Weaving a Spider Web II: Catching mosquitoes

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Weaving a Spider Web II: Catching mosquitoes

First I thought I would have another go at making a spider web pattern – this time using Geogebra.  I’m going to use polar coordinates and the idea of complex numbers to help this time.

Parametrically I will define my dots on my web by:

Here r will dictate the distance of the dot from the origin, p will dictate how many dots I will have before I return to the starting point, and then I will vary n from 1 to p.

For example, if I have a distance of 1 from the origin then r = 1.  I decide to have 10 dots in one cycle, therefore p =10.  This gives the following:

Now if I plot all these points as n varies from 1 to 10 I will get the following graph:

Here I joined up all the dots to their neighbors and also to the origin.  If you know about complex numbers you might notice that we can represent these points as complex numbers, and these are the 10th roots of unity.

Following the same method I now change the distance from the origin (so r is changed).  This then will give me the following web:

Finding the optimum web design

I’m now going to explore some maths suggested by the excellent Chalkdust magazine (recommended for some great exploration ideas).  In their “Spider Witch Project” they suggest the following idea to work out how a spider can make sure they catch their prey.

We start with lots of assumptions and simplifications.

Say we have the web above, distances in cm.  Let’s say that the inner green circle is no use for catching flies (this is where the spider wants to stay, and so flies will avoid this area).  So excluding this circle we have 2 additional concentric circle-like rings, and 10 radial lines from the centre.  This gives us a total of 20 areas (like the one shaded red).

Changing to concentric circles 

Now let’s change our web into perfect circles.  We still have 20 areas in which the spider aims to catch their prey.  Let’s say that a spider will only catch their prey if the area it flies into is such that it is caught from all sides (i.e if the area is too large it may fly through the middle, or simply touch one strand but be able to escape).

For this particular design – with 2 concentric circles outside the centre area, the question is how many radial lines through the centre should the spider spin?  If it spins too few radial lines then its prey will not be caught, but if it spins too many then it will be wasting precious time (and energy) spinning its web.

Next lets work out the average of one of these red areas (in cm squared).  This will be:

Next let’s assume that our spider wants to catch a mosquito.  Let’s say the mosquito is 3mm by 4mm with a 2D plan cross sectional area of 12 mm squared.  And let’s say that the mosquito will not be caught in the web unless the area of the red area is the same size or smaller than the mosquito’s 2D plan cross sectional area.  In this case we can see that the mosquito will escape:

So, let’s see how many radial lines the spider needs to spin if it only has 2 concentric rings outside the “green zone”.  Here n will represent the number of radial lines from the centre:

So, we can see that a spider wanting to catch a mosquito might need to make at least 105 radial lines to catch its prey.  This doesn’t seem to realistic.  So, let’s modify our model to see how this compares to real life.

Real life

This particular spiderweb has around 20 concentric circles (counting double strands) and around 38 radial lines.  Let’s see how many radial lines we would predict from our model.

If we have a radius of around 6cm for this web (around the average for a spider web), and take an inner radius of 1cm for simplicity, then with 20 concentric circles we would predict:

So a spider would need at least 46 radial lines to catch their prey.  This is not too far away from the real life case in this case of 38 radial lines.

Further study

This could make a very interesting exploration.  You can get some more ideas on this by reading the Chalkdust article where they go into ideas of optimisation for the time taken.

This topic shows how a good investigation should progress.  Start with a very simple case, make lots of assumptions, then see what happens.  If you reach a conclusion then go back and try to make your initial case more complicated, or revisit your assumptions to see how realistic they were.

Essential Resources for IB Teachers

1) Intermathematics.com

If you are a teacher then please also visit my new site.  This has been designed specifically for teachers of mathematics at international schools.  The content now includes over 2000 pages of pdf content for the entire SL and HL Analysis syllabus and also the SL Applications syllabus.  Some of the content includes:

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Essential Resources for both IB teachers and IB students

1) Exploration Guides and Paper 3 Resources

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