Replying to LO27792 --
Dear LO'ers,
What I have understood from the contributions of all of you on the subject
" holism, a product" , but particularly of Don Dwiggens and At de Lange is
that my expression ' balance' is too narrow, not to say too unique. A
certain amount of necessary tolerance seems excluded from this expression.
However, my humming (humming? there were no witnesses, apart from Terra
and my thinking was too intense to hear myself :-)) thoughts during that
stormy day were more flexible than what might be concluded from my
original contribution. But I had not thought of the range of flexibility
or tolerance. At made it clear for me in LO27792 with a briljant proposal:
the golden ratio, directly related to Fibonacci.
Please see for the mentioned figures
http://www.learning-org.com/graphics/LO27891_harmbal.gif
But let me go back to the balance. The classic balance or pair of scales,
a hinge point in the middle of an arm and where at both ends of this arm
various weights could be mounted. Thus a balance where relative weights
could be measured - which weight is the lightest? If the balance is in
balance, both weights are equal in weight and the hinge point in the
middle of the arm. A perfect symmetrical situation is realised (Fig. 1).
As soon as both weights on both sides are not equal, the pair of scales is
in unbalance (Fig. 2). Now here comes the crux: how could the balance
brought into balance again when the weights are unequal? There are three
possibilities:
a) put extra weight on the lighter side until both sides are equal again;
b) take some weight away from the heavier side until bith sides are equal
again.
Options a) and b) are common practice. We may remember it from the
chemical labs, or from the old-fashioned grocery store where the grocer
manipulated with the verified and stamped brass weights.
c) we could change the hinge point of the arm in such a way that both
sides become 'floating', demonstrating an apparent equilibrium.
Think of the situation that you play with one of your children (or
grandchildren) on a seesaw. If you want equilibrium it may be necessary
that you should move closer to the hinge. The heaviest is closer to the
hinge. In the old days there were also shops were such a balance where the
hinge could be moved on a scaled arm was practice. This is illustrated in
Fig. 3.
And now comes Dwig in the picture: working with ratios. The arm of the
balance is divided by the asymmetrical hinge point in two sides of
different length. The side where the heavy weight is mounted is the
shortest of the two. And the reciprocal ratio of the lenths of both sides
is the ratio of the weights! As long as we keep the ratio of the weights
constant, the hinge point could stay at its place.
And what has At sketched: the ratio of Fibonacci, the GOLDEN RATIO of
1.61803
What I understood from At's contribution is that as long as the ratio
between two numbers (or weights) is between 1 (equal) and 1.61803 (or
0.61803; golden ratio) then there is harmony. If the ratio is outside this
range the relationship is disharmonious (Fig. 4)
So we could define a range or window of harmony: the ratio R must be
between 0.61803< R < 1.61803 (limits included) (Fig. 5). It is as if we
are allowed to displace the hinge point of the balance between these two
minimum and maximum values. As long as we could manage to balance between
these values the relationship is harmonious (check if you have a
harmonious relationship with your children :-)).
A balance as sketched above is just a simple instrument, although you have
seen that it could not only be used for measuring weight ratios, but also
for length ratios.
Let us try to compose a complex balance. I have tried to sketch one in
Fig. 6. No, dear readers, it is not the usual organogram of an
organisation, but it is a special balance, a so-called mobile. This mobile
is 'loaded' with Fibonacci-weights and the various hinge points are at the
golden ratios of the various arms. Note that even the weight 'number 1' is
connected with weight 'number 13'. If you pull on weight 1 the whole
mobile starts moving and becomes in unbalance. If one weight is changed,
the whole becomes in unbalance. Note too, that if you relieve weight 1,
the whole becomes in a harmonious balance again.
For me this mobile is a strong working imagination of At's 7 E's. I have
sketched a mobile with 6 weights, but the number could easily extended to
7 ofcourse. Please label the weights instead of numbers with the namings
of the different E's. Note that in this mobile some E's are directly
connected to each other, the relationships with others could be more
distant. I had great fantasies playing with different labelling and
combinations. Also the whole configuration of the mobile could be changed,
although the fractal pattern is obvious.
But the strongest thought of this picture is that each one, whatever its
position, is related to the rest.
Maybe it is worth to think of this sketch as an organogram too (of a LO
??).
I realise that this dialogue becomes somewhat number-fixed. Whether it are
weights, lengths, ratios are something else. I realise too that the
Fibonacci numbers and the golden ratio are very deeply related to all
kinds of forms and contents in nature. So I could imagine that there is
something very important within this range of harmony
(0.61803..<R<1.61803..) (the mysterious force of nature?).
But the numbers and the picture of the mobile work for me more as a
metaphor, because I realy don't know how to quantify each of the 7 E's.
As a puppet on a string I have a great feeling, floating in the air.
dr. Leo D. Minnigh
l.d.minnigh@library.tudelft.nl
Library Technical University Delft
PO BOX 98, 2600 MG Delft, The Netherlands
Tel.: 31 15 2782226
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Let your thoughts meander towards a sea of ideas.
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--Leo Minnigh <l.d.minnigh@library.tudelft.nl>
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