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It is the human condition.

Scarfolk!

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Where t=.6141866…; a = ¼; b = 0, 0405353 + 0, 0255082i ; such that the rotation number of f(t,b,a) on the unit circle is (√5−1)/2 . The image has been rotated.

The Julia set of the cubic rational function e2πitz2(z−4)/(1−4z) with t=.6151732… chosen so that the rotation number is (√5−1)/2, which has a Herman ring (shaded).

Herman ring

If f is a rational function, defined in the extended complex plane, and if it is a nonlinear function (degree > 1) then for a periodic component of the Fatou set, exactly one of the following holds:

U contains an attracting periodic point.

U is parabolic.

U is a Siegel disc.

U is a Herman ring: In the mathematical discipline known as complex dynamics, the Herman ring is a Fatou component, where the rational function is conformally conjugate to an irrational rotation of the standard annulus.

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Solutions for auto-parallel curves on a sphere-like surface.

Finding the auto-parallel curves on a potato-like surface shall be left as an exercise for the student.

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Full Size & Source:

1. Fibonacci Numbers are generated by the Pascal’s Triangle.

2. Odd and Even numbers form the Sierpinski Triangle.

3. Number’s Symmetry and the”binomial coefficient expansion” numbers.

4. Powers of 2 and 11 series.

5. Triangular Numbers.

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Visualize temperature changes on earth for the last 200 years

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Boiling lava in Marum crater

Ambrym is a large basaltic volcano with a 12-km-wide caldera. It is not only one of the most active volcanoes of Vanuatu, but also in the world. Ambrym’s caldera is a wide, impressive moon-like landscape containing an ash plain, cut by innumerable erosion gullies and containing several active craters and recent lava flows.

The caldera is believed to have formed during a major plinian eruption with dacitic pyroclastic flows about 1900 years ago. Activity after the caldera formation was concentrated around two vents that have become large complex craters called Marum and Benbow. Often, both Marum and Benbow (as well as other craters) contain small lava lakes, which occasionally erupt lava flows onto the caldera floor, or even exit the caldera through erosion gaps.

To capture close-up footage of this lava lake on the Pacific island of Vanuatu, Geoff Mackley and his colleagues had to brave excruciating heat and some rather precarious-looking abseiling.

You can see more incredible shots at Mackley’s website.

Credit: Geoff Mackley, Bradley Ambrose, Nathan Berg

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Jupiter’s Irregular Satellites

The planet Jupiter has 67 confirmed moons. This gives it the largest retinue of moons with “reasonably secure” orbits of any planet in the Solar System. In fact, Jupiter and its moons are like a miniature solar system with the inner moons orbiting faster than the others. Eight of Jupiter’s moons are regular satellites, with prograde and nearly circular orbits that are not greatly inclined with respect to Jupiter’s equatorial plane. The remainder of Jupiter’s moons are irregular satellites, whose prograde and retrograde orbits are much farther from Jupiter and have high inclinations and eccentricities. These moons were probably captured by Jupiter from solar orbits. There are 17 recently discovered irregular satellites that have not yet been named.

Image Credit: NASA/ESA/Lowell Observatory/J. Spencer/JHU-APL

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Orbital Mechanics by Tatiana Plakhova 

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Click and drag.

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This interactive 3D tool shows where the European Space Agency's comet-chasing Rosetta spacecraft is on any given day – from launch in 2004 to the end of its nominal mission in 2015.