Fractal geometry transcends the Euclidean dimensions in mind-blowing ways. Quantify your knowledge of fractals with this brain-bending quiz!
Though the pattern was noticed by artists early on in human history, and the mathematical core began to be formed in the 19th century, the term itself was not used until 1975.
Fractals are definitely redundant, and more than complex, but the property of self-similarity is their most defining characteristic.
The Sponge and the Sausage would be useful if they could work as a yardstick, but the Koch Snowflake, with its progressively smaller "yardsticks," redefined the practice of measuring coastlines. Because of this, it's now generally acknowledged that any given coastline length needs to give the length of its yardstick in order to have real meaning.
Gaston Julia developed the idea of the feedback loop, where the value produced by each iteration of a formula was recycled and used as the input value for the next iteration. Benoit Mandelbrot built on Julia's work and summed it up nicely with his Mandelbrot Set.
The CIA World Factbook gives the coastline's length as around 7,723 miles, but Lewis Richardson, using the Koch Snowflake, realized that if you use a small enough ruler the coastline becomes infinite.
If only mountain shapes were part of Euclidean geometry. Triangles work well in a pinch though, as Loren Carpenter found out working for Boeing in the late 1970s. He created an amazingly (for the time) realistic mountain range using computers a lot less powerful than the one in your cell phone.
In fact, it stands for two real numbers, as the variable c is what's called a "complex number." In this case, this number is a coordinate on the Cartesian Grid. By the way, Einstein also used this variable in his famous formula explaining the relationship of mass to energy, where c was the speed of light.
The term came about as a way to describe the fractional aspect of the dimension the shapes that became fractals were taking, since they defied description by traditional Euclidean terminology. Luckily for him, the artistic quality of fractals did wonders to help promote his efforts.
For those brought up in traditional geometry it takes a little mind bending to reach outside of it. But fractal dimension provides a mathematical way to describe just how rough the surface of a shape is. The rougher the surface, the higher the dimension.
One of the most important and mathematically rigorous ways to measure fractal dimension, the Hausdorff-Besicovitch Dimension is always greater than its correlated topological, or Euclidean, dimension. Mandelbrot once claimed that if Hausdorff is the father of non-standard dimension (which he is), then Besicovitch must be the mother. You get half credit if you answered A.
Some artists definitely like fractals, but the colors are in fact representative of how fast the equation escapes the Mandelbrot Set to infinity. Each color represents one iteration, so the simplest Mandelbrot illustrations aren't very colorful. Throw in more and more and pretty soon you have a work of art.
Telecommunications rely on the physical properties of radio waves, which like their name suggests travels in a wave-like way (think of a sine curve). This means that any antenna designed to pick up these waves should include wires that are equidistant from each other. The Menger Sponge achieves this property in a way that actually allows it to receive multiple wavelengths, making it ideal for for modern mobile devices.
Like real mountains and trees, fractals too have a limit of three dimensions. After that, you're off into a whole new world of space-time stuff.
Most people couldn't name the artist, but they recognize the iconic work of Katsushika Hokusai, created in 1820. And you can even see Mount Fuji in the background.
Letterman was still studying algebra in junior high in 1960, but probably thinking more about television. Edward Lorenz wanted to re-run a weather prediction calculation, but shortened his input variable by a fraction of a percent, and was shocked when the end result was vastly different than before.
You have to wonder how many great ideas suffered in the past umpteen thousand years for want of a good computer. In fact, the traditional meaning of "computer" was "one who computes." No wonder they didn't get very far.
It doesn't seem to make any sense when you say something has an infinite perimeter surrounding finite space, but you can't escape it. The weird thing is that it doesn't even converge toward a specific number. It's a true paradox. So next time something blurts out how long a coastline is (don't hold your breath), one-up them and ask them how long their yardstick is.
Some really hard-core mathematicians might cringe, but the Minkowski-Bouligand Dimension didn't bother Mandelbrot one bit. It's a fantastic and relatively simple way to get an idea of the fractal dimension of something, and it can be as accurate as you need. What more could you ask? Really hard-core mathematicians need not reply.
This fractal and others like it get their beauty by flying around Strange Attractors, like a moon around its planet in some loopy orbit. Even the Julia and Mandelbrot sets are "attracted" to infinity to a certain degree, but apparently this is less strange to those naming these things.
At last, a little geography/history question. Although he spent most of his formative years in France, including a few years evading German occupiers on the prowl for Jews, he was born in Poland in 1924. Eventually, not finding his unconventional theories welcome among French academia, he joined IBM in America and discovered the wonderful abilities of computers.