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M-set as metaphor
'To see a World in a Grain of Sand And a Heaven in a Wild Flower Hold Infinity in
the palm of your hand And Eternity in an hour'
William Blake's implied relativity of spatial and temporal scales is intriguing
and, given the durability of this worlds-within-worlds concept (he wrote in
1803) in art, literature and science, the blurring of distinctions between the very
large and the very small must strike some kind of harmonious chord in the human
mind.
Could this concept apply to the physical world? To be honest, we cannot be
absolutely sure. Most cosmological thinking still retains the usual notions of a
finite universe and an absolute size scale extending from smallest to largest
objects. In the boundless realm of mathematics, however, the story is quite
different.
The M-set was discovered by the French mathematician Benoit Mandelbrot in 1980,
created by just a few simple lines of computer code that are repeated recursively.
As in Blake's poem, this 'world' has no bottom – we have an almost palpable
archetype for the concept of infinity. I would use the word 'tangible', but one of
the defining features of the M-set is that nowhere in the labyrinth can one find a
surface smooth enough for a tangent. Upon magnification even surfaces that appeared
to be smooth explode with quills and scrolls and lightning bolts and spiral
staircases.
And there is something more, something truly sublime. Observe a small patch with
unlimited magnifying power and, as you observe the M-set on ever-smaller scales,
down through literally endless layers of ornate structure, you occasionally come
upon a rapidly expanding cortex of dazzling colour with a small black structure at
its centre. The black spot appears to be the M-set itself! There is no end to the
hierarchy, no bottom-most level, just endless recursive worlds within worlds within
worlds. Scale is no longer fixed and absolute, but is purely relative. These
beautiful symmetries convey an immediate aesthetic pleasure and also compel one to
think about these strange concepts of self-similarity, infinity and relativity of
scale.
Our present science tends to favour reductionism. We surmise that the physics of
our world has a most fundamental level and all phenomena are built up from these
quarks or strings. Mathe-matics need not be so limited: here the mind is set free to
dream of universes with the most exquisite symmetries and infinities. I urge you to
explore the M-set. The epiphanies you experience will be worth the effort.
Robert L OldershawPhysics Department, Amherst College, Amherst, MA 01002, USA rlolders@unix.amherst.edu
Video copies of The Colors of Infinity are available from Humanities, Inc.
Princeton, New Jersey, priced $30. There are also several websites such as www.softlab.ntua.gr/mandel/mandel.html or tqd.advanced.org/3288.
The abuse of algebra
What a pleasure it is to read the work of students whose reasoning is easy to
follow, who observe the rules of grammar in all their writing, and who remember that
an algebraic equation is and must be a sentence in their native language, albeit
written in a universal shorthand.
About thirty years ago the ASE encouraged us all to use 'Quantity Algebra'
consistently rather than to muddle on with inconsistent (and therefore incorrect)
hybrids of 'Number' and 'Quantity' Algebra.
Number Algebra is tedious if used correctly in physics. But Quantity Algebra seems
to petrify Maths departments, whose incoherent practices undermine the efforts of
Physics teachers to persuade their pupils to reason both logically and clearly.
When I read a pupil's work, the final answer (or conclusion) interests me far less
than the reasoning that leads to that conclusion. I want to be able to check the
work as I read it, and it helps greatly if units are included when values are
substituted for symbols. Textbooks which set out their worked examples in Quantity
Algebra are especially appreciated, not only for illustrating the 'good practice' we
want to encourage, but, of course, in helping the student keep sight of the physics
throughout. Physics texts which do not use Quantity Algebra in their worked examples
invariably demonstrate faulty logic ... besides hiding the physics.
Here is a very simple example:
Good practice: Force = 70 kg × 10 N/kg
= 700 N
Bad practice: Force = 70 × 10
= 700 (or Force = 700
N)
The final 'slide-rule' manipulation is of numbers, of course; but we should keep
sight of the route to those numbers.
Years ago the Head of Maths at a large comprehensive school described how he
persuaded all departments to convert to Quantity Algebra. But he ended with an
admission: that such an initiative must come from the Head of Maths. That
enlightened man understood the problem: his fellow mathematicians.
Tim WatsonWorcester
