Art/Maths: Strange circles

 

Standard and non-standard round shapes.

Once upon a time, astronomers were certain that all the moving bodies in space travelled in circles, “because circles are perfect”. In many ways, modern science began when Johannes Kepler saw that the orbits of planets were ellipses. Or maybe science emerged when Isaac Newton proved that the orbits had to be that shape, because of the way gravity worked. Whichever way it happened, those odd squashed circles called ellipses were involved.

A 19^th century engraving of a Gatling gun: notice the shape of the wheels.

To me, ellipses are important, because in perspective, circles look like ellipses, but I am no artist, and I need help to get my ellipses right. When I am drawing on paper, I use plastic templates to draw my ellipses, but with a simple graphics program like Paint.Net, I can draw ellipses of any shape and size.

If you want to work on shading and stippling geometric shapes, use a colour printer to print out pale sky-blue ellipse outlines. Make just enough fine black points on the paper to show the outline, then photocopy it: pale blue (often called “dropout blue”) usually fails to show in a photocopy, and away you go.

We will meet Piet Hein again in other parts of this blog, but now we need to look briefly at his superellipses, which were adopted as a suitable shape for rounding-off a space in the centre of Stockholm, rather more nicely than the rounded rectangle above. If you look online for <Sergelstorg>, you can see the result in maps and aerial photos of Stockholm. By an odd chance, Hein came up with his solution in 1959, the year in which I encountered the two cultures, and C. P. Snow published a book about his theme. Surely, if anybody ever showed how the Two Cultures notion breaks down, it must be Hein. And now, we need to venture into mathematics of a Heavy Kind.

There is a whole family of curves with the formula (^x/_a)ⁿ + (^y/_b)ⁿ = 1, and as a group, they are called Lamé curves, after Gabriel Lamé, who discovered them. If n is between 0 and 1, the figure is a four-pointed star. If n= 1, it is a parallelogram, and for n between 1 and 2, it is a rounded-off rhombus. If n=2, we get an ellipse or a circle (depending on the values of a and b), and above that, we get squircles, or superellipses.

Sergels torg in Stockholm (look it up on the web) has n=2.5, and a/b=1.2.  Over to you, but look around on the internet for 3D supereggs and ellipsoids…

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