If you are a Platonist, you will believe it is discovered. The inference is that it exists in nature and therefore is incredibly deep and meaningful.

If you are a Formalist, you will believe that it is an invention of man like a game such as chess. It has rules and is logical but it is artificial.

At 1:22 into the video we see a Unicorn, an EBE and a werewolf or zombie all on the Moon or some other lifeless looking ball. Makes one wonder whether ole Jeff has ever been to Nevada.

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So what possible transformations are isometries of the plane? You might think that there were a whole bunch of them: rotations, reflections, and translations. (A reflection followed by a translation is sometimes called a glide reflection.) In fact the picture is simpler than that: it turns out that reflections and translations can be built up from rotations alone! A translation in a certain direction is simply a reflection in two axes perpendicular that direction, while a rotation about a point O is a reflection in two axes through O.

Hence any translation is a reflection in two axes perpendicular to the direction of translation whose distance apart is half the distance to be translated.Hence any rotation is a reflection in two axes through the centre of rotation whose angular separation is half the angle to be rotated through.

]]>It is always possible to slice a three-layered ham sandwich with a single cut of a knife in such a way that each layer of the sandwich is divided into two exactly equal halves by the cut.

The ham-sandwich theorem can be proved using the Borsukâ€“Ulam theorem.

]]>I learn that it is possible to make a tetradecahedron with just regular hexagons and squares.

It is also possible to make a truncated icosahedron with regular pentagons and hexagons.

Let , and suppose that is a continuously differentiable function on . Then:One of the many applications of this theorem is using it to show that:

by writing

where denotes the greatest integer part of x, and is the Euler-Mascheroni constant.]]>or, more precisely:

where M is the so-called Meissel-Merten's constant, with value approximately equal to 0.2614972128476427837554268386086958590516...

This result is known as Merten's second theorem.

]]>It has been a while now, how did everything go? Looked at Numberphile page, it looks pretty good.

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