scienceisbeauty: Fractals are sets of points that have a self-similar geometric or statistical structure upon magnification of any part of the set. An especially intriguing and modern set of points that is partly fractal is the positions of 200,000 galaxies up to two billion light years away from Earth (about 1/6 the diameter of the known universe!) as measured recently by the Sloan Digital Sky Survey. Each dot represents a separate galaxy and the color of the dot represents that galaxy’s luminosity. An analysis of these data indicate that the baryonic matter that Earth is made of constitutes only 5 percent (!) of the mass of the universe. The rest of the mass consists 25 percent of “dark matter” and 70 percent of “dark energy”. What these dark matter and energy consist of are two of the most interesting unsolved questions of current science. And how did the expansion of the universe and the gravitational coupling of matter to dark matter and energy produce this unusual clustered geometric structure? Source: Duke Physics, link 

scienceisbeauty:

Fractals are sets of points that have a self-similar geometric or statistical structure upon magnification of any part of the set. An especially intriguing and modern set of points that is partly fractal is the positions of 200,000 galaxies up to two billion light years away from Earth (about 1/6 the diameter of the known universe!) as measured recently by the Sloan Digital Sky Survey. Each dot represents a separate galaxy and the color of the dot represents that galaxy’s luminosity. An analysis of these data indicate that the baryonic matter that Earth is made of constitutes only 5 percent (!) of the mass of the universe. The rest of the mass consists 25 percent of “dark matter” and 70 percent of “dark energy”. What these dark matter and energy consist of are two of the most interesting unsolved questions of current science. And how did the expansion of the universe and the gravitational coupling of matter to dark matter and energy produce this unusual clustered geometric structure?

Source: Duke Physics, link 

elegantly constructed proofs through visual inspection

elegantly constructed proofs through visual inspection

Math is truly beautiful.

Visual proof for Euler’s Identity: Starting at e0 = 1, travelling at the velocity i relative to one’s position for the length of time π, and adding 1, one arrives at 0.

Visual proof for Euler’s Identity:

\displaystyle e^{i \pi} + 1 = 0.

Starting at e0 = 1, travelling at the velocity i relative to one’s position for the length of time π, and adding 1, one arrives at 0.

Arthur Benjamin does “Mathemagic” (TED Talk)

bestofwikipedia:

The Feynman point is the sequence of six 9s which begins at the 762nd decimal place of the base 10 representation of π. It is named after physicist Richard Feynman, who once stated during a lecture he would like to memorize the digits of π until that point, so he could recite them and quip “nine nine nine nine nine nine and so on.”

Graphical representation of Collatz conjecture depicting all numbers with an orbit length of 20 or less

Graphical representation of Collatz conjecture depicting all numbers with an orbit length of 20 or less

From Wikipedia:

The Collatz conjecture is an unsolved conjecture in mathematics named after Lothar Collatz, who first proposed it in 1937.  

Take any natural number n. If n is even, divide it by 2 to get n / 2, if n is odd multiply it by 3 and add 1 to obtain 3n + 1. Repeat the process indefinitely. The conjecture is that no matter what number you start with, you will always eventually reach 1. It has been called “Half Or Triple Plus One”, sometimes called HOTPO.

(full article)

Penrose tiles in ancient Islamic art (full article)

From Wikipedia:

A Penrose tiling is a nonperiodic tiling generated by an aperiodic set of prototiles named after Sir Roger Penrose, who investigated these sets in the 1970s. Because all tilings obtained with the Penrose tiles are non-periodic, Penrose tiles are considered aperiodic tiles. A Penrose tiling may be constructed so as to exhibit both reflection symmetry and fivefold rotational symmetry, as in the diagram at the right.

A Penrose tiling has many remarkable properties, most notably:

  • It is nonperiodic, which means that it lacks any translational symmetry. More informally, a shifted copy will never match the original exactly.
  • Any finite region in a tiling appears infinitely many times in that tiling and, in fact, in any other tiling. This property would be trivially true of a tiling with translational symmetry but is non-trivial when applied to the non-periodic Penrose tilings.
  • It is a quasicrystal: implemented as a physical structure a Penrose tiling will produce Bragg diffraction; the diffractogram reveals both the underlying fivefold symmetry and the long range order. This order reflects the fact that the tilings are organized, not through translational symmetry, but rather through a process sometimes called “deflation” or “inflation.”

Various methods to construct Penrose tilings have been discovered, including matching rules, substitutions, cut and project schemes and coverings.

“When you get math involved, problems that you solve for aesthetic value only, or to create something beautiful, turn around and turn out to have an application in the real world.”

Robert Lang folds way-new origami | Video on TED.com (via crackingkettles)