links is certianly possible. The links alternate in dimension, being glomolatric and glomohedric (circle, sphere)

In four dimensions, there is still one kind of hole (hc), but the hole can either be inside the body (ch is what happens when you stick a pin through something), or hc (when you glue a pole or handle on something).

The successive links of the chain are then hc ch hc ch hc. in three dimensions, the links are hh hh hh hh hh

In five dimensions, you have two different kinds of chain: ht th ht th and the cc cc cc cc cc form.

As long as the dimensions of the spheres add to N+1, the chain won't fall apart. so cc is 3+3 = 6, forms a chain in 5d.

There are polytopes where the primary hole is hhh, but this reduces to a combination of ch and hc holes, which i have not fully identified.

W

[Keiji]

Care to explain that post more? It's quite interesting, but I have no idea what you are saying.

[wendy]

When one forms a link or hole, it has an inside-hole and an outside hole.

Consider the common or garden 3d earring. It has one circle, lieing inside the metal, which can not shrink to zero, and another circle outside (eg a peice of cotton used to keep it from being lost.) These circles are spanned by 2d surfaces, or hedrices. So a ring in 2d is hedrix-hedrix.

When you put the earring on, it makes a link with a similar hole in the ear.

In four dimensions, if you made a ring by connecting two ends of a hole, you get a h-type hole. But the h-type hole makes a c-type hole in the air: the result of stabbing the interior of a sphere out. So to make an effective link, you need to have holes spanned by hedrices, and holes spanned by chorices, ie hc and ch links.

In any higher dimension, you can have links with holes that add to N+1.

If you use links and cavities, you can have e-type holes (eg a bar with two bulbus ends), and cavity-type links (eg spheres with little holes for the bar). One find bar-and-ball type chains holding pens and sinks, these are ec-ce chains in 3d, and et-te chains in 4d.

W

[Keiji]

Argh, I'm still confused... this is beyond my grasp

[wendy]

In three-dimensions, there are two kinds of chain

1. The chain made out of circular links.

2. A chain made out of point pairs, and of hollow spheres.

One sees the second kind of chain as a chain of linked spheres usually, usually designed to hold pens to counters, and plugs in sinks.

The hole designation is two bits: the sphere that exists that can't be made to shrink on the outside, and the same for the interior.

So a normal link has an outside circle (h), and an inside circle, (h), and becomes a hh kind of hole. A pair of points o o can be thought of as a line o-----o minus its interior o- -o. Likewise a cavity in a sphere can be thought of as a solid sphere with a hole inside it.

If one has two points held exactly an inch apart, to stop it from escaping, you have to put one of them inside a cavity. This means that we're dealing with a missing edge (e) coupled with a missing body choron (c).

With our cavity, we can see that it's a missing body (c) coupled with a filled in exit for a point: ie one needs an edge to break the formation of spheres from outside (--- ---( inside. So it's a ce hole.

You can make a chain of ce and ec links, eg ce ec ce ec. In practice, one implements the e links with ball-ends and a thin bar, eg 0===0, and make the cavities have holes to let only the bar, and not the ball through.

One can take any 3d chain, ie ec-ce or hh-hh, and make one of the two links into a higher dimension, eg ec-ce => hc-ch or et-te, and hh-hh => ch-hc.

[Keiji]

Thanks, that makes much more sense now!

[Batman3]

Wendy said about 3d chains that a second type was
2. a chain made of point pairs and hollow spheres

Are you referring to a 4d link of a sphere and a circle perpendicular to its surface? 2 circles will not link in 4d: They will fall apart.

In 5d can I expect to find that a sphere and a circle will fall apart but that two spheres can be linked? Weird I usually can't visualizes 5d. Am I doing it this time? Or is it just counting numbers?

[wendy]

A chain in one dimension, can be formed by linking even numbers E to E+3. This chain can be imagined as going in the z-direction, like so:

Code: Select all


    /-----------\   /----------\    /-----
    0   1   2   3   4   5   6   7   8   9
   ----/    \-----------/    \----------/


The links would need to be held together by some kind of imaginary force.

Such a chain would be an ee chain, since there is a missing Edge between 0 and 3, for example.

In two dimensions, yz, one would have to close off either the top links or the bottom links, to give eh linkage. If we close off the top, we get circle links (h), that can be joined by e links, and eh links, eg

Code: Select all

    /-----------\   /----------\
    0       2   3   4   5      7
    \-----------/   \----------/


There is two circle-links through 0-3 and 4-7 and a edge link 2-5.

In three dimensions, x-y-z, we can either make 0-3 and 4-7 into spheres, held together by points 25, or we can make 2-5 into circles. Either way, we make a chain, either ec or hh.

In four dimensions, we simply up one of the letters, by closing one set of links in the w-direction, ie et (1d-4d), or hc (2d-3d).

And so it goes...

[Deltrax]

I've emailed the author a few years ago about this error, as 2D chains are possible. No update yet to the site
A few possibilities as unbreakable (in 2D) chains are seen in this illustratation:

2dchain.gif

2D chain

(1.63 KiB) Downloaded 616 times

There might be a few other variations. Hope someone can persuade the author to update his page (as the last update was more than 4 years ago)

Thanks, Del.

[Keiji]

Ahh, the author of this site would be Alkaline, but he's said he's too busy to check/update the site any more, so he's entrusted it to me.

The reason that page says chains aren't possible is because it's considering chains formed with shapes topologically equivalent to a torus, which certainly aren't possible in planespace. Your type of chain is of course possible, but I think stating that there would be a bit confusing...

[Nick]

I was just reading the tetraspace explanation (http://tetraspace.alkaline.org/page4.htm) and noticed it said 2-dimentional chains aren't possible. Of course they are possible. Each link would be like the letter "S" and would hook onto the ends of each other.

That's certainly possible in 2D but relies on gravity to hold it together.

What if those chains were magnetic? Are we assuming electromagnetism is impossible in 2D?

[d.m.falk]

Gravity has nothing to do with it, but inertia does. Since there is no depth, the ball/peg can't slip up or down, but try to slip past the socket enclosure that's smaller than the ball. A ball & socket 2D chain is practical.

d.m.f.

[Nick]

Gravity has nothing to do with it, but inertia does. Since there is no depth, the ball/peg can't slip up or down, but try to slip past the socket enclosure that's smaller than the ball. A ball & socket 2D chain is practical.

d.m.f.

Actually I was referring to the "S" chain from earlier in the thread.

[papernuke]

But then how would a chain work? You obviously can't stick the chain "around" the being because that would go into the 3D,
so if you put it "over" the 2D being, then it would just suffocate them, unless you somehow put holes in it for air,
and if there were enough holes for air to go though,
then the chain would be too weak.
so in a sense, it wouldn't "work"

[Halfbaker]

Maybe the chains would be used as body armor? One could also use 2D chains to suspend stuff from overhangs, but I see little use for that.

[wendy]

chainmail requires 3-3 links. Essentially, more than linkage is needed.

[Halfbaker]

Sorry, I don't grok what your saying, please explain furtherer.

What I'm suggesting is a ball-socket chain that is laid across the dionian's body like a blanket for protection. One could say that plate mail and chain mail are the same in two dimensions.

[wendy]

In order to weave, each thread must act as a marginix (or surface-divider). You can do this in any dimension, as long as the threads are N-2 dimensions.

In order to knot, all that is sufficient is that the two spaces must span a surface, that is, the sum of dimensions add to at least N-1. So you can't restrain a line with another line, but need a hedrix or better to do this job. A totally bound knot, i suppose, requires the use of closed loops whose surfaces add to a dimension-surface (eg circles = line + line = surface in 3d).

[Halfbaker]

Ok, thank you.

[AmitX123]

Uploaded with ImageShack.us

Refer to the figure above showing 2d links A and B. Assuming that the two links are rigid, it is not possible to "separate" them without resorting to 3rd dimension. Such links can be used to make a chain in 2d.

[anderscolingustafson]

This is one kind of structure that a 2d chain could have in, which it would be impossible to break the chain without either moving the links into the third dimension or breaking one or more of the links.



Also this is a structure that a 2d digestive system might have. Just like how the links on the chain I have a picture of above would hold together do to their shape the two halves of the digestive system below would hold together because of their shape.

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Of course the kind of chain that is made up of rings in which each ring goes threw the middle of the other could not exist in 2d but another kind of chain could exist in 2d like the above one. This kind of chain could also be used to make 2d armor that would be flexible but tough and that would be the closest 2d equivalent to 3d chain mail and armor maid of this type of chain would have the advantage over 3d chain mail of having no holes so that would allow a weapon to penetrate it.

[Pentoon]

Here is a comic strip showing two 2D chains and a 3D being interacting with a 2D being.

[quickfur]

+1 funny