Re: What is Orthogonal (Exactly)?

James wrote:

> I would like to know what orthogonal means as exactly as possible.
> And what is the minimum number of thing(s) required in the concept of
> orthogonality.

Please stop this rabid cross-posting. My newser has selected a follow-up group that I don't read. And please crack a book.

> The following definition from searchStorage.com as a
> seed for dicussion:
>
> [ In geometry, orthogonal means "involving right angles" (from Greek
> ortho, meaning right, and gon meaning angled). The term has been
> extended to general use, meaning the characteristic of being
> independent (relative to something else). It also can mean:
> non-redundant, non-overlapping, or irrelevant. In computer
> terminology, something - such as a programming language or a data
> object - is orthogonal if it can be used without consideration as to
> how its use will affect something else.
>
> In itself, a programming language is orthogonal if its features can be
> used without thinking about how that usage will affect other features.
> Pascal is sometimes considered to be an orthogonal language, while C++
> is considered to be a non-orthogonal language. ]

It means given an R1 vector (a line), you have 1 degree of freedom. You are a dot on the line and can travel left or right.

But if you are an R2 vector (a plane), you now have 2 degrees of freedom. You can travel left or right, and at the same time up or down. This means a walk of "left left up up up" puts you in a place where an R1 left-right line can't go, and where an R1 up-down line can't go.

"Orthogonal" means you can combine two sets to form a number of possibilities equaling the count of one set times the count of the other set. But the two sets must have no elements in common ("non-overlapping"); if they did then the number of result possibilities would be less than the count of one set times the count of the other set.

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  Phlip
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  --  "Probably one of the toughest times in anyone's life is when
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