Fun With Fractals - A Quiz About Roughness

Answer: Geometry

Fractals are infinitely complex patterns that are self-similar across different scales The term "fractal" itself comes from the notion of breaking a line or a figure up into smaller but similar instances of the original. Conceptually, this is done over and over, ad-infinitum resulting in infinite sets.

Combinatorics deal primarily with counting and sets. Game theory deals primarily with strategies between rational decision makers (often in zero-sum games). Probability deals with the chances of indeterminate outcomes (e.g. odds & gambling)

Answer: determining optimal pathing in highway traffic situations

Loren Carpenter of Pixar Studios used fractals on low speed computers to simulate mountain ranges when employed at Boeing and then again at Lucasfilm in developing the world's first fully CGI sequence for a feature film, "Star Trek II: The Wrath of Khan". Mandelbrot initially presented his work in a paper "How Long Is the Coast of Britain?." Repeated branching as in tress, the placement of leaves in ferns, and blood vessels in the circulatory system have all been successfully modeled using fractal functions.

Highway traffic problems don't involve the sames aspects of natural roughness in which fractals excel. They tend to be more related to probability.

Answer: Benoit Mandelbrot

Known for his talent for practical research, Mandelbrot and computer affinity, born a Polish Jew, became an IBM employee in 1958. He was one of the first to use computer graphics to create and display fractal geometric images which display "the art of roughness."

Answer: all of these

Fractal problems have been touched upon since as early as the 17th century but most of the earliest work in the area was hindered by the amount of repetitive calculation that was required. In fact some of the earliest formulae in this area were referred to as mathematical "monsters" because of their unwieldiness.

Answer: cell phones

Most cell phones use antennas that are based on fractal designs. The recursive nature of self-similarity that is required of fractal values pretty much guarantee a wide variety of effective lengths meaning that they can handle a wide spectrum of signal frequencies in a smaller footprint than other antenna designs.

Answer: the stripes on a zebra

The stripes on a zebra are basically regular and mirror the growth patterns of grasses. If viewed in closer detail they do not exhibit the "self similar" aspect of fractals.

Conversely, zooming in on snowflakes, coastlines and tree branches all yield images that exhibit similarities to their zoomed out images.

Answer: an equilateral triangle

To construct a Koch snowflake, you need to do the following:

1) Start with an equilateral triangle.

2)Now divide each of the legs of that triangle into three equal parts and replace the middle third with two line segments of that length that form their own equilateral triangles (with no base), pointing outwards from the original shape.
This should be a six pointed shape similar to "the Star of David." Not that the perimeter of the resultant shape is 4/3rds the length of the original perimeter.

3) Dividing each of the resulting line segments into three parts and repeating
this process will yield a figure that has 18 points and begins to resemble a snowflake

4)The fourth iteration will yield a figure with 54 points, then 162, 486 and so on..

Each iteration yields a figure that has a perimeter 4/3s of the length of the previous figure and it quickly becomes apparent that, as iteration continues the perimeter will approach infinity even though the enclosed area remains finite.

Answer: the Sierpinski Triangle

The Sierpinski Triangle is also sometimes called the Sierpinski gasket or the Sierpinski sieve. It was first conceived by Wac³aw Sierpiñski in 1915. Though it's clearest when started with an equilateral triangle, other starting shapes can be used in this transformation. Most modern cell phone antennas are based on one of these types of functions.

Answer: noise in incoming signal transmissions

Mandelbrot graphed the noise rate and noticed that the graphs looked almost identical regardless of the time scale. It reminded him of the self similarity that he'd heard about with "the monsters."

Answer: Tower of Hanoi puzzle

In Tower of Hanoi puzzles the player must move disks from one rod to another one at a time subject to rules. In the standard version, each move must take only one disk from a stack and place it atop any other stack and no larger disk may be placed on top of a smaller disk.

In a full solution of this puzzle, each disk will move only half as often as the disk directly above it and plotting the movements by the height of the disk moved creates a fractal-like shape (although it would only be a true fractal for an infinite height puzzle).