spiderwebs

[Tamfang]

A 4D spider has to arrange a series of threads in a 3D region such that every point in that region is not far from the nearest thread. I wonder what's the most efficient way to do that.

[quickfur]

No idea. Maybe concentric geodesic domes with radial strands? (Just a naïve extrapolation from the 3D case.)

[anderscolingustafson]

The most efficient way to do that would be for the spider to first spin at least 4 threads that would be attached to each other in the center arranged in some kind of 3d shape. The spider would then go to the center and weave a series of threads that would be the edges of each side of the platonic solid and the original threads would go through the corners of that platonic solid. The spider would then go a little further from the center and again it would weave from one thread to the next to connect them. When it would get to a certain distance from the center it would start to weave a series of circles on each sphere. It would have a group of circles along each of the three dimensions of the sphere as that would be the best way to hold each sphere together. As the spider would continue to make new spheres it would also make new stands to connect those spheres to each other. It would also make a few strands that would come from the main stands and lead into the spheres to give the web strength. And that's how a 4d spider would make a web.

[Tamfang]

The classic radial-plus-spiral pattern simplifies construction, but cross-links are not necessary to the web's function; a simple array of parallel strands would suffice, but be harder to build.

[Tamfang]

Cross-strands also help the spider reach the prey.

[quickfur]

One thing is clear: the web needs to span a 3D region, otherwise the area it covers is too small to have a reasonable chance of catching any prey. (The 3D equivalent of a 2D web is a single strand: not very likely that any prospective prey will bump into it.)

In order to span a 3D region, the amount of thread needed must be proportional to the cube of its average radius. And since the spider will presumably wait at the center of the web, the web must allow easy access from the center to any other point: therefore, it should have a radial access path. For stability, it would seem that radial strands would likely be employed for this purpose.

Once you have radial strands, you need to interconnect them to form a net that can catch prey. This may or may not be in the form of concentric spheres, but that would seem to be the simplest way to cover a 3D region given existing radial strands. As to what form those concentric shells would take, I'd say geodesic domes may be best for maximum stability and simplicity of construction, since a partially-constructed web will already be gaining stability and strength as more and more geodesics are put into place. Geodesic domes are also better by being triangle-based, as that covers a spherical area more evenly than if other types of polygons are used (covering each shell evenly is important to impart strength to the resulting web; otherwise it may acquire weak regions from which stronger regions are prone to tear from).

Of course, just as a 3D spider's web isn't perfectly concentric, so a 4D spider's web may have incomplete or overlapping shells that aren't perfectly concentric. But I'd say that would be the general pattern of it.

[wendy]

It is probable that the spiderweb is woven of hedra (2D patches), not lines.

[anderscolingustafson]

It is probable that the spiderweb is woven of hedra (2D patches), not lines.

I don't see how a spider would actually spin out a hedra of silk when the direction it would be traveling in would be linear. Hedras would require more silk than lines and so would require more work and energy to build. Also if hedras were so much better than lines than why would 3d spiders weave their webs from lines instead of just make one continuous web with no holes in it? A web made of lines would still have a good chance of catching insects so long as it was woven tightly enough but would only require a fraction of the silk that a web made of hedras would require.

[quickfur]

It is probable that the spiderweb is woven of hedra (2D patches), not lines.

I don't see how a spider would actually spin out a hedra of silk when the direction it would be traveling in would be linear. Hedras would require more silk than lines and so would require more work and energy to build. Also if hedras were so much better than lines than why would 3d spiders weave their webs from lines instead of just make one continuous web with no holes in it? A web made of lines would still have a good chance of catching insects so long as it was woven tightly enough but would only require a fraction of the silk that a web made of hedras would require.

Furthermore, once you get to 5D, would the resident arachnids spin spiderwebs woven of chora (3D patches)? The difference in cost between a web based on lines and a web based on chora is huge: the amount of matter needed to cover a teron with choron patches is orders of magnitude greater than the amount of matter needed to cover a teron with a skeleton of linear strands.

In 4D, tiling a choron with strands, say the skeleton of a cubical tesselation (tiling), would only require O(3n) strands. Assuming each strand has unit length and a thickness of 0.01 per tiling unit, that's a mass of 0.03n. To cover the same region with a solid choron requires a mass of n³.

In 5D, tiling a teron with strands requires O(4n) strands, which amounts to a mass of 0.04n given unit strand length and the same strand thickness. To cover the same region with a teron requires a mass of n⁴. The cost is even more exorbitant here.

In k dimensions, you're talking about the difference between O(kn) and O(n^k). Strands are definitely favored here.

[wendy]

What is apparent in four dimensions, is not really clear. You can't tie nots in latrices (string, eg), because the dimension is too small.

You need at least (n-1)/2 to make a self-knotting structure, and (n-2)? to weave.

[anderscolingustafson]

and (n-2)? to weave.

Although we usually say how spiders "weave" their webs, the word weave for spiderwebs is a misnomer. Spiders don't actually weave their webs in the true meaning of weave they just lay strands on top of each other and the strands stick together because the stickiness of the strands holds the web together. So wile weaving with stands would not work in 4d making a spiderweb out of strands would still work in 4d therefor it would still be most efficient for 4d spiders to make their webs out of strands.

[Halfbaker]

In four dimensions webs probably wouldn't work as a way to catch food, There's too much space to passively catch food.

In two dimensions the opposite is true. A dionien spider would just spew one big strand across the landscape and feast on anything that walks or lands there. However, such a creature would need non-stick feet. Perhaps they instead would just leave a trail of closely spaced beads that they could step in between but that prey would stumble into. This would take even less energy and web material and would have the added advantage of suction on large/legless prey.

Also in two dimensions since spiderwebs rest on the ground they need not be strong. In fact, they need not be solid, they could be made of something more like white (non-powdery) earwax.