Spiky Feathers and Geometric Fish

The astroid curve is an example of a hypocycloid – a curve generated by rotating a fixed point on a small circle around a larger circle. Specifically, it is a hypocycloid with four cusps.

Astroid curve

The curve can be defined parametrically by the equations $x=\cos^3(t),y=\sin^3(t)$. It’s fun to play with these equations by changing the coefficients of $t$. The curve stops being a hypocycloid and becomes something…else.



These curves are beautiful just as they are, but they can also be used for generating string art. Start by plotting points on the curve for equally spaced values of $t$. The next image shows points plotted for $t=0$ to $2\pi$ in steps of $\frac{\pi}{13}$ on the curve $x=\cos^3(7t), y=\sin^3(6t)$.

Then choose a value of $n$ and join every $n$th point with a straight line. Note this this is not counting in the order that the points appear on the image, but in the order of the value of $t$ that generated the point. Joining every $6$th point in the image above gives the following.

Changing the number of points and how they are joined produces a huge variety of images, some chaotic, some bland, some beautiful. There is a Geogebra applet here where you can experiment for yourself. I’ll leave you with a couple of my favourites – I call them Spiky Feathers and Geometric Fish. You can buy T-shirts featuring the Spiky Feathers design.

Spiky Feathers
Geometric Fish





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Hello! I’m Sam Hartburn, a freelance maths author, editor and animator. I also dabble in music and write mathematical songs. Get in touch by emailing or using any of the social media buttons above.

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