The Koch Snowflake has finite area but infinite perimeter… yeah that happens with fractals. This abstract curve requires an infinite process (depicted in the gif) to construct and is an example of a fractal–a mathematical set (usually a curve or geometric figure) which exhibits a repeating pattern that displays at every scale. (More about fractals here https://en.wikipedia.org/wiki/Fractal)
But How? It seems clear that the area would be finite since the figure encloses a finite amount of space. To grasp why the Koch Snowflake has infinite perimeter, notice how as the iterations progress, the edges become more and more intricate. Now imagine trying to draw the edges with a pen. Since the construction of the snowflake continues indefinitely, the edges become infinitely intricate and you could never finish detailing these intricacies with your pen (that is the intuitive argument at least. I’ll leave the precise calculations up to you).
Fractals may seem so abstract and impractical but they actually have many useful real-world applications. For example, Benoit Mandelbrot (considered the “father of fractals”) found that stock market prices could be modeled with a factual curve. Check the wiki page for a long and diverse list of applications.
Fractal geometry may seem more abstract than traditional geometry but Mandelbrot argues that fractals are “the geometry of nature”. Objects in nature have random irregularities and are seemingly infinite in their intricacies. Attempting to incorporate this in drawings or animations is extremely difficult. Movie special effects and CGI often use fractals to make objects appear more natural looking. Since fractals can be made with mathematical formulae they are easy to generate with a computer. The first Star Wars movies were renowned for their special effects and were some of the first to use fractals to generate life-like explosions and landscapes of other worlds.
The term “Art Deco” comes from the Exposition Internationale des Arts Décoratifs et Industriels Modernes (International Exposition of Modern Industrial and Decorative Arts), the World’s fair held in Paris, France from April to October 1925. It was derived by shortening the words “Arts Décoratifs” in the title of the exposition.
Spirotechnics are related to two types of mathematical curves called hypotrochoids and epitrochoids, which are created by tracing the path of a point on one circle while it is rolled around the perimeter of another.
The distinction between the two is that hypotrochoids are created by rolling a circle inside a stationary circle, and epitrochoids are created by rolling a circle on the outside.
In Spirotechnics, multiple hypotrochoids and epitrochoids are nested within or outside each other, producing an endless variety of shapes.
In Ancient Greeks, up until Greece fell to Rome, geometry was the main mathematical science. Scientists practiced geometry to explore the relationships among points, lines, angles, and shapes.
Numbers was their religion, they thought of numbers as simple, pure and elegant. Therefore, they only believed in whole numbers (i.e, 6, 14, etc.) and ratios that are made up of whole numbers (i.e, 8/9, 2/3, etc.).
If a value wasn’t whole, or couldn’t be represented as a ratio of whole numbers, then it wasn’t a true number to them.
Pointing out that geometry actually needed other kinds of numbers was a kind of blasphemy. Legend has it that a Mathematician named Hippasus was thrown from a ship, condemned to drown at the bottom of the sea, for having proved what was contrary to their belief: That numbers can be imperfect.
Later Greek scholars had to confront the reality of the faults in their beliefs. But rather than make room in their understanding for numbers that weren’t whole, or weren’t ratios of whole numbers, they instead decided that those numbers were “ἄλογος”, meaning both “not a ratio” and “not to be spoken.”
The “not a number” fallacy prevented Greek algebraists of the time from advancing. For them, numbers had to be spoken, or at least written out. They followed the Pythagorean rule because they were fearful of joining Hippasus at the bottom of the Mediterranean Sea.
I used the cover of “The Web Design Book Volume 5″ for the shapes
So anyone else have an idea that the number 20 is important in the Bureau of Balance design scheme?
- The buildings are described as looking like geodesic domes; possibly resembling icosahedrons; a kinda roundish shape made of triangles.
- The training arena is formally known as the Icosagon.
- The emblem of the Bureau resembles two X’s when turned on its side- 20 in Roman numerals.
-Probably a coincidence, but on the TAZ Wiki, exactly 20 named characters are listed as bureau members.
Even if unintentional, it still works very well symbolism-wise, as the d20 is arguably the most important piece in a D&D game. What’s more, the icosahedron shape is the Platonic solid (3D shape where all the planes have equilateral faces and the same angle between each face) with the most facets, and such a mathematically perfect shape would likely be symbolic of order.