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enter image description here

Sometimes I see spider webs in very complex surroundings, like in the middle of twigs in a tree or in a bush. I keep thinking “if you understand the spider web, you understand the space around it”. What fascinates me, in some sense it gives a discrete view on the continuous space surrounding it.

I started to wonder what are good mathematical models for spider webs. Obvious candidates are geometric graphs embedded in surfaces, or rather in space. One could argue that Tutte’s Spring Theorem from 1963 is the base model: a planar geometric graph, given as the equilibrium position for a system of springs representing the edges of the graph. It is the minimum-energy configuration of the system of springs (see the picture for illustration). There are generalizations of such minimum-energy configurations for convex graph embeddings into space (Linial, Lovász, Wigderson 1988), where you place, for example, four vertices of the graph at the vertices of a simplex in $\mathbb R^3$.

I think such systems of springs are good models, because the threads of the spider web are elastic. However, when viewed as models for spider webs, I wonder whether these minimum-energy spring models are missing two aspects:
The purpose of spider webs is to catch prey, so I feel the ideal model should also consider
(A) maximizing the area covered (or the volume of the convex hull) and
(B) minimizing the distances between the edges.

To me, formalizing (A) and (B) and combining it with the minimum-energy principle for a system of springs would be the ideal mathematical model for spider webs.

Now, it is not obvious to me whether the minimum-energy principle alone determines a geometric graph satisfying (A) and/or (B)? Asking differently, if you add conditions like (A) or (B) to the minimum-energy principle, will this lead to different geometric graphs?

My second, broader question: Are you aware of any mathematical models developed explicitely to model spider webs? I checked MO and MSE and searched on the internet, but could not find anything. Maybe I am looking in the wrong fields, I wonder. Any help would be greatly appreciated!

References:
Tutte, W. T. (1963), "How to draw a graph", Proceedings of the London Mathematical Society, 13: 743–767, doi:10.1112/plms/s3-13.1.743
Linial, N.; Lovász, L.; Wigderson, A. (1988), "Rubber bands, convex embeddings and graph connectivity", Combinatorica, 8(1): 91–102, doi:10.1007/BF02122557
The picture is from Daniel Spielman’s lecture notes pdf on the web

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    $\begingroup$ OK I read it now. Guaranteed that I will come for an answer to your spiderquestion. Meanwhile, as an appetizer, are you familiar with the SPRING LAYOUT for graphs? That goes a long way... $\endgroup$ – Mirco A. Mannucci Sep 13 at 17:52
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    $\begingroup$ Regarding your last points (structure and capturing), real webs often have different types of silk to attain those different purposes. In particular, in orbital webs the radial threads are structural (stronger) while spirals are designed for capture (only they are sticky, as you can test by touch). Webs are truly impressive. $\endgroup$ – leonbloy Sep 15 at 19:28
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In response to the second question (which I interpret as asking for math models of spider webs as they appear in Nature): There exist several distinct types of spider webs. The most common type, the orb web of araneids, has been modeled in Simple Model for the Mechanics of Spider Webs (2010).

A key property of the orb web model is that the web is free of stress concentrations even when a few spiral threads are broken. This is distinctly different from usual elastic materials in which a crack causes stress concentrations and weakens the material.

The model highlights the mechanical adaptability of the web: spiders can increase the number of spiral threads to make a dense web (to catch small insects) or they can adjust the number of radial threads (to adapt to environmental conditions or reduce the cost of making the web) – in both cases without reducing the damage tolerance of the web.

Left panel: Construction of the orb web described in the cited paper.
Right panel: Naturally occurring orb web (Wikipedia).

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    $\begingroup$ Great reference, thanks a lot! I have not thought about stability conditions yet, that’s a great add! $\endgroup$ – Claus Dollinger Sep 13 at 18:25
  • $\begingroup$ I have seen a spider web in Florida that looks like a vertical funnel, suspended in mid air. very large, maybe 4 feet tall. Unlike the funnel example in the link, the one I saw has walls you can easily see through. $\endgroup$ – HenryM Sep 14 at 13:59
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    $\begingroup$ @HenryM: there are a wide variety of web types. Most people think of the orb web, which this answer focuses on. There are funnel webs, cob webs (eg: black widows), sac webs, etc. Some spiders only use webbing for their egg sacs, or as a climbing apparatus (jumping spiders). There are an enormous number of web types. Florida has some spiders that make some quite novel web types, like the bolas spider or the ogre spider. $\endgroup$ – Ryan Budney Sep 17 at 17:05
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In topology there is the notion of an infinite spider's web in the (complex) plane $\mathbb C$ that was introduced in 2010 https://arxiv.org/pdf/1009.5081.pdf.

A set $E\subseteq \mathbb C$ is an infinite spider’s web if $E$ is connected and there exists a sequence of bounded simply connected domains $(G_n)$ with

  • $G_n \subset G_{n+1},$
  • $\partial G_n\subset E,$ and
  • $\bigcup _{n\in \mathbb N}G_n = \mathbb C.$

In certain cases we also have that $E$ is closed and nowhere dense, and each $\partial G_n$ is a simple closed curve (Jordan curve), so that $E$ more closely resembles a traditional spider's web. These sets can be generated by iteration of entire functions such as $f(z)=\frac{1}{2}(\cos z^{1/4}+\cosh z^{1/4})$. The image below shows a spider's web consisting of the points $z\in \mathbb C$ such that $f^n(z)\to\infty$ at a certain rate (see https://arxiv.org/pdf/1009.5081.pdf for details).

Every locally connected Julia set of a transcendental entire function also has this form (shown in https://arxiv.org/pdf/1110.3256.pdf).

enter image description here

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    $\begingroup$ This seems to go the other direction from what the OP is asking. This is not a mathematical model for real-world spiderwebs; rather it's a mathematical object which we mentally model by thinking about spiderwebs. I do not think Julia sets of transcendental entire functions shed much light about the spiderwebs I see in the garden. Nice picture though. $\endgroup$ – Mike Miller Sep 13 at 18:47
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    $\begingroup$ @MikeMiller Yes the definition I gave is a pretty vague mathematical abstraction. However I think there are certain geometric regularities that we observe in actual spider's webs that may be be captured by simple iterative processes (iteration of entire functions is just an example). This could be considered when designing mathematical models. $\endgroup$ – D.S. Lipham Sep 13 at 19:08
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    $\begingroup$ Fair point! ${}$ $\endgroup$ – Mike Miller Sep 13 at 20:11
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So, I promised I would come up with some answer, but looks like there is already a great deal in the great answers above.

Anyway, I find it impossible to resist the temptation, especially because I think there is so much math that we can learn from Nature, particularly from our little friends, the spiders.

SPIDERWEBS AS SMART SENSORS (MORPHOLOGICAL COMPUTING)

Spider webs are not only for catching preys. They are, in a sense, an extension of their sensory apparatus, in that they help a spider detect at least three types of objects: a prey, a predator, and a potential mate.

How? Well, the web is kind of elastic, and it acts as a strange non linear filter: by "measuring" the perturbations on the web, our friends can isolate some frequencies which give them the clues.

See here and here and also here for details.

As far as I know, the Theory of Morphological Computation is still undeveloped, particularly from the mathematical standpoint. Perhaps some smart fellow here on MO can enlighten us. Meanwhile, just wish to point out that the changes of configurations mentioned by Carlo above are also done as "tuning" the morphological computing capabilities of the web

SPIDERWEB AS QUANTUM GRAVITY MODELS

As we all know, quantum gravity is the holy grail of modern physics. Among the most intriguing attempts so far, there is Fotini Markopoulou Kalamara's Quantum Graphity. see here.

To summarise Fotini's brilliant idea is not easy, but here is the gist: start from a universe in which there is no space-time, and try to build it as a graph. Create a quantum system, that basically is a quantum superposition of many graphs, and associate to this beast an hamiltonian.

Set it to some default eigenvalue of energy, say HOT. That corresponds to a fully connected graph, where every point is one step from any other. Too many connections to make up our space time! But now suppose it "cools off": the edges get disactivated, till it settles into something like our ordinary space time (the full theory is, as far as I know, still undeveloped, needs some really good mathematician to work it out). Now, I would suggest you toy with that theory, precisely because , as you suggested, spiderwebs give you an insight into the nature of space (and time too). For quantum gravity, you do not simply need to replicate the topological properties of space-time, but also its metrics (say volumes, areas, etc. Essentially approximate general relativity).

So, perhaps the DEMIURGOS is a giant super inteligent spider!

a toy black hole in quantum graphity: in proximity of the hole the graph is fully connected and loses its low dimensionality

SPIDER WEBS AS WEIGHTED SIMPLICIAL COMPLEXES

Now, after the double detour, back to the question and the answer: I suspect, in fact I am pretty sure, that the best way to model spider webs are Weighted Simplicial Complexes, ie simplicial complexes in which all simplices have a weight (either a real number, or even a complex one, in case we wanna formalize quantum spiders): see here.

Why weighted simplicial complexes? Because they generalize weighted graphs, and they have an entire artillery (laplacian, persistent homology, etc) which can be put to use here. Example: you ask for minimal area, that to me means that when you write your energy, you must add a term which tries to minimize the total area covered by the web. I also think that these complexes could add some light into spider webs as morphological computers (I have a small paper with some folks on diffusion of information on weighted simplicial complexes. The core idea is that higher simplices are hubs of information broadcasting, which in case of graphs are missing)

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  • $\begingroup$ Mirco thanks for a fantastic answer with so much extra perspective. Are you able to give the link to your paper about weighted simplicial complexes as information hubs?Very interested in that. And will also follow on on Quantum Graphity. Plus, never heard of morphological computing, want to learn about it $\endgroup$ – Claus Dollinger Sep 20 at 5:51
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    $\begingroup$ +1 for "perhaps the DEMIURGOS is a giant super inteligent spider" and giving a reason for that! Your answer is a great read. $\endgroup$ – MaryS. Sep 20 at 6:02
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    $\begingroup$ @Claus the article was meant for communication researchers, not mathematicians (though I have plans to morph it into a real math article) Here it is: digitalcommons.chapman.edu/cgi/… $\endgroup$ – Mirco A. Mannucci Sep 20 at 12:41
  • $\begingroup$ @MaryS. thanks! spiders are very smart beings... $\endgroup$ – Mirco A. Mannucci Sep 20 at 12:42
  • $\begingroup$ @MircoA.Mannucci Just saw your article, really interesting!! I am a big fan of such applications of math. Please let us know when your math background paper Is ready. This whole area is fascinating $\endgroup$ – Claus Dollinger Sep 20 at 15:26
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A biologist friend told me about this question on MathOverflow, so I wanted to contribute a useful link to a related article that appeared in NATURE.

Published: 01 February 2012
Nonlinear material behaviour of spider silk yields robust webs
Steven W. Cranford, Anna Tarakanova, Nicola M. Pugno & Markus J. Buehler
Nature volume 482, pages72–76(2012)

This is the link https://www.nature.com/articles/nature10739

The mathematically interesting feature investigated here is the nonlinear response of silk threads to stress:

From the abstract of this article: Here we report web deformation experiments and simulations that identify the nonlinear response of silk threads to stress — involving softening at a yield point and substantial stiffening at large strain until failure — as being crucial to localize load-induced deformation and resulting in mechanically robust spider webs. Control simulations confirmed that a nonlinear stress response results in superior resistance to structural defects in the web compared to linear elastic or elastic–plastic (softening) material behaviour. (...) The superior performance of silk in webs is therefore not due merely to its exceptional ultimate strength and strain, but arises from the nonlinear response of silk threads to strain and their geometrical arrangement in a web.

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