# HLF Blogs – Fractals for dinner

This week, Katie and Paul are blogging from the Heidelberg Laureate Forum – a week-long maths conference where current young researchers in maths and computer science can meet and hear talks by top-level prize-winning researchers. For more information about the HLF, visit the Heidelberg Laureate Forum website.

The HLF, like all good conference events, has involved a large number of extravagant dinners, serving a variety of delicious food and drink to sustain the high levels of serious mathematical and research conversation. At last night’s Bavarian evening, I noticed a particularly mathematically interesting foodstuff was on the menu, and it’s inspired me enough to write about it.

Alongside the traditional Bavarian wurst and noodles, there was a pile of cooked vegetables, most of which were pretty standard – broccoli, and cauliflower. But in amongst was also a very special type of vegetable – known in German as Pyramidenblumenkohl, or “pyramid cauliflower”, and in English called Romanesco broccoli. In fact, even botanists can’t agree whether it’s really a broccoli or a cauliflower, but everyone who’s ever seen one agrees that it’s beautiful, and mathematically interesting.

### Fractal Food

One of the first things you notice when you look at this plant is that it’s made up of a repeated pattern of shapes – each part is a small version of the whole plant, and they get smaller and smaller as you go up to the top.

Credit: Image from Wikipedia

This property of a shape is called self-similarity, and it simply means that small parts of the object have the same structure as the whole shape. This is a property which fractals have, and in fact is one of the defining characteristics of a fractal – if you look at any example of a fractal you should be able to see smaller copies of the shape inside itself, repeated at different sizes all the way down.

Examples of Fractals – L-R Sierpinski Triangle, Mandelbrot Set, Sierpinski Carpet. Images from Wikipedia: Sierpinski Triangle CC-BY-SA Beojan Stanislaus; Mandelbrot Set CC-BY-SA Wolfgang Beyer

In the case of a fractal, this property needs to continue to infinity, and sadly in the case of food it doesn’t carry on forever – but it does mimic the structure of a fractal.

### Pre-Spiralised Vegetables

Based on an image from fermilab.ch

As well as having the excellent fractal property of self-similarity, the Romanesco broccoli also contains a beautiful spiral structure. The spirals go from the tip of each part of the plant and continue around to the base of each floret.

The shape of the spiral is a logarithmic spiral – a spiral whose rate of curving changes along the length of the curve in such a way that the shape of the curve stays the same. So, if you zoom in on a small part of this spiral, it’ll look like the whole curve – it’s also self-similar.

There are many examples of logarithmic spirals in nature, including in nautilus shells, and in the way spiral galaxies form. But this is obviously the most delicious.

### But wait, there’s more

Just when you thought you’d discovered all the interesting mathematical properties contained within your dinner – there’s another. If you count the number of spirals running around the florets in each direction, with a small amount of variation due to natural mutations, you’ll find there’s a Fibonacci number of spirals running each way.

The Fibonacci series is a list of numbers dating back to 1202, when Leonardo of Pisa (aka Fibonacci) discovered them as a pattern in various systems. Each Fibonacci number is the sum of the previous two, so if you start with 0 and 1 as the first two numbers, the sequence continues 1, 2, 3, 5, 8, 13, 21, 34, 55 and so on.

The same property of having a Fibonacci number of spirals running in each direction crops up in many other types of plant also – pine cones, pineapples, flower petals and the seeds in sunflower heads have all also exhibited this property – the latter being noted by Alan Turing, and thoroughly researched in a large-scale sunflower growing and observation project that took place in 2012.

Mathematician Vi Hart has produced an excellent series of YouTube videos on the subject, starting with this one.