Interactive visualisation of the Mandelbrot set using the Genie app

Introduction

The Mandelbrot set is one of the most famous fractals, first studied by the French mathematicians Pierre Fatou and Gaston Julia in the early 20th century, but widely known for the work of Benoit Mandelbrot in the 1970s and 80s. This is the set of points on the complex plane for which the iterative process remains bounded when .

Mathematically, the Mandelbrot set is defined as follows. For each complex number c, consider the sequence:

Where:

• — the current value in the iterative process, which is calculated step by step using the formula .

• — the complex number under investigation (the coordinate of a point on the complex plane, checked to see if it belongs to the Mandelbrot set).

• — the number of the current iteration, indicating how many steps have been taken in the calculation process.

If the module remains bounded (does not exceed ) when , then point c belongs to the Mandelbrot set. In practice, we limit ourselves to a finite number of iterations and consider a point to belong to the set if, within a given number of steps, the sequence has not extended beyond the radius circle. .

The configuration of the set is striking in its complexity and infinite detail: the main cardioid, the kidney circle, and countless ‘threads’ and ‘pollen’ along the boundary. As the scale increases, more and more new structures are revealed that resemble the basic shape — this property of self-similarity is characteristic of fractals. This set has become a classic example of fractal geometry and is used in mathematics, physics, computer graphics and even in art.

This application implements an interactive Mandelbrot set browser. By changing the initial parameters, you can explore the fractal in real time, navigate across the complex plane, zoom in on areas of interest, and observe the infinite complexity of the set’s boundary. Different colour schemes allow you to visualise the depth of the divergence of points — that is, how many iterations it took to exceed the threshold value.

Launching the app

The script for this application is in the file mandelbrot.jl. Launch the application and open it in a new browser tab.

This application is a digital model of a mathematical microscope for studying fractal structures.

By adjusting the parameters on the control panel, you can control the visualisation settings:

• The dimensions of the axes determine the resolution of the final image and allow you to optimise it for your screen (the number of pixels horizontally and vertically).

• The number of iterations sets the maximum computational depth — the higher this is, the more detailed the fine structures on the boundary of the set will be.

• The coordinates of the centre allow you to move along the complex plane, selecting the area to explore.

• Zoom enlarges or reduces the image, allowing you to view the fractal with varying levels of detail.

• The colour scheme determines how the number of iterations before divergence is displayed, ranging from monochrome to rainbow palettes.

Observe how, as you approach the boundary of the set, more and more new details are revealed, repeating the basic shape in endless recursion. The colour gradients will allow you to see various aspects of the dynamics of the iterative process.

By moving the cursor to the top-right corner of the image and clicking on the camera icon, you can save an image of the fractal.

Once you have finished using the application, close it.

Conclusion

This application is a digital tool for studying one of the most famous objects in modern mathematics. Virtual laboratories of this type allow you to study fractal geometry without having to programme the visualisation yourself. By modelling various regions of the complex plane and adjusting the calculation parameters, the user can experimentally explore the properties of fractals: self-similarity, infinite detail and boundary complexity. The application can serve as a textbook on fractal geometry, the theory of dynamical systems and complex analysis. The combination of Engee and Genie opens up opportunities for creating interactive materials that make abstract mathematical concepts visual and accessible to a wide audience.