My final project is a spotlight on the mathematical ingenuity that orb-web-weaving spiders have been endowed with. The typical orb web usually appears droopy if visible to the human eye at all. Due to its width, spider silk cannot be seen unless light is reflecting off of it, or it has been caused to sag from the weight of morning dew. Essentially, in its perfection, an orb web is much more complex than the below depiction.

The spider web is actually comprised of numerous radii, a logarithmic spiral (given by the polar equation r=ae^{bθ} ) and the arithmetic spiral (given by the polar equation r = a+bθ ). The constants a and b are adjustable.

Construction begins with the spider’s identification of two vertical, structural supports. After ascending one of the supports, the spider perfectly predicts the length of silk that it will need to reach the other vertical support. A gentle gust of wind is all the spider needs to carry it to the other side. Once the very first strand is constructed and strengthened by multiple strands, the spider will build three radii and a web-frame. More radii are constructed to the frame and then a temporary logarithmic spiral is built. Keith Devlin describes the logarithmic spiral that increases distance between successive turns by a constant growth factor (Devlin 89). View my graph of a logarithmic spiral below as well as further reading about the logarithmic spiral.

http://www.desmos.com/calculator/cis4o1sxc8

Logarithmic Spiral

http://mathworld.wolfram.com/LogarithmicSpiral.html

Next, the spider will turn around and spiral inward with an arithmetic spiral while simultaneously eating/ destructing the temporary spiral. According to Keith Devlin:

*In an arithmetic spiral, a line drawn outward from the center crosses each successive turn at the same distance, and the tangent angles increase at a constant rate, approaching 90 degrees as the number of turns increases (Devlin 89).*

Below is a picture of my graph of an arithmetic spiral as well as some further reading.

Arithmetic Spiral

http://en.wikipedia.org/wiki/Archimedean_spiral

At the completion of the web, the spider retreats to the the center of its web, suspends itself upside-down, motionless while awaiting its prey. The final product of the web, if undisturbed and created perfectly, would look something like this.

Pattern of Final Product created by Alicia Bautista

As part of my experiment, I decided to attempt a time logged construction of an orb web. I am certainly much slower than this guy.

My construction, which is incomparable due to my lack of engineering capability, took me about three hours. However, I bypassed the logistic spiral and went straight for observable end product of radii and arithmetic spiral. The average garden spider, an orb-weaver, takes between one and three hours to construct its web. Spidy Speed!

In order to stimulate the stronger, structural silk that spiders use, I used fishing line. In place of the arithmetic silk spiral, which is used to catch the prey, I constructed a spiral of hot glue. I got to “be the spider” so to speak. The properties of hot glue are actually pretty similar to the “sticky” silk that spiders use since both materials solidify almost instantly.

However, my web is very fragile and was only sticky for about 2 seconds. If I were a spider, I would die.

It just goes to show that spiders are true engineers by nature. I mean just look at them!

Video

https://www.youtube.com/watch?v=DwLn11KJS9Y

For further reading on the the spider and its construction methods:

http://ednieuw.home.xs4all.nl/Spiders/InfoNed/webthread.html

http://grandpacliff.com/Animals/Spiders/OrbWebCon.htm

Blog by: Alicia Bautista