### Introduction

The ability of spiders to spin their webs with great precision, strength, and weaving of the threats at the equidistant interval is an intriguing occurrence that frequently goes unnoticed. To common observers, spiders construct the webs for the sake of catching their preys. However, mathematicians have dug deeper to investigate the shape and patterns of construction and make various conclusions. According to evolution theory, organisms develop mechanisms to enable them to survive various conditions and establish feeding habits that ensure the continuation of generations (Maciejewski, 201). Some organisms have simple methods of blending in the environment to ease hunting while others rely on wit and industriousness to establish traps that ease hunting process. One of such predators is a spider. According to Alicia (2015), spiders can correctly determine the distance between two objects that can support the web before they start the construction process. Due to such findings, it is essential to investigate the mathematical concept of the construction process of a spider web.

### Rationale

Through independent thinking and personal interest in the application of symmetry in real life, I observed various shapes and counted the numbers of symmetrical shapes. In the process, I encountered spider webs that were more intriguing and challenging. For this reason, I started investigating how various disciplines explain the construction process of the webs, their strength, and how spiders manage to retain their shapes and symmetry throughout the construction process. Based on the shapes of spider webs such as the existence of radii, spiral shapes of the loop, I knew that there must be mathematical ideas behind the building of spider web.

### Aim of Exploration

Primarily, this project aims to understand the mathematical interpretation and concept behind the building of spider web. I was sure that the nature of spider webs could not be random due to their uniformity, dependence on the nature of the environment, and its symmetry. Through reading various publications and mathematical journals and books, it would be possible to discover the ideas behind spider web building.

### Analysis of the Spider Web-Building Behavior

Although to spiders, the act of building a web is an inbuilt trait that does not require learning, these organisms vary their behavior to facilitate construction of an optimal structure in different environmental conditions. I have identified that the nature of spider webs differs depending on their location and the size of the organism (Nature’s Math, 2008). Big spiders have stronger and larger webs than the smaller ones. They also construct the webs in isolated places with big insects (Nogueira & Ades, 2012). It is evident the environmental conditions and adaptability of the creatures determine the nature of the web. Spiders erect the two-dimensional trap vertically to easily trap flying insects by the use of vertical support while considering environmental interferences such as wind and rain (Krink & Vollrath, 1997). I identified that spiders are intelligent and positioning of the web is not random but a strategic move that facilitates durability of the trap and increases the chance of catching prey.

In essence, mathematicians have established that “the spider web is composed of numerous radii, a logarithmic spiral, and the arithmetic spiral” (Alicia, 2015).

The logarithmic spiral equation is given by the polar equation r = ae^{bθ} while the arithmetic spiral equation is given by the polar equation r = a + bθ (Alicia, 2015).

### The first equation becomes:

Logarithmic spiral is r = aeθcotb where ‘r’ represents the radius of each spiral turn, ‘a’ and ‘b’ are constants depending on a given spiral loop, ‘e’ is the base of natural logarithm, and θ is the rotational angle (Encyclopedia Britannica, n.d.).

### Mathematical Perspective of Web-Building

To understand the mathematical perspective of web-building process, it is important to understand the stages of construction in details. The process passes through four stages namely “first radii, other radii and frame, first spiral loop, and following spiral loops” (Krink & Vollrath, 1997). Spiders are free to choose its location for placing the hub which in turn determines the number of radii to use (Merciejewki, 2010). However, that location will dictate the frame, shape, and size of the hub (Chandler, 2012a). Overall, there is a close association between the environmental factors, location, frame, and radii of the web (Herberstein, 2000). Mathematicians have identified that the construction process obeys some mathematical concepts and rules.

### Stage one

The first stage of web building is the construction of the initial radii whose end-point connections determines the web’s frame. The mathematical concepts that govern the process are the Equal Spacing Rule and the Length-Cross Rule.

Equal spacing rule. After the spider chooses the location, it decides on the vertical structures to use as support. The equal spacing rule will ensure that all the radii are equally-spaced which helps in determining the number of radii the web and their orientation (Chandler, 2012b). Spiders construct the first radius which faces the north direction. Its modification results from the influence of the phase angle (Nature’s Math, 2008). The construction is possible through determination of the base angles between the radii by dividing 360 degrees by the total number of radii of the web (He & Xie, 2006). In addition to the base angle, the variation factor also helps to determine the spacing of the web’s radii.

Length-cross rule. The spacing between the vertical supports does not automatically determine the length of the radii of the web. The orientation in the north, south, east, and west are the base points of the length cross rule that spans the curve of the frame to determine the length of the radii (He & Xie, 2006). An interpolation function determines the shape between orientation points within the four quadrants (He & Xie, 2006). Therefore, the use of the equal spacing rule and length cross rule determines the number and lengths of the radii required in the web after dividing the hub into four equal quadrants.

Addition of more radii between the initial radii forms the major part of the second phase of the web-building process. Essentially, this stage also has rules that govern the addition of the radii namely Angle Cross Rule and Length to Frame Rule (Powers, 2013). Each of the rules uses various methods of interpolation to ensure equidistant construction of radii.

Angle cross rule. This rule determines the angle between an already constructed radius and the new one that spider intends to build (Watanabe, 1999). The four values of the orientation directions and interpolation assists in the determination of the angle.

Length of frame rule. It sets the length and the angle of the radii from the center to the frame of the web (Powers, 2013). The repeated application of these rules leads to the construction of all the radii of the web leaving only the final states building the spirals to complete the web.Slope rule. Slope rule determines the point of attaching the capture spiral loop on the radii and explains the displacement of each subsequent loop from the center of the web (Zschokke, 1999). The distance between the loops is determined by the first radius’ initial distance and increases up to the last radius of the slope. In the construction of radius from the center to the frame, the spider needs to spin an equal length of silk (Blackage & Gillespie, 2002. The process of constructing the loop is more complicated than building the loops (Zschokke, 1995). As a result, this stage involves four rules namely Last Ins. Dist. Rule, Mesh-Size Rule, No Room Rule, and No End Rule. `

ast Ins. Dist. Rule. Its main purpose is determining the base location of the first loop on the first radius and increases the length of the subsequent loop due to the circular nature of the web (Cranfold, 2012). The distance to the next capture node is determined by multiplication of the last distance from the which the loop begins with the variation factor initially determined by equal spacing rule.

Mesh-size rule. After the Last Ins. Dist. The rule determines the position of the next loop from the center of the web; this rule modifies the rule by determining deviation value. Deviation value less than 100 requires reduction of the mesh while values greater than 100 reduces the size of mesh along the vertical orientation (Barghusen et al., 1997). The deviation value usually depends on the orientation of the body of spider (Vollrath, 1987). Hence, this rule considers the species of the spider which plays a vital role in the determination of the environmental factors.

No room rule. The first two rules foresee the construction of the spiral loops until they reach the frame of the web. The no room rule is used to check on the effectiveness of the web regarding its ability to trap small insects. In particular, this rule identifies the lengths that require reinforcing to eliminate the room for escape (Wit et al., 1977). It ensures that the frame becomes so small that it is not possible to fix another thread.

No end rule. After the no room rule ensures that there is no space to fix another loop, the no end rule prevents the spider from producing more silk to continue spinning the web (Wit et al., 1977). The chart below illustrates all the mathematical rules that interpret the building of web at a glance.

Source: Krink & Vollrath (1997). Analyzing spider web-building behavior with rule-based simulations and genetic algorithms.

### Mathematical Expressions of Spider Silk Strength

It is surprising to note that spider silk is stronger than several materials. The table below shows comparison of spider frame silk with some selected materials in terms of it modulus, strength, and amount of energy required to break the material.

Source: Bonino (2003), Material propeerties of spider silk.

It is evident that spider silk is stronger than most materials such as rubber and tendon. It also requires a lot of energy to break the silk.

The equation for calculating the tensile strength of silk is given by:

where P_{MAX} is the maximum load it can carry and A_{0} is its original cross-sectional area (Bonino, 2013).

The silk percentage elongation when carry a load is given by:

where l_{0} is the silk’s original length and l is the extension length (Bonino, 2013).

### The energy required to break the silk is given by:

where l values indicate initial and final extension while P values are the weights of the loads used during the experimenation (Bonino, 2003).

Spider silk extends to a certain limit until it reaches a breakage point. Comparatively, spider dragline silk extends by a greater value compared to kelvar as shown in the graph below:

The material undergoes a great stress before it breaks as shown in the graph below:

Under a compressional force, spider silk responds to stress more than most materials. The graph below compares the silk’s stress with other materials such as wool and nylon among others.

The difference in response of the materials to stress results from the difference in diameter as shown in the table below:

In summary, the spiders undergo a long procedure to construct their web in an efficient manner that maintains its symmetry. From an observant’s point of view, the process does not appear complex but a normal operation of the organism as it attempts to catch its prey. Interestingly, mathematicians have investigated the process and identified that most of the procedures during the web-building process agree with various mathematical rules. Spiders apply these rules without knowing and establish astonishingly symmetrical webs that endure the environmental conditions in their respective surroundings.

## References

Alicia, B. (2015). Spider web: Creepy or cool. Math Project. Retrieved from https://recursiveprocess.com/mathprojects/index.php/2015/06/09/spider-webs-creepy-or-cool/

Barghusen, L. E., Claussen, D. L., Anderson, M. S., Bailer, A. J. (1997). The effects of temperature on the web-building behavior of the common house spider. Functional Ecology, 11, 4-10.

Blackage, T. A., & Gillespie, R. G. (2002). Estimation of capture areas of spider orb web in relation to asymmetry. The Journal of Arachnology, 30, 70-77.

Bonino, M. J. (2003). Material Properties of Spider Silk, (Unpublished Dissertation), University of Rochester, Rochester.

Chandler, D. L. (2012a). From spider web tangled proteins, researcher finds connection between mathematics, molecules, and materials. Massachusetts Institute of Technology. Retrieved from https://phys.org/news/2012-04-spider-webs-tangled-proteins-mathematics.html

Chandler, D. L. (2012b). How spider web achieves its strength. MIT News. Retrieved from https://news.mit.edu/2012/spider-web-strength-0202

Cranfold, S. W. (2012). Non-linear material behavior of spider silk webs. Nature, 482(7383), 72-76.

Encyclopedia Britannica. (n.d.). Spiral mathematics. Retrieved from https://www.britannica.com/science/spiral-mathematics

Frank, K. K., Kawabata, S., Inoue, M., Niwa, M., Fossey, S., & Song, J. W. (n.d.). Engineering Properties of Spider Silk. Retrieved from https://web.mit.edu/3.064/www/slides/Ko_spider_silk.pdf

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