# Patterns of Nature with MathLapse

### MathLapse Software

This MathLapse was realized by *Alessandro Cattaneo*, C*ristina Cattoni*, *Filippo Favale* and *Riccardo Moschetti* (members of Curvilinea - "*Our cooperative company was born for our common passion for scientific popularization of mathematics*") for the Imaginary Conference 2016. The IMAGINARY conference on open and collaborative communication of mathematical research (Math, imagination, discovery): *The community is invited to think along, to work along, to help in design, programming or general implementation*. IMAGINARY is a non-profit organization for open and interactive mathematics.

- https://imaginary.org/galleries - Galleries are the visual heart of the platform.

- https://imaginary.org/films - Here you can find films and their background to be watched online and used for public screenings at exhibitions or in museums.

- https://imaginary.org/texts - Here you can find a variety of mathematical texts on many different topics.

- https://imaginary.org/projects - Projects are open for collaboration and need support. Projects can be a new exhibition, new software, new ways of communicating maths, a publication, etc.

EXAMPLES

1. https://imaginary.org/film/mathlapse-patterns-of-nature - A wonderful way to describe natural shapes using the language of mathematics is provided by self-similar patterns.

2. https://imaginary.org/film/lpdjlqh-d-vhfuhw - An artistic journey into the world of elliptic curve cryptography.

3. https://imaginary.org/film/the-math-of-the-shells - This film illustrates how the great majority of seahells existing in nature can be generated by a fixed set of equations by simply varying some parameters.

4. https://imaginary.org/film/mathlapse-cybernetic-ant-watching-its-own-creation - The video shows the construction of an algebraic curve by a mechanical linkage.

5. https://imaginary.org/film/triangulating-unknown-environments-using-robot-swarms - How can a robot swarm explore an unknown area and understand its layout? In this video, a swarm of R-one mobile robots is used for such a task.

### Patterns of Nature

A wonderful way to describe natural shapes using the language of mathematics is provided by self-similar patterns. The idea is to repeat the same base module on different scales and positions, following an iterative process that gives rise to very complicated and amazing results. By iterating the process infinitely many times one can often obtain a very interesting mathematical object: a fractal.

The protagonists of the Mathlapse are: Acer palmatum, Echium vulgare, Verbena Bonariensis, Pteridophyte, Hordeum vulgare L., Acer palmatum leaf.

In order to realize these shapes, one starts from a base module, composed by different lines. In the first iteration, one has to draw, for each segment, a copy of the base module; one has also to scale, translate and rotate the module depending on the position of the segment. In this way one obtains a different figure, composed by a certain number of copies of the initial module. Now it is possible to iterate the process using the same rules applied to each new segment. If the initial module is carefully chosen, the shape that is obtained after a few iterations can be very similar to some plants, or other objects which can be found in nature. By iterating the process infinitely many times one can often obtain a very interesting mathematical object: a fractal. For example, the Sierpinski triangle or the Cantor set can be obtained in this way. Is it even possible to calculate the fractal dimension of some of these fractals created with such a rule once the length of the lines and then the ratio of the scalings are known.

By adding simple rules one can obtain a complete new variety of objects. For example, a base module can have some “special” segments which “stops” the recursion; in other words they are not used for the further iterations. In this way one can create the trunk of a tree, or the stalk of a plant. By using these rules, and by considering also the length of the lines it is possible to color these self-similar shapes in a very realistic way.

Source: https://imaginary.org/film/mathlapse-patterns-of-nature