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#2 Euler Avenue » My theory of the Universe » 2016-08-29 00:29:41

Replies: 3

I'm convinced the Universe is a 3-dimensional manifold embedded in 4-dimensional hyperspace.

Mathematically, a manifold is a space embedded in the space of the next higher dimension in which every point is locally "flat". For example: the surface of the sphere is a 2-dimensional manifold (embedded in 3 dimensions). Close up, the surface appears "flat". The technical definition is this: An n-dimensional (real) manifold is a topological subspace
such that given any point x in
there exists a real number
(depending on x) such that the open ball
is homeomorphic to

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).

#4 Re: Help Me ! » Difference between Mod operation and Mod in histogram » 2015-05-20 23:21:37

I have no idea what Mod in histogram is. As an operation, the MOD button on a scientific calculator gives the remainder when one number is divided by another. E.g.

gives 5 (i.e. 21 divided by 8 leaves a remainder of 5).

#8 Re: Help Me ! » derivative of 2D funciton by the taylor series » 2015-05-15 20:07:01

The Taylor series to the first order of f(x,y) at the point (a,b) is

If you replace (x,y) by (x+u,y+v) and (a,b) by (x,y), you are done.

#9 Re: Help Me ! » Which of the following are subrings of C? » 2015-05-12 19:11:08

The first one.

The second one is not even a ring – it's not closed under multiplication.

#10 Re: Jokes » Very short jokes! » 2015-05-12 00:02:03

Q: Why did the mushroom go to the party?

A: Because he was a fun guy.

#11 Puzzles and Games » You rang, mallard? » 2015-05-10 16:47:06

Replies: 1

Q: Why did the mallard cross the street?

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#14 Re: Dark Discussions at Cafe Infinity » 2015 UK General Election » 2015-05-07 23:39:09

And the Conservatives have got their majority. With 7 seats still to be declared, they have 326 seats.

#15 Dark Discussions at Cafe Infinity » 2015 UK General Election » 2015-05-07 21:13:15

Replies: 30

With 636 of 650 seats declared (as of time of posting), David Cameron looks set to remain in office in 10 Downing Street for another five years. The Conservatives currently have 320 seats and need a further 5 of the remaining 14 seats to get the overall majority. But even if there is a hung parliament, a simple power-sharing deal with the DUP will be enough for Cameron to retain his government.

It all means Nick Clegg will no longer be Deputy Prime Minister of the UK – hooray! Who can forget or forgive what that big liar did in the last election? Five years ago, the Liberal Democrats promised that they would not increase tuition fees; then, as soon as they got to share power with the Tories, they did a complete U-turn and said tuition fees should rise. That was what did it for the Lib Dems this time round. Just look at the Lib Dems now! They have lost 46 seats and gained none – NONE – in their worst ever performance in a general election in their whole history. Serve them right!

#17 Computer Math » Principal roots » 2015-05-07 20:09:04

Replies: 1

If z is real and positive, the principal root lies on the positive real axis and so is real.

If z is real and negative, the real root (if any) will not be on the positive real axis but there will still be a complex root in the first quadrant; the principal root in this case is complex.

This, at any rate, is Wolfram's definition; Wikipedia has a slightly different definition.

#18 Re: Help Me ! » quadratic polynomial » 2015-05-07 15:18:10

Wolfram always returns a complex number as the principal value of the cube root of a negative real number. Presumably this is the complex number in the first quadrant of the Argand diagram.

#19 Re: Help Me ! » quadratic polynomial » 2015-05-07 15:06:38

When you allow complex values, then every real number will have exactly three cube roots: one of them is real and the other two conjugate complex numbers. If you want only real solutions, you should tell Wolfram explicitly – otherwise it will give you its "principal value", which is not always the real root.

#21 Re: Help Me ! » quadratic polynomial » 2015-05-05 10:34:53

You are taking values of x and p independently of each other – but they are not independent of each other! x and p are related by

In other words your <cube root c> minus <cube root a> must be equal to 1 – but as you can see they are not. shame

#22 Re: Help Me ! » Vectors » 2015-05-05 06:51:40

Now interchange u and v and get

Result follows.

#25 Re: Help Me ! » quadratic polynomial » 2015-05-02 23:06:06

bob bundy wrote:

If p = -0.25 then y = -0.5 and so x = -0.7937.

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