The Gif Animation Of Folding The Net Of The Tesseract(8-cell). ;-)

A picture from the un-official German Grand Prix held at the Solitudering, Stuttgart, in 1964. Jim Clark was the winner.

"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

Bedruthan Steps, Cornwall | England (by Ian Percival)

Dog Mountain- Washington

May 2013

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.”

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