From time to time I'll add some images and movie files (though only small ones due to server space) which I feel are interesting, significant, or in some way cool.

I'll try to keep them grouped, with little explanations.

 

 

Fractals

I find that mathematics has an aesthetic quality to it which we often overlook. One great example of the beauty that mathematics can express is fractals. Each of these is a particular type of fractal, created with various color groupings. Click to see a larger one (they make great screen backgrounds as well!). Some natural examples of fractals are snowflakes, patterns in tree branches, frost on glass, branching of our lungs and circulatory system, and leaves. I love that you can zoom into a fractal infinitely, and there is always more and more detail to be found. It is said that 'the entire universe is contained within one flower' and perhaps that is an intuitive observation of the fractal concept.

Fractals are increasingly used not only for amazing imagery but also in science and engineering. Fractal Analysis is a growing field for modeling complex datasets, and there are numerous applications. The more I learn about Fractals, the more inspired I am about them as a profound concept for modeling the world around us.

This part of my website was originally created to inspire a student who I was a teaching assistant for in a psychology class focusing on statistics. The student loved digital art but did not feel a connection to mathematics and was having difficulty drawing connections to their interest. We had a discussion about Fractals, and I discussed how simple equations can generate incredible digital imagery. Fast forward to now, about twenty years later and that student succeeded in the course, graduated, completed a graduate degree and has a related business applying mathematical concepts to help people using psychology.

Good teaching helps the student bridge between their background as well as interests and the concepts being taught. Then the concepts themselves can be the beginning for the student and open doors to a fulfilling life. May these images inspire you as well.

 

"Wings of Freedom"

 

Mandelbrot Set fractal - possibly the most significant of all, named after the person who coined the term 'Fractal'

 

Fractal based on electromagnetic equations

 

Fractal based on trigonometric equations (tangent in particular)

 

"Nautilus"

 

"Comma"

 

Fractal Mountainscape

The random heights of this group of mountains are created by choosing a random height value, then breaking that up into groups of random height values, and so on. Pretty soon you have an interesting mountain! The water surface is created with a randomized set of combinations of the wave equation. The clouds are created by a similar method, using the equation of a sphere. Random points are created which represent the centers of overlapping spheres. Each one is partially transparent, and depending on the characteristics you give each quantity, you get various types of clouds.

 

Fractal trees.

One way of modelling trees is to use equations. A simple one is to begin with a randomly sized and shaped trunk, then at the end, break it up into 2 or 3 branches of 1/2 or 1/3 the size, in random directions, and then to break each of those up into 1/2 or 1/3 the size, and so on until the final end, which is leaves. No leaves suggests a tree in winter, as below is an example:

 

Shapes

Shapes can be used, even basic shapes to create powerful imagery and for powerful applications. Basic geometric shapes essentially define spaces. These can be representative of a solid object, a set of values, scores, a region of interest mathematically, etc. Even a basic group of a cube with a sphere subtracted creates a compelling image.