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Research
Form and Mathematics -
The Play
of Complexity: Growing Forms from Equations
The desire to create 'dynamic surfaces' probably arose out of some haptic,
kinesthetic sensibility. This might be in strong contrast to employing
a technical, mathematical route of exploration, in attempting to grasp
and visualise the evasive complex forms. The attempt to acquaint myself
with the mathematical language has not been an easy process but a very
rewarding one.
Three mathematical approaches were investigated:
Initially the forms were inspired by a series of diagrams
that are similar to three-dimensional 'Lissajous's curves', that
arise from an interaction of simple harmonic motions. Though not intrinsically
three-dimensional, the same principles could be simulated mathematically
in three dimensions and the resulting figures were expected to help in
the visualisation of complex curvature transitions in the desired forms.
Secondly, the global, dynamic principles further related
to the behaviour of dynamical systems as it is investigated in physics.
The mathematical principles are very different and the visual plottings
from this domain are more abstract in nature. Only in certain cases can
they be compared to actual surfaces. Nevertheless, the plots give some
insight into the possible dynamic continuity of three-dimensional curves
or surfaces. These will be dealt with on the Dynamics
page.
The most practical benefit was expected to come from the
investigation of minimal surfaces, which have been of particular
interest to mathematicians in the areas of differential geometry and topology.
Non-plane minimal surfaces are broadly speaking identical to anticlastic
surfaces, and a familiarisation with the related geometrical vocabulary
promised to be very useful.
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3D Lissajous Curves
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Strange Attractors
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Minimal Surfaces
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Three-dimensional lissajous curves
One way of generating Lissajous figures is via
the coupling of two pendulums, in a device called a Harmonograph.
Though it can be defined as simple harmonic motion, a Harmonogram could
be interpreted as tracing an orbit around the centres of gravity of the
pendulums, under the specific circumstances of phase and ratio. The
variables underlying these forms can be imitated mathematically, to draw
similar three-dimensional loops.
It was the harmonograms that triggered my fascination
with fluid surfaces, and the attempt of expressing the forms in metal,
the medium I was used to. The connection to the dynamic principles active
in the strange attractors proposed itself afterwards.
Harmonogram created with home-built Harmonograph
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Minimal Surfaces
The geometry of anticlastic curvature is directly
related to non-planar minimal surfaces.
Minimal surfaces occur naturally when a soap film
spans a non-planar three-dimensional boundary. The surface tension seeks
to minimise the energy required for maintaining itself and thus minimises
the surface area. The mathematical interest is based on the special topological
properties of such surfaces, and partially their economic elegance. I
often found the boundaries of the mathematically rendered minimal surfaces
to be rather restrictive and contrived, artificial - for they are still
tied to x - y - z .. . and not an iteration - which is time-based and
has a much more relaxed sense of origin (0,0,0), by referring back to
the previous point rather than the orthogonal axes. .. (excuse this simplification
- please send me an email if you can explain it . . .) ........ a thought
on 'reference points' . . .
I found that most minimal surface definitions
are either simple and easily visualised without the computer, or very
complex to the point where their mathematical definition is beyond my
scope of understanding.
The metaphor in this is that minimal, anticlastic
surfaces embody the 'negative' curvature of hyperbolic space, which is
a Non-Euclidean space, and is also the complementary principle to spherical
form, which is the basis of 'volume'. The anticlastic surfaces occur in
transitional phases, and are more open to change than the self-enclosed
sphere. Similarly, the transitional sections, between attractors, in a
dynamic diagram (phase portrait) are often such saddle shapes, where the
movement of flow goes up in one plane, and down in the perpendicular plane.
As such they could imply an opportunity for change, open to outside impressions
but also indecision.
Heikki Seppä (an inspired silversmith) also
pointed out that it is those spaces between things/solids which we easily
overlook....
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Heltocat
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'Seahorse'
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Meander
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copyright©Benjamin Storch2004
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