Hamid Naderi Yeganeh, a 24-year-old maths student at the University of Qom, Iran, spends his time looking for mathematical shapes.
His method is a bit like fishing with a large net. So it is particularly appropriate that some of the shapes he has found look like fish.
Hamid wrote a computer program that trawls for interesting figures. It produces print outs like this:
This is what’s going on: the two formulae (see caption) with the variable k each produce a dot on a graph. So, for each value of k, you get two dots. The computer then draws a line between the two dots for each value of k from 1 to 100, creating an image made up of 100 lines.
Then he tweaks the original two formulae (by changing the values of a, b, c and d), and does the same thing all over again, producing another image made up of 100 lines. Hamid’s programme tweaked the original formulae more than a thousand times producing a feed of 100-line diagrams like the one above.
Eventually, and unexpectedly, Hamid started to see shapes that looked like fish.
Like this one.
Yup, kind of bonkers, but it does look like a fish.
By redrawing this image with all real values of k between 0 and 100 (so, in effect, there are an infinite number of lines) the shape became the black silhouette pictured at the top of this post.
“My goal was to find a figure like a real thing or a beautiful symmetric figure,” he said. “I was surprised when I saw the fish. It showed me the power of mathematics.”
The more he looked, and he ran programmes that created tens of thousands of images, the more ichthyic outlines he found. Do you see an angel fish? Or the profile of a cartoon character with big lips and a quiff?
These distinctive-looking mutant fish are the result of combining two mathematical areas - trigonometry and complex numbers.
Trigonometry is all about circles, and complex numbers are numbers that include a multiple of the square root of minus one.
(The formulae include the basic trigonometric functions sine and cosine and the imaginary number i, the symbol for square root of minus one. Each formulae provides a number of the form x + iy. The dot described by x + iy is positioned x along the horizontal axis and y along the vertical axis, a coordinate system commonly referred to as the complex plane.)
I find the silhouettes bold and distinctive, and quite unlike any other mathematically produced curves I’ve seen before. Quite what it tells us about marine biology, I’m not so sure.
Yeganeh also found a bird:
And a boat! Seen here made up of 2,000 lines.
Inevitably, they make a wonderful ocean scene:
Look! Land Ahoy! Over on the shore! Is that an oryx, capt’n?
Or maybe Bart Simpson with a punk haircut.
An oryx is an antelope that lives in the Arabian peninsula and Africa.
Which is a perfect segue-way to another of Hamid’s mathematical images, a simplified outline of Africa.
Look at how it stacks together.
It’s a fractal Africa, since there are subsets that are identical to the overall shape.
(Fractafrica? Afractica?)
Also, if you count the number of Africas of different sizes you get the Fibonacci sequence: (0, 1,) 1, 2, 3, 5, 8, 13, 21, 34, ... where each number is the sum of the previous two.
Afractifabulous.
Hamid Naderi Yeganeh’s website has more of his mathematical images.
Shameless plug alert! If you want to know more about trigonometry, the square root of minus one, the complex plane and fractals, I’ve written chapters about them in my latest book Alex Through the Looking-Glass: How Life Reflects Numbers and Numbers Reflect Life. (The Grapes of Math in the US).
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