Maintaining the structural integrity of a spider web^{1} is important for spiders, since it is both a residence and a hunting tool.^{2} Here, we analyzed the robustness of a spider web under various conditions using a simple rubber band model of a spider web, with an eye toward its utility as an integrative science experiment for students and teachers of physics. We perturbed the model web in many ways and monitored how its shape and its tension distribution change. We also discussed what kind of damage is more critical and should be avoided, in terms of survival and fitness of spiders.

## A spider web model made of rubber bands

In order to compare the effects of various perturbations, it is necessary to keep the initial conditions identical for all experiments. It is thus not practical to use a real spider web, because any damage would be permanent. Therefore, we designed a simple spider web model so that we can easily restore its unperturbed status after testing damage or a load.

While a spider web in nature is spiral in form, we made an equilateral octagon model [Figs. 1(a) and (b)] following a theoretical work^{3} that studied a radially symmetric, equilateral polygon web. The authors called the lines from the center to each vertex “radial” and the sides of polygons “spiral,” and we follow their naming convention in this study. Making an octagon web is easier, and repeating the same experiments for each radial direction is not necessary because of the symmetry.

For our purposes, damaging a component and restoring the damage should be easy. Adding a weight to the web (representing caught prey) and removing it should also be easy. Moreover, to understand why the Young’s moduli of radial and spiral segments are different in nature,^{4} the spring constant of the radial and spiral segments should be adjustable. Using rubber bands, key rings, and cell-phone hooks, we designed a spider web model that meets all of the aforementioned requirements. Using only one kind of rubber band, we controlled the spring constants with the number of rubber bands connected in parallel: multiple bands for radial segments and only one for spiral segments. Hereafter, let *K* and *K*′ denote the spring constants of the 1-m-long radial and spiral rubber bands, respectively. *K*′ was constant in all experiments, while *K* was varied. More details of the web construction and tension measurement of each segment can be found in the Appendix.^{5}

To explain our results clearly, we named the components of the web in the following manner [Fig. 1(b)]. The eight radial directions and four spiral orbits are labeled *n* = 1–8 and *m* = 1–4, respectively. In addition, *m* = 5 is used to designate the outermost radial segments connected to the posts standing on the eight vertices of an octagon. The point of intersection, the radial segment, the spiral segment, and the area surrounded by radial and spiral segments are termed P(*n*, *m*), *R*(*n*, *m*), *S*(*n*, *m*), and *A*(*n*, *m*), respectively. For example, P(2, 4) is the intersection of the *n* = 2 axis and the *m* = 4 orbit. *R*(3, 4) is the radial segment that connects P(3, 3) and P(3, 4). *S*(2, 3) is the spiral segment that connects P(2, 3) and P(3, 3). For the area, *A*(1, 4) is the area of the trapezoid surrounded by *R*(1, 4), *R*(2, 4), *S*(1, 3), and *S*(1, 4). The tensions on *R*(*n*, *m*) and *S*(*n*, *m*) are termed *F*_{n,m} and *f*_{n,m}, respectively. To indicate an average or a sum over a few selected *n*’s or *m’*s, we used parentheses. For example, $F(1,5),3\xaf$ denotes the average tension of the radial segments *R*(1, 3) and *R*(5, 3). For an average (or a sum) over all axes (*n*’s) or all orbits (*m*’s), an asterisk is used, such as $F*,2\xaf$ (or $\Sigma \Delta f3,*$). For an average (or a sum) over all (*n*, *m*)’s, we made the denotation even simpler, as in $F\xaf$ (or Σ *f*).

## The properties of the unperturbed spider web model

The unperturbed web has the following characteristics. First, the radial tension increases moving farther from the center. This tendency is diminished with a higher *K*/*K*′; i.e., the radial tensions on different spiral orbits become more uniform [Fig. 2(a)]. Second, the spiral tension increases moving farther from the center with *K*/*K*′ = 2; however, the trend is reversed with *K*/*K*′ > 2 [Fig. 2(b)]. Finally, the radial tensions account for most of the tensions on the spider web: they explain more than 80% and the proportion becomes even larger as *K*/*K*′ increases [Fig. 2(c)].

## Spider web with a damaged radial segment

To view the effect of a damaged radial segment, we removed *R*(3, 2) and *R*(3, 4), which represented damage near the center and near the perimeter, respectively. Removing a radial segment resulted in the following changes (Fig. 3). The tensions in the radial segments next to the damage increase [Figs. 3(a) and (b)], while those on the opposite side decrease [Figs. 3(a) and (c)]. The spiral tensions just outside the damaged segment increase [Figs. 3(a) and (d)], while those just inside it decrease [Figs. 3(a) and (e)].

The radial tensions change more when the outer segment is damaged [Figs. 3(b) and (c)], while the spiral tensions change more when the inner segment is damaged [Figs. 3(d) and (e)]. As *K*/*K*′ increases, the radial tensions change more [Figs. 3(b) and (c)], while the spiral tensions change less [Figs. 3(d) and 3(e)].

## Spider web with a damaged spiral segment

To examine the changes with a damaged spiral segment, we removed *S*(3, *m* = 1–4) one segment at a time. Both the spiral tensions just inside (*f*_{3,m−1}) and outside (*f*_{3,m+1}) the damaged segment increase [Figs. 4(a)–(c)], while they decrease next to the damage [*f*_{3±1,m}, Figs. 4(a) and (d)]. The change in the spiral tension becomes smaller as the outer spiral orbit is damaged [Figs. 4(b)–(d)] and as *K*/*K*′ becomes higher: the changes converge at zero with *K*/*K*′ = 16.

## Spider web with prey

Figure 5 summarizes the changes of the web when a 200-g mass (representing the prey) is located at P(0, 0) and P(3, 2), which corresponds to prey at the center [Fig. 5(a)] and away from the center [Fig. 5(b)], respectively. Hereafter, let *w*_{prey} denote the weight of the mass. As *K*/*K*′ increases, the tension change decreases, which happens faster with *w*_{prey} on the center than away from the center [Fig. 5(c)]. *w*_{prey} (∼2 N) was chosen to be comparable to the maximum tension with *K*/*K*′ = 2 [Fig. 1(c)], so the tension increase due to *w*_{prey} is considerable with *K*/*K*′ = 2 but diminishes dramatically as *K*/*K*′ increases. This suggests that a spider web with a higher *K*/*K*′ handles a load better. With the weight away from the center, the total radial tension increase on the axis with the weight (*n* = 3) is much larger than that on the other axes [Fig. 5(d)].

The spiral tensions behave differently. The total spiral tension over the web decreases when the weight is on the center, while it increases when the weight is away from the center [Fig. 5(e)]. The changes in the spiral tensions also become smaller with a higher *K*/*K*′. Figure 5(f) shows that the spiral tensions on the orbits inside (*m* ≤ 2) and outside (*m* > 2) of the weight are dramatically different [Fig. 5(f)]. The former is much larger than the latter, which stays close to zero. On the other hand, the spiral tension increases more on an axis closer to the weight [Fig. 5(g)]. It is remarkable that all these changes become smaller as *K*/*K*′ is increased.

## The area changes in the web

We defined *A*(*n*, *m*) as the area of a trapezoid (*m* > 1) or a triangle (*m* = 1) surrounded by radial and spiral segments. Figure 6 depicts how these areas change under various conditions. To be precise, the area here is the projected area on the ground, looked at from above vertically. When *R*(3, 4) is damaged, the areas of trapezoids adjacent to the damage increase [Fig. 6(a)]. When *S*(3, 3) is damaged, the areas of trapezoids just inside and outside the damage also increase [Fig. 6(b)]. When the weight is added, the total area of the web reduces [Fig. 6(c)].

The areas adjacent to the damage increase more for a radial segment loss than for a spiral segment loss. For both cases, the areas increase more for the loss of the outer segment than for the inner segment. On the other hand, adding a weight to the web decreases the web area, whether it is on the center or away from the center. It is also notable that, as *K*/*K*′ increases, the change in the affected areas becomes larger for a radial damage, but it becomes smaller for a spiral damage and an added weight [Fig. 6(c)].

## Discussion

We handcrafted a simple octagon spider web model and automated the measurement of the three dimensional positions of all 41 vertices of the web using three cameras. Using this model, we examined the changes in the web under various conditions, varying *K*/*K*′. In nature, *K*/*K*′ is about 14,^{7,8} which lies within our range (2–16).

When the spider web is not perturbed, the radial segments on the outer spiral orbits have higher tension than those on the inner orbits. This tendency is less pronounced with a higher *K*/*K*′; in other words, every radial segment shares the burden more uniformly. Moreover, a spider web with a higher *K*/*K*′ exhibits less spiral tension [Figs. 2(b) and (c)]. It is reasonable to presume that losing a weakly pulled segment would have a small impact. Therefore, a spider web with a higher *K*/*K*′ has less risk of losing a radial segment due to excessive tension and would suffer less when a spiral segment is damaged. Figure 4 shows that the changes in the spiral tension due to a damaged spiral segment indeed converge on zero as *K*/*K*′ increases. This explains why a high *K*/*K*′ is found in nature.

On the other hand, a spider web with a higher *K*/*K*′ would suffer more heavily when a radial segment is lost [Figs. 3(b) and (c)]. Therefore, a spider web should prevent damage to its radial segments, which is in accordance with radial spider silk having higher strength (1.1 GPa for the spider *Araneus diadematus*) than most biomaterials.^{7}

When prey is caught, the radial axis where it is trapped sustains most of the load [0.3–0.9*w*_{prey}, Fig. 5(d)], while the spiral orbits that are closer to the center than the prey share the burden [0.1–0.6*w*_{prey}, Fig. 5(f)]. However, the spiral orbits farther from the center do not help at all [<0.1*w*_{prey}, Fig. 5(f)]. This also supports the idea that radial segments are crucial for the mechanical integrity of the web.

The areas of trapezoids (or triangles) surrounded by radial and spiral segments increase the most when the radial segment is lost [Fig. 6(c)]. This is not desirable, because the spacing between spiral segments becomes larger, allowing prey to pass through more easily.^{2} Therefore, loss of a radial segment should be prevented to maintain hunting efficiency. Additionally, as *K*/*K*′ increases, the increase in area due to damage to a spiral segment or added weight becomes negligible. Thus, if radial segments are reliable and do not break easily, which is indeed true (a radial spider silk of Araneus has higher toughness than most biomaterials and engineering materials), then a spider web with a high *K*/*K*′ would be a reasonable choice. The loss of a spiral segment or the weight of prey would affect the web very little. Moreover, a high *K*/*K*′ suppresses the chances of radial damage by a uniform tension distribution.

Our simple model is not only easy to make but also sufficient to probe the mechanical properties of a spider web. This work is recommended for college students in physics or physics education and physics teachers who are also interested in biology and computer handling of data. Our model enabled systematic investigation of variables and efficient collection of data, which in itself could generate student interest. As future work using this model, we envision studying how a spider can sense the location of vibrating prey: the propagation of the vibration through the web may have a pattern that reveals the location of its source. This is an integrative science experiment that combines physics, biology, and computer science. We hope that more integrative studies will become available to students, showing them how useful physics can be even when applied to other fields of study.

## Acknowledgments

This work was supported by the National Research Foundation (2019R1C1C1007124, 2021R1A4A200138911), Republic of Korea, and the research grant of Kongju National University in 2021.

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