another dimension.

[reverse]

If,

 

X is a line.

 

X squared is an area.

 

X cubed is a volume.

 

What do you suppose X to the power of four is?

 

 

[Kygron]

A hyper-volume, and various other names.

 

I believe you'll find this topic well-explored.

 

For example, you know how if you draw a circle, you can also draw six other circles of the same size around it so that they all join? Well, if you extend that to 3 dimentions it doesn't work any more. But I read a Science News article once that said that in 4 dimentions it works again, and again in 10 and 26! Now try imagining 26D space if you will?

[reverse]

I’m having trouble visualising it.

I thought Hyperspace was a thing invented to allow Han Solo’s space ship to escape the Death star.

 

This is what I have dug up so far.

 

 

 

Around 200 B.C., Euclid's extraordinaire proved that there are only five regular polyhedra, polytopes in three spatial dimensions: the tetrahedron, cube, octahedron, icosahedron, and the dodecahedron. In 1901, Ludwig Schlafli showed that there are only six regular polychora1, or polytopes in hyperspace.

One may believe with more dimensions, there are more regular polytopes. However, the fourth dimension is as complex as it gets. This is because each dimension's increasing "freedom" nullifies its complexity. In fact, all higher dimensions each only have three regular polytopes.

A Geometric Approach

Differentiating between figures in the first few dimensions can be quite a task. One of the simplest ways to view higher dimensions is by slicing. A simple three-dimensional figure, a cube, can be sliced parallel to its sides to give a square, a two-dimensional figure. A hypercube, the cubic equivalent in four dimensions, therefore can be sliced to give cubes.

When you take a sheet of paper and look at it from the top, you see a rectangle. If you turn in on its side, you see a line. Between the rectangle and line, you see rhombuses, parallelograms, and other quadrilaterals.

When you try to draw a cube, notice what shapes are used to create it. If you drew a cube from a corner view, you would need parallelograms and other quadrilaterals. From that view, you won't see any squares, although squares are what make up a cube. This strange phenomenon occurs anytime a polytope is rotated. So when you look at a hypercube rotating, don't expect to see just perfect cubes, rather look for distorted ones.

 

It’s from here.

http://temporal_science.tripod.com/introduction/special1.htm

[Kygron]

Tesseract's avatar is one of those images of a hypercube.

 

Visuallizing in 4 dimentions is tricky because your mind isn't used to doing it. Also because any drawing of a hypercube, for example, has to be projected twice. By projected, I mean, "flattening" it. You could possibly put on 3D glasses and look at a computer-generated single-projected hypercube. It would look like the two cubes connected at all the corners. You'd have to have it transparent to see all the way through.

 

I'm going on and on, but the "standard" way is to use the mathematical formulas themselves and not even bother trying to "see" it.

[reverse]

I’m starting to visualise it,

with the realisation that a sphere can not really be drawn on paper-

without the use of shading - to trick the mind to flip into three dimensional mode.

 

So I’m thinking of the original use for math as to keep track of sheep, get in tax... and later to get buildings straight.

 

At what point did it spin off into imaginary dimensions,

and is that really a thing you can do rationally?

[Kygron]

As soon as there's a question, there's someone trying to find a solution. As soon as there's a solution, there's someone trying to extend it in order to have the solution apply to future problems. As soon as there's a solution without a problem, there's someone who thinks it's a game and starts playing with it.

 

 

 

Wow, I like that, I'm going to have to apply that to other things

[reverse]

agree.

 

[reverse]

"In 1901, Ludwig Schlafli showed that there are only six regular polychora1, or polytopes in hyperspace.

One may believe with more dimensions, there are more regular polytopes. However, the fourth dimension is as complex as it gets. This is because each dimension's increasing "freedom" nullifies its complexity. In fact, all higher dimensions each only have three regular polytopes."

 

 

 

so, thinking dimensionally then,

a triangle is rather special.

and in reality it is also special.

The laws of our real universe wont allow it to distort by racking over .

(A square and other shapes can.)

 

now if this whole game is about the relationships between the three dimensions of Length Area and Volume, then by understanding the above quote, the law of A squared plus B squared equals C squared becomes interesting.

 

It’s an extrusion of a triangles “area”, into the last real dimension (we are calling it “Volume”).

 

Now the same law that constrains a triangle in reality also constrains it in mathematics.

 

It’s interesting that the parabola Y = X squared, is a one dimensional representation of that two dimensional relationship.

 

It’s like a stick man you draw on paper getting up and walking off your page and saying "hey draw a picture of me now".

 

Ps, I’m not so sure that a parabola is a justifiable representation of Y = X^2.

 

I can now see the excitement and temptation preoccupying the early mystic Greek mathematicians and even modern physicists.

 

It’s like having a tool to look out the window at God.

[Bettina]

I’m having trouble visualising it.

I thought Hyperspace was a thing invented to allow Han Solo’s space ship to escape the Death star.

 

This is what I have dug up so far.

 

 

 

Around 200 B.C.' date=' Euclid's extraordinaire proved that there are only five regular polyhedra, polytopes in three spatial dimensions: the tetrahedron, cube, octahedron, icosahedron, and the dodecahedron. In 1901, Ludwig Schlafli showed that there are only six regular polychora1, or polytopes in hyperspace.

One may believe with more dimensions, there are more regular polytopes. However, the fourth dimension is as complex as it gets. This is because each dimension's increasing "freedom" nullifies its complexity. In fact, all higher dimensions each only have three regular polytopes.

A Geometric Approach

Differentiating between figures in the first few dimensions can be quite a task. One of the simplest ways to view higher dimensions is by slicing. A simple three-dimensional figure, a cube, can be sliced parallel to its sides to give a square, a two-dimensional figure. A hypercube, the cubic equivalent in four dimensions, therefore can be sliced to give cubes.

When you take a sheet of paper and look at it from the top, you see a rectangle. If you turn in on its side, you see a line. Between the rectangle and line, you see rhombuses, parallelograms, and other quadrilaterals.

When you try to draw a cube, notice what shapes are used to create it. If you drew a cube from a corner view, you would need parallelograms and other quadrilaterals. From that view, you won't see any squares, although squares are what make up a cube. This strange phenomenon occurs anytime a polytope is rotated. So when you look at a hypercube rotating, don't expect to see just perfect cubes, rather look for distorted ones.

 

It’s from here.

http://temporal_science.tripod.com/introduction/special1.htm[/quote']

 

The above link you posted is awesome!!. I clicked on the hypersphere. Check that out.....Thanks a bunch.

 

Bettina

[reverse]

That was a total fluke.

I didn’t realise there was a similar thread in the relativity section until later.

I was trying to find out what gifted mathematicians saw to keep them continuously interested in a subject some consider too abstract.

[Dave]

For me at least, it's two things:

 

1) Being able to prove things using the knowledge I've already aquired. There's nothing quite like proving things in maths - although it sounds sad, it's quite fun.

2) Passing on my knowledge and experience to others and helping them to understand the things that I had trouble understanding.

[Johnny5]

Thanks for the site. I clicked on the hypercube. Highly interesting animation. Yet there is nothing in reality which rotates in that manner.

 

Yet still... it is interesting.

 

Regards

 

PS: What is most interesting, is that anyone at all had a formula for it, which could then be worked into a program.

 

One of the ways i can see that it cannot exist, is that while it rotates, certain things vanish. And also pass through each other, as if it isn't solid.

[Johnny5]

Around 200 B.C.' date=' Euclid's extraordinaire proved that there are only five regular polyhedra, polytopes in three spatial dimensions: the tetrahedron, cube, octahedron, icosahedron, and the dodecahedron[/quote']

 

 

Who is the individual who proved this?

 

What is the name of the fellow who proved that there are only five regular polyhedra?

[reverse]

Euclid from 200 B.C.,

 

http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Euclid.html

 

Ludwig Schlafli from 1901.

 

http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Schlafli.html

 

[Johnny5]

Euclid from 200 B.C.' date='

 

http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Euclid.html

 

Ludwig Schlafli from 1901.

 

http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Schlafli.html

 

[/quote']

 

Euclid was the individual who proved that there are only 5?

 

What is a polytope?

 

poly means many in Greek, and topos, means i think it means side, or surface. I'll check on that.

 

From an online dictionary, I got this:

 

topos = locus, place, room, space, spot

 

Ludwig schlafli gave the integral representation of the Bessel function, and the gamma function. Wow, I didn't know that. Let me see if i recall the gamma function

 

[math] \Gamma(n+1) \equiv \int_0^{\infty} t^n e^{-t} dt [/math]

 

And that is equal to n factorial, which is rapidly provable using integration by parts. In other words:

 

[math] \Gamma(n+1) = n! = \int_0^{\infty} t^n e^{-t} dt [/math]

 

1814-1895 Ludwig Schlafli (switzerland)... discovered integral representation of both the Bessel function, and the gamma function.

 

325 BC-265BC Euclid of Alexandria, Egypt: Author of "Elements Of Geometry," treatise on optics, and individual who proved that there are only five regular polyhedra... the tetrahedron, which is a closed three dimensional four sided figure, the cube, which is a closed three dimensional six sided figure, the octohedron, which is a closed three dimensional eight sided figure, the dodecahedron, which is a closed three dimensional 12 sided figure, and the icosahedron, which is a closed three dimensional 20 sided figure.

[reverse]

well it's all Greek to me.

 

I'm guessing he has realised that the various forms can go into only five groups, each containing the single root form appearing as different only due to a trick of POV.

 

Yes, and utopia is in hyperspace as well.

 

utopia = no place.

[Kygron]

One of the ways i can see that [a hypercube'] cannot exist, is that while it rotates, certain things vanish. And also pass through each other, as if it isn't solid.

 

Look at a similar animation of a NORMAL cube rotating. You'll notice your description applies there as well, an I'm sure you'll agree that a cube exists.