Why do you need more dimensions (q)

>Hi all, >I have done extensive research on meta physics... and couldn't find >any single reason to support the existence of extra dimension... Its >totally an absured concept given by some stupids... >Its an advice not to think abt dimensions.... be happy with the >dimensions given to u by GOD( the only 3 dimension and not even the >fourth time dimension) >i dont know why some ppl are trying to make the nature complex...

First of all, science is not constructing nature, it's about explaining and predicting nature. And when ordinary explanations fail, u need to invent better ones that give more accurate predictions.

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multi dimensional analysis seems non-intuitive, if u seperate the math from what a dimension is or means.

A concept of dimension is related to the concept of orthogonality or independence.

Orthogonality means that the dot product of one dimension with another dimensions yeilds 0

that is

xhati dot xharj = Delta_ij

where Delta_ij is 1 if and only if i=j and 0, if otherwise.

The concept of multi dimensional analysis, also has influence from notion of independent variables.

For example,

Let's take a store like sears and TV sales.

The sales,on a single day, may be related to

the average number of people entering the store N the average salary of money they each get paid S the average number of TVs on displaysT the month of purchase M the average price of TV pTV

We can assume that these events are independent. For example, independence in this context means, that number of people entering the store does not depend on the month they enter. Ofcourse, you may realize that people do a lot more shopping during holiday seasons that they do other times of the year. For now, we will ignore this effect.

Statiscally, independence means 1 thing. No correlation

E[[ ( N- <N>) ( M - <M> ) ]] = 0

I shall come back to it expression later and how it is useful in orthogonality analysis or multi dimension analysis.

Now,the plot of the sales function SALES( N, S, T, M , pTV) requires a multi dimensonal graph (6-dimensonal) . However, it does not mean that the SALES function is non-intuitive concept.

Now, in mathematical analysis,

If you studied dot products in your vector algebra class

a dot b = ||a|||b|| cos ( heta)

heta is angle between the 2 vectors. Now if theta = 90 degrees,

a dot b = 0

In a way, 2 vectors are orthogonal. Also note that dot of a vector purely along x-axis with a vector purely along y-axis is 0.

Now, we can extend this concept to functions. Two real functions a(x), b(x) are orthogonal if

integrate[[x-infty, infty]] a(x) b(x) dx Deltaab

That means that, it is just simlar to saying that the functions are at 90 degrees to each other.

Moreover, we also require

integrate_[[xinfty, infty]] a(x) dx 0

and

integrate_[[xinfty, infty]] b(x) dx 0

So, There is no net energy inside them.

However,

integrate_[[x-infty, infty]] a(x)^2 dx 1

This follow from

integrate[[x-infty, infty]] a(x) b(x) dx Deltaab

We can treat the series of functions a(x), b(x), c(x), which statisfy the above conditions as dimensions.

Now let's look at our expression

E[[ ( N- <N>) ( M - <M> ) ]] = EN * M - ENEM

You must notice something interesting there. If N and M are othorgonal,

that means that

integrate_[[x-infty, infty]] N(x) M(x) dx 0

EN 0, EM 0 since

integrate_[[xinfty, infty]] N(x) dx 0

Here, x is supposedly a variable that influences both N and M. It could even be N or M.

So, that means, that if N is orthogonal to M, N is not correlated to M and so, N and M are independent, and can change without influencing each other.

The concept of a mutli dimensonal the sales function SALES( N, S, T, M , pTV) is not counter-intuitive. It is definitely not possible, in our 3d world to plot this functions. That does not make it unreal or ridiculous.

Moreover, you need to realize that if the SALES function is only of 3 variables for example, of

the average number of people entering the store N the average salary of money they each get paid S the average number of TVs on displaysT

The SALES functions may be compromizing some details and parts of it may appear to be random because the other variables are not accounted for as observable effects correctly.

Similarly, when physicists talk about more than 3 dimensional space, they are talking about more than 3 othogonal dimensions that each satisfy the dot product requirment

xi dot xj = Delta_ij

There is no restriction on the number of x_i, as along as they satisfy the above condition. From a mathematical stand point, multi dimensional analysis, is very real, and is in fact, already has many practical applications. For example, consider the problem of optimal packaging. My math prof told me that there are hunderds of factors that can influence the arrangement of spaces taken by boxes.

-suresh