Mathematicians like to come up with new stuff. A theorem, perhaps, a lemma or even just a corollary.

Edmund Harriss discovered a curve.

Harriss teaches maths at the University of Arkansas. He’s also an artist and his intellectual quest began with a shape that famously belongs to both science and art: the golden rectangle:

The golden rectangle divides into a square and a smaller golden rectangle.

A golden rectangle is a rectangle whose sides are proportioned according to the golden ratio, which is 1.618. In other words, the long side is 1.618 times the size of the short side.

What is particularly interesting, however, about a golden rectangle is that if you draw a square inside it, as above, the remaining section (in blue) is a smaller golden rectangle.

Let’s continue. We can divide the smaller rectangle into a square and an even smaller golden rectangle:

Cutting up a golden rectangle.

We can go on for as long as we like subdividing rectangles. And if we draw quarter circles in each square we get a spiral. The illustration below is probably one of the most famous images in mathematics, if not in all of science. The curve is called the “golden spiral”.

The golden spiral is a patchwork made up of quarter-circles. (Strictly speaking the golden spiral is a smoothed out version of this curve, so no pedantic comments below please!)

Inspired by the classic construction of the golden spiral, Harriss – who is British – began to play around with the process of subdividing rectangles in the hope that he would be able to generate other aesthetically pleasing curves.

So, rather than starting with a rectangle and then cutting out a square that leaves a similar rectangle, as we did above with a golden rectangle, he did something subversive.

“Instead of cutting a square, I cut a rectangle,” he said.

What he did was this: he found the rectangle that would divide into two similar rectangles and a square, as illustrated below.

The blue rectangle and the orange rectangle have the same proportions as the overall rectangle, which is a ratio between the sides of 1.325.

The rectangle gives birth to two similar children, of different sizes, and a square.

Since we have two of these rectangles, we can carry on subdividing.

As we subdivide we produce more rectangles with a 1.325 ratio.

And again. And again.

Let’s have a rectangle party!

Once more for the road.

Finally, the original rectangle is divided into 34 similar rectangles and 33 squares.

Remember that to make a golden spiral we added quarter circles to the squares? Harriss did that here, too:

The quarter circles go from corner to adjacent corner of each square.

Lo and behold, another spiral! But there are other squares we left out. Let’s fill them in.

The spirals branch out in a fractal pattern.

And now delete the largest arc, to reveal…a shape that I am going to call the “Harriss spiral”.

The Harriss spiral.

Harriss was overjoyed when he first saw the spiral because it was aesthetically appealing – one of his primary aims was to draw branching spirals like you might find in Islamic art or the work of Gustav Klimt. But he was particularly delighted because he arrived at the spiral using a very simple mathematical process.

“It’s not hard to make something that no one has seen before,” he said. “It’s more difficult to make something mathematically satisfying that people haven’t seen before.”

His first concern was that maybe someone else had had, in fact, drawn the spiral “One thing about mathematical discoveries and mathematical art is that even if the process is completely new there is no guarantee that someone else has not already explored it.”

It turned out that the ratio 1.325, which gives you the rectangle that creates the Harriss spiral has been written about – it is known as the “plastic number” – but Harriss could find no previous drawings of the spiral. (In fact, the ratio is a number that begins 1.32472… and carries on forever).

Now, the curve has become his signature, printed on T-shirts:

Spiral tribe: Edmund Harriss, Jesse Horton, Ella Van Horn and Ezra Van Horn (Clockwise from the one with the beard). They are holding a screen for printing T-shirts.

On cards.

If you are a friend of Edmund’s you might have received one of these gorgeous cards.

And on the plate of his car.

Special branch.

Another of Harriss’s motivations was to bring the golden ratio into a wider family of what he calls “proportion systems”.

“The golden ratio is this incredibly well explored corner of a whole city,” he said. “I wanted to give signposts to other locations in that city.”

Harriss’s proportion systems are rectangles than can be subdivided into only squares and similar rectangles.

There are only three possibilities for rectangles that can be divided into two, under this rule.

The ratio for each rectangle can be worked out by solving an equation based on the geometry of the shapes. The solution and the equation are marked for each rectangle.

The first one (divided into two similar rectangles) is the proportion of an A4 (or any A-number) piece of paper, with a ratio of √2.

The second one (divided into a square and a similar rectangle) is the golden rectangle, and the third one (two squares) is the shape of a domino.

There are 16 possibilities for rectangles that can be divided into three.

Here are the six options for where you cut off a big square and put the two squares/rectangles in a column. Remember that in every case, if there is a mini-rectangle it must have the same proportions as the overall rectangle.

The ratios of the rectangles and the relevant equations are marked for each rectangle.

Here are the six options for where you cut off a similar rectangle and put the two squares/rectangles in a column.

Bottom left is the proportion that gives the Harriss spiral.

And here are the four options where the squares/rectangles are all in a line.

Mathematicians will recognise the ratios provided by Harriss’s proportion systems as “algebraic numbers” – which are those numbers that are solutions to simple equations. Harriss thinks that a geometric approach to algebraic numbers may lead to a deeper understanding of them.

“The ratios already turn up in maths and art, which suggests that the proportion systems capture some idea of simplicity for these numbers,” said Harriss, adding that he is working on a proof that every algebraic number is the ratio of a rectangle belonging to a proportion system.

I asked if Harriss had looked for nice spirals with any of the other ratios? “Yes, but with limited success,” he sighed. “The problem is getting a set of squares that matches up in a nice order.”

Here’s what he has come up with so far:

Spirals from proportion systems.

Readers are invited to construct their own spirals – and then send me the T-shirt.

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Edmund Harriss’s blog is Maxwell’s Demon.



To read more about the golden ratio, here’s a post I did about it being discovered in the uterus, and here’s a project I was involved with that asked 55 designers to interpret the number.