Exploring the Mathematical Concept of the Roots Chain of Shadows

[Lama]

Please look at the attached pdf http://www.geocities.com/complementarytheory/Roots-Chain.pdf .

By this model we can see that √1 is the "shadow" of √2 and √2 is the "shadow" of √3.

If we can conclude that √3 is the "shadow" of √4 ... and so on, then do you think that this "Chain of Shadows" has any mathematical/physical meaning?

Thank you,

Lama

 

[Hurkyl]

No, I don't. If you picked just about any other line with length √3, those projections would not have length √2 and √1.

 

[Lama]

But I am talking about roots which each one of them is the diagonal of its dimension level, where each n_dim diagonal is the "shadow" of n+1_dim diagonal.

Don't you see the beauity here?


We Have a chain of "shadows" between infinitely many diagonals in |N| dimension levels, isn't it?

Also my diagram in the first post is a proof without words (http://mathworld.wolfram.com/ProofwithoutWords.html) of this beautiful chain of shadows.

Someone can say: "But this proof is good only for R^3, and we still need to prove/disprove it for any n>3".

In this case I think that my "chain of shadows" is an interesting question, unless you easly can disprove it.

 

[Lama]

Do you think that we can use this diagonals' chain of shadows as some kind of communication gateways between different dimensions.

 

[Lama]

Ok, let us say that “A chain of shadows” is a conjuncture.

Is there a possibility that there is no such a thing like a diagonal, in symmetrical elements (like a square in 2-d and a cube in 3-d), which have more then 3 dimensions?

For example in 1-dim any line-segment of length 1 is its own diagonal, but in 2-dim and 3-dim the diagonal is the result of the √ of its dimension level, and we get an irrational number, which is different then the 1-dim case.

In 4-dim we again get a natural number as the result of the diagonal length.

In short, since in all cases we are talking about a diagonal (which is the root of its dimension level), then to disprove my conjuncture we have:

1) To show that there is a clear way to conclude that diagonals which belong to dimension levels that are > 3, are not necessarily the root of their dimension.

2) To show that there are dimensions > 3 without diagonals.

If you have more ideas of how to disprove my conjuncture, then I’ll be glad to know them.

Here is again my diagram: http://www.geocities.com/complementarytheory/Roots-Chain.pdf

 

[Chronos]

I sense a violation of causality in that proposition.

 

[Lama]

It is interesting, please give more details, thank you.

 

[Lama]

Maybe these facts are aslo related to this conjuncture.

Square Numbers (http://www.krysstal.com/numbers.html)

Square Numbers are integers that are the square of smaller integers. For example 4 is 22 and 9 is 32 so the first few square numbers are:

1 4 9 16 25 36 49 64 81 100 121 144 169

Note that the sequence of square numbers alternates between odd and even.

Another interesting fact is that this series of square numbers can be produced by adding successive odd numbers. For example, the sum of the first two odd numbers (1, 3) is 1 + 3 = 4 (a square number). The sum of the first three odd numbers (1, 3, 5) is 9 (another sqauare). This is shown in the table below:


1 = 1
1 + 3 = 4
1 + 3 + 5 = 9
1 + 3 + 5 + 7 = 16
1 + 3 + 5 + 7 + 9 = 25
1 + 3 + 5 + 7 + 9 + 11 = 36
1 + 3 + 5 + 7 + 9 + 11 + 13 = 49
1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 = 64
...

 

[Lama]

Let us take Pi for example.

We can say that Pi is no more than some irrational number that can be found in R collection.

But we think that Pi is important because it is the result of the relations between the perimeter and the diameter of a circle.

And a circle is a very important symmetrical object.

Now, if we construct some dimension level by using delta-x , delta-x,y , delta-x,y,z , delta-x,y,z,... , then in each dimension we can find the basic building-block that is constructed by the number of its deltas, where each delta is 1.

To each such a building-block, there is a diagonal, which its length is the root of the dimension level of the building-block.

From this point of view, each diagonal of some given dimension is the shadow of dimension_n+1 and we get a chain of shadows between the building-blocks of |N| dimensions.

Maybe this chain of shadows between different dimension levels can be used as some kind of a gateway (a deep invariant symmetry) between different dimensions, which are represented by N members.

Here is my diagram (again) to help you to understand my idea:http://www.geocities.com/complementarytheory/Roots-Chain.pdf .


What do you think?

 

[Lama]

But in terms of your description √1 isn't just the shadow of √2 it's also the shadow of 75, 89, 3.1415926535897932384626433832795 etc. Likewise √2 need not just be the shadow of √3 but also the shadow of any number you pick.

Which means that n_dim is the shadow of n+1_dim, isn't it?

The root of each given dimansion is like the invariant symmetry which connect between |N| dimensions.

Don't you see the importence of this invariant symmetry as a gateway between |N| dimensions?

 

[Lama]

Not really it just means that you can project one vector onto another (which isn't very remarkable).

As for the whole dimension "issue", one can consider a 2d point (x,y) in terms of a 3d point (x,y,0) if you want to . Likewise you can do that for higher dimensions e.g. (x,y,z) --> (x,y,z,0).

I am talking about the invariant symmtery that stands in the basis of the roots of any n of |N| different dimensions.

Don't you see this invariant symmetry, and how it can maybe used as a gateway between |N| different dimensions?

(x,y,z) --> (x,y,z,0) means that the 4th dimension cannot be but constant 0, and so is x,y,z,0,0,0,... which is not a represontation of |N| free dimensional degrees.

 

[sparkster]

You really enjoy talking to yourself, don't you?

 

[Lama]

Dear ex-xian,

I'll be glad if you join to haggy and me in http://www.scienceforums.net/forums/showthread.php?t=5200


Unless you prefer to run and shout in circles.