mathematical shapes

Koch Snowflake: Finite area, Infinite perimeter.

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.

vividlasagna  asked:

question: was there as much of a coherent ideology behind art deco as there was behind modernism? and if so what was it?

Art Deco is a style not an ideology like Modernism.

Radio City Hall in New York City

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.

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To appreciate the nature of fractals, recall Galileo’s splendid manifesto that “Philosophy is written in the language of mathematics and its characters are triangles, circles and other geometric figures, without which one wanders about in a dark labyrinth.” Observe that circles, ellipses, and parabolas are very smooth shapes and that a triangle has a small number of points of irregularity. Galileo was absolutely right to assert that in science those shapes are necessary. But they have turned out not to be sufficient, “merely” because most of the world is of infinitely great roughness and complexity. However, the infinite sea of complexity includes two islands: one of Euclidean simplicity, and also a second of relative simplicity in which roughness is present, but is the same at all scales.
—  Benoit Mandelbrot
buddingsaccharomyces replied to your post “WIP”

rotate one section counterclockwise so touch is at 12 o clock, then mirror the upper dulie so it’s symmetrical to the righthand chuxas, then ditch the bottom two arms?

I understand all the words involved in this explanation, but A) actually understanding them is spinning my gears becuase my spatial sense is SHIT.  And B) it would be a pain in the ass to do because of the way the layers are set up.  

Plus C) if I’m imagining what you’re saying correctly, the final product would be symmetrical only across the vertical axis, and I like it symmetrical in the way that it is now, if you catch my drift!

I appreciate that you put the thought into figuring out how to do it, but I’ll probably just keep it the way it is and take the repeats tbqh.  TuT

The Mechanics of the Language of Love

‘Fashionista’ is one more tile in the mosaic that is the ‘Holy Robots’ sequence. Mayhap it is a tile in a puzzle, where the tiles must be shifted in their slots until they make the desired image. Mayhap. Here, Orlova jokes about nail polish, allowing her robot “one hundred hands” to experiment on, so that she can make use of her “collection” in good time, “before/Betelguese explodes.” Science-fiction meets twenty-first century vanity. Orlova is the mistress of the accurate juxtaposition. Her poems are like finely decorated nails, held out for inspection, presented to the world in a series of surprising photographs. Fashion might reduced to the particulate, the minuscule, the refined. Orlova has fine hands and daring taste in fashion, colour, life. Her living hand reaches out of the robot of the words, the mathematics of the shaped poem. Some things might be beautiful, but I am not sure of that. I am not permitted to exist in amongst the beautiful. 

@vasilinaorlova

Kirsten Fell - from The Music Antics of the Landslide of Learning - a moped manual on ‘Huggy Realists’ by Vaguely Acidic Or Not So

The Question That Could Unite Quantum Theory With General Relativity: Is Spacetime Countable?

Current thinking about quantum gravity assumes that spacetime exists in countable lumps, like grains of sand. That can’t be right, can it?

One of the big problems with quantum gravity is that it generates infinities that have no physical meaning. These come about because quantum mechanics implies that accurate measurements of the universe on the tiniest scales require high-energy. But when the scale becomes very small, the energy density associated with a measurement is so great that it should lead to the formation of a black hole, which would paradoxically ruin the measurement that created it.

These kinds of infinities are something of an annoyance. Their paradoxical nature makes them hard to deal with mathematically and difficult to reconcile with our knowledge of the universe, which as far as we can tell, avoids this kind of paradoxical behaviour.

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Neolithic Carved Stone Polyhedra.

Euclid, 300 BC and the Ancient Greeks, in their inherited love for Geometry, called the Five Solids shown above, the Atoms of the Universe.

In the same way that we today believe that all Matter is made up of combinations of Atoms, so the Ancient Greeks also believed that all Physical matter is made up of the Atoms of the Platonic Solids and that all matter has a Mystical side represented by their Connection with Earth, Air, Fire, Water and Aether.

Similar to our conventional Atom model which shows a Nucleus surrounded by Electrons in orbits creating Spheres of Energy, the Greeks felt that these Platonic Solids also have a Spherical property, where one Platonic Solid fits in a Sphere, which alternately fits inside another Platonic Solid, again fitting in another Sphere. It is fascinating to see how any one of these Solids can fit inside one another. The concept of one Sphere fitting inside another Sphere is surprisingly frequently seen in different cultures. Indeed, the Mechanism of Platonic Solids is so perfect, that perhaps as we are approaching in this study, their concept of Platonics as being the building blocks of Matter might be more evolved than our present knowledge of the Atom model.

As shown in the photograph above, as in so many other aspects of their Science and Philosophy, the Greeks were not the originators of these concepts. The photograph is of a collection Neolithic Stones, unmistakably showing the same basic “Platonic” Shapes. These are at least 3,000 years old (>1000BC). Indeed, we know that from the Vedic times, around 3000 B.C. to 1000 B.C., Indians (Indo-Aryans) had classified the Material world into the Four Elements; Earth (Prithvi), Fire (Agni), Air (Maya) and Water (Apa). To these Four Elements was added a Fifth one; Ether or Akasha. According to some scholars these Five Elements or Pancha Mahabhootas, were also identified with the various human senses of Perception; Earth with smell, Air with feeling, Fire with vision, Water with taste and Ether with sound. Whatever the validity behind this interpretation, it is true that since very Ancient times Indians had perceived the Material world as comprising these 5 Elements.

The information one can get from these carved out Shapes shows that a highly developed generation of human kind gave a lot of scientific importance to these Shapes, and perhaps carving out these Stones was one of their attempts to pass over their Knowledge to others, including us.