Let us use the surface of the sphere as an analogy. As I've said, the surface looks flat over a small area – so much so that to a flatlander living in that area, the surface appears to be an infinite 2D plane extending boundlessly in all directions. But the surface is not boundless: it is finite. In the same way, the Universe may appear boundless on a small scale – we think we see space as infinitely stretchable in every direction we look – but the Universe (so I believe) is finite in the way the surface of a sphere is finite.

Lines that are "straight" on the surface of a sphere are actually curved in the 3D space the surface is embedded in; they're arcs of great circles. Travelling on a straight line on the surface means moving along the circumference of a great circle such as the equator. If you keep travelling along this "straight" line, you will eventually end up where you started from. Indeed, no matter how you move on the surface, you can never escape from it. In the same way, I'm convinced that straight lines in the Universe are actually curved in the 4D hyperspace in which I believe the Universe to be embedded. If, starting at a point in the Universe, we could keep travelling on and on in a straight line, we would eventually end up where we started! This is because lines that are straight in 3D are actually curved in 4D hyperspace. And no matter how we move around in the Universe, we will never escape from it.

Back to the sphere analogy. If the radius of the sphere increases, the surface area also increases: the world of the flatlander expands. This, I believe, is also why our 3D Universe is expanding: because the hyper-radius of 4D hyperspace is increasing. The Big Bang was the moment at which this hyper-radius was zero.

And this is my metaphysical theory of the Universe – which I will never be able to prove. I don't expect anyone to agree with me anyway (or even understand what I'm saying in the first place).

]]>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.

]]>]]>If it ain't broke, fix it until it is.

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 Borsuk–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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