This post serves as a collection of information and work related to seashell morphogenesis and ornamentation, which is collaborative work with Professor Derek Moulton and my PhD thesis supervisor Professor Alain Goriely at Oxford, as well as Professor Régis Chirat at Lyon.
Currently, we are in the finishing stages of a publication on ammonite ornamentation, extending this recent work by Moulton, Chirat and Goriely. I will add a preprint very soon.
In summer 2013, I wrote this work on antimarginal ornamentation as a ten week research project supervised by Professor Derek Moulton. Prior to that, I wrote some personal notes to get acquainted with the topic, which you can find here and here.
As the topic of seashells naturally generates beautiful images, I am showing a few 3D seashell renderings below.
Photos and 3D renderings of Lambis truncata seashell. Our model captures both the coiling of the shell surface and the finger-like antimarginal ornamentation.
Plots of parametric surfaces serving as foundation for ornamentation (such as ribbing).
Quick summary: Thesis here, slides here, related notes here.
This post serves as a collection of information and work related to my undergraduate thesis, which I wrote in summer 2010 under the supervision of Professor Peter H. Richter at the Physics department of the University of Bremen, Germany. A brief summary aimed at a general audience can be found below in this post.
The title of my thesis is: Melnikov’s Method and the transition to chaotic behaviour in Cardan-mounted Euler tops. It is written in German. The thesis is available here and the presentation slides here. As a way of learning about Melnikov’s method, I wrote these notes prior to the thesis. The Cardan-mounted Euler top to which the method is applied is depicted in the picture below.
The Cardan-mounted Euler top to which Melnikov’s Method is applied in my thesis. The reference coordinate system is (x,y,z), the current system is (1,2,3).
Animations and source code
I wrote a little Matlab program which visualises the motion of the Cardan-mounted Euler top. It is hosted on Matlab File Exchange. Below, you can see two videos, one showing regular (oscillating) motion and the other showing irregular (chaotic) motion.
A brief summary
Here is a brief summary aimed at a general audience:
Euler’s top is a rigid body (with arbitrary shape, say, a potato) suspended from its centre of gravity (CoG). Its motion is regular and well-known. In most cases, however, the CoG cannot be reached by a regular suspension method like a thread or rod (one would have to reach inside the potato, i.e. pierce a hole through it). But the problem can be solved using a more sophisticated suspension mechanism: a ‘Cardan mounting’ (e.g. helicopters or gyro compasses are similar to Cardan-mounted rigid bodies).
But as the dynamic properties of the Cardan-mounted Euler top are different from those of the normal Euler top, the resulting motion is much more complex: the spectrum of possible motion ranges from regular (i.e. harmonic, predictable) to chaotic (i.e. unpredictable) in a multitude of varying degrees. My thesis provides a description of the transition from regular to chaotic motion, using methods from nonlinear dynamics and Melnikov’s Method from perturbation theory.