Bedruthan Steps, Cornwall | England (by Ian Percival)

If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.

(via brittbutterfly21)

P-61 Black Widow cockpit.

four-bar linkages, and how changing the length of the bars affects the motion in one particular case

Fifty years of space exploration. Well, American space exploration at least. Originally published by National Geographic back in 2008, a few things have changed:

Several new missions have embarked to our Moon since 2008, including the Lunar Reconnaissance Orbiter, GRAIL (mission complete, RIP) and LADEE, just launched last week.

The Mars Curiosity rover, duh!

Juno, launched in 2011, is only halfway to Jupiter today. That big planet is far away.

The Solar Dynamics Observatory has given us beautiful solar spectra to stare at.

New Horizons is set to be the first spacecraft to fly by Pluto and its moons in 2015. It rode on the fastest launch vehicle ever in order to begin its epic journey.

And Voyager has left the building …

Embigginate this beautiful piece of work here.

Another animation of the fractal Harriss spiral. This exploits the plastic number (1.3247…) a number whose cube is itself plus one. This number is considered by some to be the forgotten cousin of the golden ratio (whose square is itself plus one).

I made a gif animation of the Binomial Cube. :D

http://en.wikipedia.org/wiki/Montessori_sensorial_materials

http://hyrodium.tumblr.com/post/85901458594/do-you-know-the-maria-montessori-method-of-education

"We humans long to be connected with our origins, so we create rituals. Science is another way to express this longing. It also connects us with our origins, and it too has its rituals and its commandments. Its only sacred truth is that there are no sacred truths. All assumptions must be critically examined. Arguments from authority are worthless. Whatever is inconsistent with the facts, no matter how fond of it we are, must be discarded or revised. Science is not perfect, it’s often misused, it’s only a tool, but it’s the best tool we have. Self-correcting, ever-changing, applicable to everything. With this tool we vanquish the impossible. With the methods of science we have begun to explore the cosmos." - Carl Sagan

Lagrange's theorem

Theorem: The size of a subgroup of finite group is divisible by the group’s size. That is, if H is a subgroup of G, |H| is a divisor of |G|.

Proof: Let’s start by saying we have a group G and a subgroup H.

This proof will count cosets. Specifically, I’ll use left cosets, but right cosets work the same way. Also, this proof will rely on a few properties of the integers.

I’ll prove this through lemmas, which are theorems used to prove other theorems. The distinction between a lemma and a theorem is only based on how we use them, and so historical reasons might leave some theorems as “lemmas.”

Keep reading