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Conformal Geometry


We are used to studying conformal geometries in physics/maths but some times they turn up in nature in remarkable and rather beautiful ways. An example of this is the so called conformal lattice, the geometry of which I have studied in collaboration with Professor Denis Weaire. 

To construct a conformal lattice consider a regular lattice in the z-plane, upon applying an analytical mapping f(z) to these points we end up with a new set of points in the w-plane. The resulting structure in the image plane, due to the conformal mapping, is known as a conformal lattice.  An example of this is shown in the image below where in (a) we see the original regular lattice and the results of using the applying the mapping (b) z^(-1/2) and (c) z^(-1) are shown in the corresponding images on the below.

Papers 1.9.8ScreenSnapz001

An example of such a conformal lattice in nature is the phyllotactic design of a sunflower:


They can also be realised by sandwiching an ordered, monodisperse, soap foam between two non parallel glass plates (image courtesy of S. Cox):

Papers 1.9.8ScreenSnapz002

Here, image (a) shows  two glass plates at an angle to each other, sandwiched between them is a dry foam forming a honeycomb structure. Since the bubbles towards the top are more elongated than the bubbles at the bottom of the glass plates there is a variation in the area of the bubbles when they are observed from above. By this means specific variations can be imposed on the aerial area of the bubbles (although their total volume remains conserved). 

The result is shown in (c), this is an actual image from an experiment by W. Drenckhan et. al. - for more details see this paper. Here the authors have highlighted in grey three  lines of bubbles which were initially straight but now are curved into arches.

Also shown for comparison (b) is the transformation of the honeycomb structure by a complex logarithmic mapping.

To find out more, see 

[1]  Curvature in conformal mappings of 2D lattices and foam structure. A. Mughal and D. Weaire (2008), Proceedings of the Royal Society A, also see arxiv/0801.2474. 

© Adil Mughal 2012