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Weaving a Spider Web

I often see some beautiful spider webs near my house, similar to the one pictured above (picture from here).  They clearly have some sort of mathematical structure, so I decided to have a quick go at creating my own.

Looking at the picture above there are 2 main parts, an inner spiral, then a structure of hanging threads from lines which radiate from the centre.

Firstly I will use the general parametric equation of a hypocycloid:

and take the special case when a = 10 and b = 9:

This gives the following graph:

I can then vary the value of n in the following equations:

Which generates the following:

Next, I can generate the spiral in the centre by using an Archimedean spiral, plotting the curve in polar form as:

Which now gives:

Lastly, I want to have straight lines radiating from the centre going through the vertices of the graphs.  I can notice that at these vertices the gradient will be undefined (as we can’t define the gradient at a sharp point).  Therefore I can differentiate and look for when the gradient will be undefined.

I can see that this will be undefined when the denominator is zero.  Therefore:

I can notice that all the vertices are are on the same lines, therefore I can simply choose n =9 to make my life easier, and then solve for t.   I use the fact that sine is an odd function to help here.

Here p is an integer.  I’ll then rearrange the first of these two equations for t to show how I can then find my equations of the lines.

If I now substitute this value of t back into my parametric equations I get:

So, this will tell me the coordinates of the vertices of the “sharp points” of the graph.  Therefore the equation of the straight lines through these points and also through the origin are given by the first equation below. I can then choose my values of p (with p an integer) to get specific solutions.  For example when I choose p = 1 above I get the equation of a line which will pass through one of these vertices:

Let’s check that this works:

Yes!  So, we can use this method to find the other lines radiating from the centre.  This gives us our final spider web:

So, there we go, a quick go at making a spider web – quite a simplistic pattern, but still utilising parametric equations, polar coordinates and also calculus and trigonometric equations.