Algae
Aristid Lindenmayer introduces a system of substitutions that would later bear his name. He did this in a set of two articles; Mathematical Models for Cellular Interactions in Development. As the title implies it offers a mathematical theory to model growth in certain type of cells. Stripping away the mathematics his article described an example that modeled algae. A more modern description follows below.
We will study words, i.e. a sequence of symbols, over an alphabet. In our example we take our alphabet to be the symbols A, and B. We will examine a series of words. In the algae example we are starting with the word A, it is our axiom. With the current word in the series, we replace each symbol with a sequence of symbols and concatenating the sub-sequences in a new word. In our example we will replace each occurring symbol A with the sequence AB, and each occurring symbol B with the sequence A.
Below you find the first few iterations of this process.
- A, our starting word.
- AB, because the single A is replaced with AB.
- ABA, because the single A is replaced with AB and the B is replaced with A.
- ABAAB, because each A is replaced with AB and each B is replaced with A.
In the above model the symbol A is a model for a mature cell, ready to divide itself. The division however is asymmetric. It allows the original cell to comfortable remain in place, making place for a young cell B. In its stead, the young cell B first must mature to and become an A cell before it can start reproducing itself.
Although this is a simple model. A lot can be learned from this. For an few examples, see the exercises.
Exercises
- Extend the example with the a few iterations.
- Count the number of symbols in each word of the series. Guess what number comes next.
- Count the number of A's and the number of B's separately. What do you get.
- We will number the words in our series, \(W_{0}\) for our start word, \(W_{1}\), for the next, \(W_{2}\) for the one after that, etcetera. Pick any number, let's say 4. Notice how \(W_{4}\), i.e. ABAAB is the concatenation of ABA, which is \(W_{3}\), and AB, which is \(W_{2}\). In other words \(W_{4}=W_{3}W_{2}\).
- Check if something similar holds for \(W_{3}\) and \(W_{5}\).
- Does this property, i.e. \(W_{k} = W_{k-1}W_{k-2}\) always hold?
- The above properties remind us of the Fibonacci sequence. For the Fibonacci sequence there is a formula that instantly calculates the the value in the sequences. Robert Dimartino found out that a similar property holds for our words. Search for Fibonacci word for more information.