Mathematics can help us to draw real-life objects. The regularity of natural patterns can lead artists to use mathematical concepts in works of art. Many plants have very interesting and beautiful leaves and some mathematical patterns can be found in their structures. For example, Aloe polyphylla is a plant species from Lesotho. The leaves of an Aloe polyphylla form very beautiful spirals.

There are various ways of generating leaves shapes with mathematical concepts. A famous example is the Barnsley fern. The British mathematician Michael Barnsley (born 1946) described this beautiful fractal in his book Fractals Everywhere. His fractal resembles the fern leaves. He created this fractal by using the iterated function system method.

Drawing a Japanese maple leaf with mathematical formulas. Credit: Hamid Naderi Yeganeh

You might imagine that the leaves in the images below are drawn by hand. But they are actually computer-generated mathematical figures. These shapes are generated with the trigonometric functions sine and cosine. Here I would like to talk about the method I used to create these drawings.

When I want to draw a real-life object, I try to find a mathematical formula that produces the drawing. I use a step-by-step process to find such a formula. In each step of the process, I try to increase the resemblance of the drawing to the real-life object by adding a mathematical expression to the formula. Usually, I search through the expressions that are generated by the sine and cosine functions. The properties of these two mathematical functions (especially periodicity, boundedness and smoothness) make them very useful in the process. Indeed, I need to solve the problem of finding an appropriate mathematical expression in each step. So some steps may be very difficult or even impossible. For example, in the image above, you can see the process of generating a mathematical curve that resembles a Japanese maple leaf. The graph shows how the process transforms a circle into the drawing.

These images show three mathematical curves that are created with the mentioned method. They are described by parametric equations:

Maple Leaf

Maple Leaf. Credit: Hamid Naderi Yeganeh

Japanese Maple Leaf

Japanese Maple Leaf. Credit: Hamid Naderi Yeganeh.

Oak Leaf

Oak Leaf. Credit: Hamid Naderi Yeganeh

Also, below you can see two mathematical figures created by drawing thousands of circles. The radiuses and centers of the circles are determined with trigonometric functions.

Olive Branch

Olive Branch. Credit: Hamid Naderi Yeganeh

Palm Branch

Palm Branch. Credit: Hamid Naderi Yeganeh