A chain of shadows

[Guest Doron Shadmi]

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.

 

I think that we can conclude that √3 is the "shadow" of √4 ... and so on.

 

In short, 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.

 

We have a chain of "shadows" between infinitely many diagonals in |N| dimension levels.

 

Do you think that this "Chain of Shadows" has any mathematical/physical meaning?

[e(ho0n3]

I think that we can conclude that √3 is the "shadow" of √4 ... and so on.

It would be nice to see a proof of this (if you can).

 

Do you think that this "Chain of Shadows" has any mathematical/physical meaning?

Until I find one, the answer will remain NO.

[Guest Doron Shadmi]

It would be nice to see a proof of this (if you can).

My diagram (http://www.geocities.com/complementarytheory/Roots-Chain.pdf) is a proof without words (http://mathworld.wolfram.com/ProofwithoutWords.html)

 

 

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

[bloodhound]

ur proof without words is valid only for [math]\mathbb{R}^3[/math]. you havent proved the general case of n dimensions

[bloodhound]

this is very interesting. i might have a go at proving/disproving a general case. maybe using some stuff from vector spaces. mathematical induction here seems to be the most obvious way to go

[Guest Doron Shadmi]

Don't you think that my diagram is also a proof by induction?

[e(ho0n3]

Don't you think that my diagram is also a proof by induction?

As bloodhound said, we need a proof for the n-dimensional case. And how is your diagram a proof by induction (note that by induction, I mean MATHEMATICAL INDUCTION)?

[Guest Doron Shadmi]

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

[Guest Doron Shadmi]

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

...

[jordan]

Wow. I've never seen that table at the bottom before. It's really neat that it works like that.

[bloodhound]

thats really easy to show. an odd number is of the form [math]2k-1[/math]

 

so if u sum that i.e

 

[math]\sum_{1}^{n}(2k-1)[/math]

[math]=2\sum_{1}^{n}k-\sum_{1}^{n}1[/math]

[math]=n(n+1)-n[/math]

[math]=n^2[/math]

[Guest Doron Shadmi]

Wow. I've never seen that table at the bottom before. It's really neat that it works like that.

You are right dear jordan, the language of Mathematics is much more then only an elegant way to write things.

 

It is first of all the filling of wonder when you see a beautiful thing.

 

Keep this attitude and do no let any person to get your filling of wonder and your sense of beauty down.

[jordan]

Awesome. Nice proof Bloodhound. I hope I can come up with stuff like that someday.

[haggy]

What exactly are you trying to show with √1 being "shadow" of √2, etc?

 

By the same sort of illustration you could "show" √1, √2 to be the shadows of any number you please.

[Guest Doron Shadmi]

Hi haggy,

 

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.

[haggy]

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.

[Guest Doron Shadmi]

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?

[haggy]

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

 

[/quote']

 

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).

[Guest Doron Shadmi]

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

 

As for the whole dimension "issue"' date=' 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).

[/quote']

 

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.

[jordan]

Now' date=' 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.[/quote']

I got lost on this part. Can you explain it a little more?

[Guest Doron Shadmi]

I got lost on this part. Can you explain it a little more?

 

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

[bloodhound]

think he is talking about the basis vectors of the vector space R^n

[Guest Doron Shadmi]

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.

 

Again:

 

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

[bloodhound]

sorry doron. i am not familiar with these terms. what is invariant symmenty?

what is "basis of the roots" what is gateway? etc.

[Guest Doron Shadmi]

Invariant symmetry is a constant which stand in the basis of infinitely many variations of it.

 

For example:

 

A ball is the invariant symmetry of a closed cube, closed cylinder, closed cone, closed pyramid, .... , and so on.

 

The ball is the "gateway" to make a transformation from one shape to another.

 

So my idea is that this 'chain of root's shadows' of N members, is maybe the gateway between different |N| dimensions.