Thursday, December 4, 2008

Reciprocal Spaces and Negative Dimensionality

The logarithm function (among other things) measures (approximately) the lengths of numbers expressed in our standard place-value notation - for example (base 10 logs rounded to the nearest whole number) :

4 : length = 1 log = 1

12 : length = 2 log = 1
90 : length = 2 log = 2
100 : length = 3 log = 2
150 : length = 3 log = 2
446 : length = 3 log = 3
165462454 : length = 9 log = 8

( ....this off the reciprocal space / negative dimensionality topic but....I wonder what are the bounds on the lengths of the names that can be given to numbers, under different possible naming schemes.

There is the trivial naming scheme whereby the names of numbers are the same length as the numbers themselves, so that the length of the name of a number increases identically and linearly as the number it names :

* = 1
** = 2
*** = 3

And easy to construct names whose length increases as, say, the square of a given number :

* = 1

** = 2

*** = 3

- but what about the more useful schemes, whereby the length of the name of a number increases sub-linearly ?

Roman Numerals can be more compact

C = 100
M = 1000

....or less

XXXIV = 34

....but the Roman system is "non deterministic" - yes it is able to compress orders of magnitude to a single symbol as with C and M , but there is no automatic compact Roman symbol for 1,000,000 unless and until we intervene and assign one.

So then a question - is there a deterministic naming system for integers, under which the length of the name of a number N increases at less than log(N) ? )

(see for interest the Berry paradox relating to lengths of names of numbers - )

But to return to the topic - there are some parallels between the log function and the concept of dimensionality.

  • the dimensionality of a space gives the length of the "names" of points in that space, in a similar way in which the log of an integer gives the length of the name of that integer. For example points on a 2-D plain have names like (1,3) , of length 2 ; points in a 3-D space have names like (3,4,8) , of length 3 etc

  • When we multiply numbers we add their logarithms ; and when we "multiply" spaces - i.e. take a cross product (aka cartesian product, direct product) - we add their dimensions. So for example, the dimension of the Cartesian plane is two , the sum of the dimensionality of the x and y axes that are "multiplied together" to create the plane.

We can use this parallel to motivate an interpretation of negative dimensionality, by considering what is the dimensional analog of a negative logarithm, via these correspondences.

Negative logarithms are obtained from positive reciprocal numbers.

Log(10) = 1
Log(1/10) = -1

This is so since the log of 1 is zero , and we must have

log(10) + log(1/10) = log ( 10 * 1/10) = log (1) = 0

This might suggest that :

  • a space with negative dimensionality is in some sense perhaps a "small" space , just as a number with a negative logarithm is a smallish number

  • a space S with negative dimensionality -D is in some sense a reciprocal space in that , if T is another space with dimension +D , it seems that we should have, operationally, and as with the logarithm analogy :

    dim ( S X T )
    = dim(S) + dim(T)
    = D - D

    = 0

So without further ado (there is far too much ado in this blog as it is ! ) I will take this as my working concept of a negative dimensional space : I will call it a reciprocal space , meaning that on cross multiplication with a positive dimensional space it yields (in some strange black-box way yet to be specified) a product space whose dimension is the dimension of the positive space, less the negative dimension of the reciprocal space - and if these dimensions are equal, the product space is a zero dimensional point.

The obvious fact that we cannot yet specify what actually goes on in this multiplication, and that we cannot visualise what a negative dimensional reciprocal space looks like, need not worry us for the time being. It is similarly impossible to visualise a negative length (and I do not count the metaphor of oppositely directed lengths as such a visualisation - this is just a model of a negative number (see also Roger Penrose on negative numbers, page 65 in "The Road to Reality") - yet we can still discover how negative numbers should behave, and find a use for them. (I would argue that fractional numbers are equally abstract and it is in fact impossible to visualise 1/2, but that is for another blog !)

So far this is all pretty harmless. Next blog will leave planet earth entirely and start to see reciprocal spaces everywhere - the brain as a reciprocal space ; consciousness as the lower dimensional product of this reciprocal space, with the very high dimensional space of the flux of experience and sensation - thus "explaining" certain aspects of our conscious experience. And making a few predictions though probably not testable.

(That will be pretty harmless as well , apart from the small carbon footprint made by the disk space used up in the post)

(I am dimly aware of previous characterisations of negative dimensional spaces - there is a fractal one from Mandelbrot I briefly encountered - but its good to just go Sunday driving without a map with these things sometimes - its of course impossible you will actually discover part of the countryside that hasn't been mapped already (you have to be an intellectual mountaineer for that , which I am not), but you might get to see some interesting places you would not otherwise have seen had you been better prepared ! )

(And a "reciprocal space" in crystallography is a Fourier transform - I do not mean anything like a Fourier tansform in my use of this term however)