mathmajik

ryanandmath:

Imagine you wanted to measure the coastline of Great Britain. You might remember from calculus that straight lines can make a pretty good approximation of curves, so you decide that you’re going to estimate the length of the coast using straight lines of the length of 100km (not a very good estimate, but it’s a start). You finish, and you come up with a total costal length of 2800km. And you’re pretty happy. Now, you have a friend who also for some reason wants to measure the length of the coast of Great Britain. And she goes out and measures, but this time using straight lines of the length 50km and comes up with a total costal length of 3400km. Hold up! How can she have gotten such a dramatically different number?

It turns out that due to the fractal-like nature of the coast of Great Britain, the smaller the measurement that is used, the larger the coastline length will be become. Empirically, if we started to make the measurements smaller and smaller, the coastal length will increase without limit. This is a problem! And this problem is known as the coastline paradox.

By how fractals are defined, straight lines actually do not provide as much information about them as they do with other “nicer” curves. What is interesting though is that while the length of the curve may be impossible to measure, the area it encloses does converge to some value, as demonstrated by the Sierpinski curve, pictured above. For this reason, while it is a difficult reason to talk about how long the coastline of a country may be, it is still possible to get a good estimate of the total land mass that the country occupies. This phenomena was studied in detail by Benoit Mandelbrot in his paper “How Long is the Coast of Britain" and motivated many of connections between nature and fractals in his later work.

fuckyeahfluiddynamics

fuckyeahfluiddynamics:

Soap bubbles are ephemeral creations. The slightest prick will send them tearing apart in the blink of an eye. It may come as a surprise, therefore, that dropping a water droplet through a bubble will not break it. Instead, the bubble will heal itself using the Marangoni effect. In a soap bubble, the soap molecules act as a surfactant, lowering the surface tension of the water and allowing the fragile structure to hold together. When the water drop impacts the bubble, the local surface tension increases because of the relative lack of soap molecules. This increase in surface tension pulls at the rest of the bubble, drawing more soap molecules toward the point of contact. The effect evens out surface tension across the surface and stabilizes the bubble. You can test the effect at home, too. If you wet your finger, you can poke a soap bubble without popping it. (Video credit: G. Mitchell; via io9)

proofmathisbeautiful

proofmathisbeautiful:

Mesmerizing Interiors Of Iran’s Mosques Captured In Rare Photographs By Mohammad Domiri

Mohammad Domiri, a talented architectural photographer from northern Iran, takes stunning photos of grandiose mosque architecture throughout the Middle East.

Middle Eastern architecture is often recognized by its elegantly curved arches and spiraling columns, which feature heavily throughout Domiri’s photos. Many of the historic sites Domiri shoots are decorated with colorful stained-glass windows, geometric decorations and painstakingly detailed mosaics, so he shoots with special wide-angle lenses to make sure that he captures all of these details. Because they are historic structures, many of these mosques also impose heavy restrictions on photography – making photos like Domiri’s very rare.

mathmajik

underthesymmetree:

Fibonacci you crazy bastard….

As seen in the solar system (by no ridiculous coincidence), Earth orbits the Sun 8 times in the same period that Venus orbits the Sun 13 times! Drawing a line between Earth & Venus every week results in a spectacular FIVE side symmetry!!

Lets bring up those Fibonacci numbers again: 1, 1, 2, 3, 5, 8, 13, 21, 34..

So if we imagine planets with Fibonacci orbits, do they create Fibonacci symmetries?!

You bet!! Depicted here is a:

  • 2 sided symmetry (5 orbits x 3 orbits)
  • 3 sided symmetry (8 orbits x 5 orbits)
  • sided symmetry (13 orbits x 8 orbits) - like Earth & Venus
  • sided symmetry (21 orbits x 13 orbits)

I wonder if relationships like this exist somewhere in the universe….

Read the Book    |    Follow    |    Hi-Res    -2-    -3-    -5-    -8-