Old notes 1992, for Tommy Teacakes

hi tom,

from my archives, the following...

Airdrie, 15.09.1992: James Gleick, "Chaos": p97 "..what is the dimension of a ball of twine? Mandelbrot answered, It depends on your point of view. From a great distance, the ball is no more than a point, with zero dimensions. From closer, the ball is seen to fill spherical space, taking up three dimensions. From closer still, the twine comes into view, and the object becomes effectively one-dimensional, though the one dimension.. makes use of three-dimensional space... A weakness in Mandelbrot's argument seemed to be it's reliance on vague notions, 'from far away' and 'a little closer'. What about in between? Surely there was no clear boundary at which a ball of twine changes from a three-dimensional to a one-dimensional object. Yet.. the ill-defined nature of these transitions led to a new idea about the problem of dimensions.
Mandelbrot moved beyond dimensions 0,1,2,3.. to a seeming impossibility: fractional dimensions...
Fractional dimension becomes a way of measuring qualities that otherwise have no clear definition: the degree of roughness or irregularity in an object. A twisting coastline, for example, despite it's immeasurability in terms of length [dd: i.e. in 1-dim measurement], nevertheless has a certain characteristic degree of roughness. Mandelbrot specified ways of calculating the fractional dimension of real objects, given some technique of constructing a shape or given some data..."

This is an "extension of the meaning" of the word 'dimension' - and certainly dimensionality is dependent on the possibilities of measurement: if we can measure in n directions at mutual right angles to each other, what we measure has n dimensions. .. what the fractal-dimension measures is not another quantity normal to the other 3 usual directions, but something different altogether. (Reason why time shouldn't be called "the 4th dimension"). - Although time measures too, and we can speak of it as a dimension - so long as we don't assimilate the different senses we have then given to the word "dimension".

"effective dimension" - interesting concept. Since we do seem to use it e.g. maps as essentially two-dimensional, meaning here that the dimensions of a map which represent do not include the thickness of the paper on which it is printed (nor the thickness of the ink which prints it) - "effective dimension" is a concept involved in subjectivity (and also in perception) - the twine is zero-dimensional for somebody (far away).

[Undated scribble at side]: No fractional time dimensions.

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Airdrie, 16.09.1992: Gleick (op cit), p218: "[For the equation: x cubed - 1 = 0] ..Given any complex number as a starting point, the question was to see which of the three solutions Newton's method would lead to... [p219] Starting points that led to one solution were all coloured blue. Points that led to the second solution were red, and points that led to the third were green... A boundary between two colours never quite forms... no point serves as a boundary between just two colours... Impossibly[dd - !?], every boundary point borders a region of each of the three colours... [p220] magnified segments [of the boundary] reveal a fractal structure, repeating the basic pattern on smaller and smaller scales"

- Although there are areas of the complex plane not in the Mandelbrot set, it is not possible to draw the boundary - i.e. give the completed (spatial) description (picture) - between the points that do and don't lie in the set.

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[Undated list]: "research interests"; how pictures represent e.g. maps. how dimension affects representation. measurement and dimension. Topology, and what shapes can represent what others. "infinite complexity", boundaries, chaos theory. Fractal geometry/shapes, dimensions. different sorts of space: topological space, n-dimensional space, phase space (space representing time); spatial metaphors. space and the logical possibility of movement. modal logic, geometry, and experience. photography, maps, pictures. projection and pictures (and Wittgenstein on).

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Old musings posted on deckchair blogspot according to your orders SIR!