Formula unlocks secrets of cauliflower's geometry

October 23, 2012

The laws that govern how intricate surface patterns, such as those found in the cauliflower, develop over time have been described, for the first time, by a group of European researchers.

In a study published today, 24 October, in the Institute of Physics and German Physical Society's New Journal of Physics, researchers have provided a mathematical formula to describe the processes that dictate how cauliflower-like patterns - a type of fractal pattern - form and develop.

The term fractal defines a pattern that, when you take a small part of it, looks similar, although perhaps not identical, to its full structure. For example, the leaf of a fern tree resembles the full plant and a river's tributary resembles the shape of the river itself.

Nature is full of fractal patterns; they can be seen in clouds, lightning bolts, crystals, snowflakes, mountains, and blood vessels. The fractal pattern of the cauliflower plant is ubiquitous and can be spotted in numerous living and non-living systems.

The properties of fractals, such as their shapes, sizes and relative positions, have been studied extensively; however, little is known about the processes involved in their formation.

To identify this, the researchers, from Comillas Pontifical University, Universidad Carlos III de Madrid, Instituto de Ciencia de Materiales-CSIC, École Polytechnique and Katholieke Universiteit Leuven, firstly grew thin films using a technique known as chemical vapour deposition (CVD).

CVD is a technique used to grow a solid, in which a substrate is exposed to a number of precursors that react and/or decompose on its surface to create a specific thin film. The researchers tailored the CVD process so the film would grow into shapes similar to those seen on a cauliflower, but limited to the submicron scales.

From this the researchers were able to derive the formula which described how the cauliflower-like patterns develop over time. They proved that the formula was able to successfully predict the final cauliflower-like patterns by comparing them to actual cauliflower plants and combustion fronts, both of which occur at much larger scales.

Co-author of the paper, Mario Castro, said: "In spite of the widespread success of fractal geometry to describe natural and artificial fractal shapes, purely geometrical descriptions do not provide insight into the laws that govern the emergence of the shapes in time.

"We believe that by knowing the general laws that dictate how these patterns form and grow, it will help to identify the biological and physical mechanisms that are at play."
Notes to Editors


1. For further information, a full draft of the journal paper or contact with one of the researchers, contact IOP Press Officer, Michael Bishop:
Tel: 0117 930 1032

Universality of cauliflower-like fronts: from nanoscale thin films to macroscopic plants

2. The published version of the paper "Universality of cauliflower-like fronts: from nanoscale thin films to macroscopic plants" (Castro et al 2012 New J. Phys. 14 103039) will be freely available online from Wednesday 24 October.

New Journal of Physics

3. New Journal of Physics publishes across the whole of physics, encompassing pure, applied, theoretical and experimental research, as well as interdisciplinary topics where physics forms the central theme. All content is permanently free to read and the journal is funded by an article publication charge.

IOP Publishing

4. IOP Publishing provides publications through which leading-edge scientific research is distributed worldwide. IOP Publishing is central to the Institute of Physics (IOP), a not-for-profit society. Any financial surplus earned by IOP Publishing goes to support science through the activities of IOP. Beyond our traditional journals programme, we make high-value scientific information easily accessible through an ever-evolving portfolio of community websites, magazines, conference proceedings and a multitude of electronic services. Focused on making the most of new technologies, we're continually improving our electronic interfaces to make it easier for researchers to find exactly what they need, when they need it, in the format that suits them best. Go to

The Institute of Physics

5. The Institute of Physics is a leading scientific society promoting physics and bringing physicists together for the benefit of all.

It has a worldwide membership of around 40 000 comprising physicists from all sectors, as well as those with an interest in physics. It works to advance physics research, application and education; and engages with policy makers and the public to develop awareness and understanding of physics. Its publishing company, IOP Publishing, is a world leader in professional scientific communications. Go to

The German Physical Society

6. The German Physical Society (DPG) with a tradition extending back to 1845 is the largest physical society in the world with more than 59,000 members. The DPG sees itself as the forum and mouthpiece for physics and is a non-profit organisation that does not pursue financial interests. It supports the sharing of ideas and thoughts within the scientific community, fosters physics teaching and would also like to open a window to physics for all those with a healthy curiosity.

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