Form

The mathematics of form as an exploration of parameter spaces defined by minimum inventory/maximum diversity systems.

An oscillator

This formula describes an simple oscillator and serves as a vehicle for expanding my knowledge about Sage:

The first step is to make it a bit more intresting. I add two oscillators and choose some arbitrary constants:

Now I know the position of the oscillating particle, then I calculate the first derivative to get the particle's velocity:

The next logical step is to do a graph of the functions:

The green lines indicate the two parts of the oscillator; the red line is the compound function; and the blue is the velocity.

The last step is to make a graph of position versus velocity and that graph looks really neat:

The change of color indicates the pasage of time. The sequence starts as red, and turn into yellow; green; blue and purple.

The whole thing is pretty simple to do even though a few Python constructs are needed.

Winter trees

The trunks of the trees along the railroad line are greenish black, only a few stubborn leaves cling to the twigs. It is the time of year where the structure of trees are clearly revealed against the grey sky. I'm enjoying the diversity of branching patterns and wishing that I could collect all the patterns in one drawing.

Sage a beginning

I have made some rather basic calculus experiments in the online version of Sage, the open source mathematical system. It works really well, the Notebook interface is easy to use, and the mathematical typesetting is beautiful. I think a have found a new tool, as there is no need for a local installation, a browser is all that's needed. It is also a way to get introduced to Python programming.

What's in a good model

After reading Joshua M. Epstein comprehensive explanation of why we make models, I asked my self which elements does a good model contain? and well this is my list:

• Prose: because the use of a informal human language creates a mental image of the system.
• Illustration: because a drawing acts as a map of the system.
• Numbers: because numbers, originating from either measurements or assumptions, are inherent to system and they can be calculated up on.
• Algorithm: because to make even the simplest algorithm that describes a system the connections within the system has to be identified.
• Mathematics: because mathematical notation demands formality in thinking.
  

The spectrum of systems

I have read an article from New Scientist, about the financial system viewed as an ecological system. It made me think about heterogeneous versus homogeneous systems as the end points of a spectrum; where:

• Heterogeneous systems are robust but not optimal in an economic sense.
• Homogeneous systems are vulnerable but optimal in an economic sense.

Shape diversity

This weekend I did spend some time at the shores of the Great Belt. I have realised that the diversity of possible plant shapes are even larger than the diversity of shapes I created in my sketches of imaginary plants. I found a wide variation of beautiful entangled form, the result of an evolution guided by the harsh environment of salt, wind and sea.

OS X mathematics

I have just discovered a marvelous piece of software called Grapher. It is bundled with OS X and it creates beautiful graphs. As a start I created a graph of some of the solutions for a very simpel second order ODE:

Branching - the scientific way

When a a biophysicist model a pattern in nature, and that includes the branching of plants, it is most often done by estimation a system of diferential equations. Like the Gierer-Meinhardt model system which models an activator-inhibitor process. The graph illustrates my modest first try out in the estimation of such a model. Incidentally I used SAS, to estimate the model, well now I have added PROC IML to my SAS knowledge.

Plant (3) - where do plants branch?

Often when I have a minute or two, maybe waiting for a train, I study the structure of plants - how they branch, how they grow. One thing that I have noticed is, that behind the apparent symmetric structure of a plant, looking carefully, a haphazard structure is found. A structure with a lot of false starts and runaway tries aiming nowhere.