Fractals in Virtual Reality

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Logo programming with its recursive procedure capability enables easy creation of fractal geometry. In VRMath2, many of the existing fractals by tranditional 2D Logo programs can be viewed in 3D and reconstructed to be 3D fractals. This project will explore existing and new ideas about fractals, and use the 3D Logo programming environemnt in VRMath2 to generate and share 3D fractals online. Below are some classic examples of fractals.

Fractals in the nature

1. Fern leaf

Perhaps one of the most popular fractals in the nature, the fern leaf has a recursive structure, which can be described and coded using the Logo turtle geometry.

In VRMath2, the third dimention can be utilised to cretae a curved 3D fern leaf, which is based on the 2D recursive procedure above.

2. Trees

There are a few popular trees created in traditional 2D Logo. In this project, we can experiemnt based on these 2D procedures and make 3D trees. Below are two tree examples, and you may copy the codes into the VRMath2 Editor to make them 3D.

Another tree (binary).

Mathematical fractals

Fractals are recursive and self-similar. Using mathematical ideas with recursive programming could produce a variety of fractals. Here are some well known fractals.

1. Koch snowflake

Start with a triangle

First iteration

Second iteration

Third iteration

2. Sierpiński triangle

Generated by recursive procedure

Start with a triangle

First iteration

Second iteration

Thrid iteration

Generated by random procedure

2D, 3000 points

3D, 5000 points, tetrahedron

3D, 5000 points, Egyptian pyramid