 The magazine of the Melbourne PC User Group The Pecten: An intriguing sculpture Ken Holmes kholmes@melbpc.org.au |
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As you swing from Wellington Parade into Spring Street, your
passengers (you are too occupied with traffic) will note a largish arrangement of curved metallic panels, on
the forecourt of the corner building, which suggests the shape of a seashell. Perhaps that is the Shell
building? An even larger fanned, facetted column of polished granite blocks, supporting a wide portal in
front of the entrance, also suggests the structure of a
seashell. You might well arrange your life to one day place yourself in the vicinity, on foot, with time to
contemplate the sculpture and, indeed, Mr Harry Seidler's acclaimed building. Mr Seidler has granted his kind
permission to publish the images here, based on the two structures.
At first sight of the sculpture, different people may react in different ways. Some may prefer to accept it
as a piece of modern art, perhaps too geometric for their tastes, with some vague symmetries too difficult to
fathom. Others would feel the challenge to discern the system behind its geometry. Even a cursory inspection
reveals that, from two directions, you see an extremely formalised shape of the Shell Pecten, or Scallop,
considered to be one of the most widely recognised commercial emblems around the world. The sculptor, C O
Perry, deserves full praise for his ingenious transformation to three dimensions of the two dimensional
shape.
Figure 1 is a wireline stereo to illustrate the shape in its most symmetrical position which makes it easier
to see that it actually has rotational symmetry, in that it is the same if rotated 180 degrees about the
vertical axis. Also, we see the upright shape of the pecten in the frontal view and can see that the side
view is an inverted pecten. The basis of the structure is two semicircles, about six metres in diameter,
crossed at 90 degrees and joined by concave, twisted bands. The surface in each quadrant is divided into four
strips, with two alternate ones present and the other two absent. Some years back, I produced a similar
diagram but had much difficulty defining the edges and dividing the strips into smaller rectangles. Any maths
seemed too clumsy and did not easily capture the shape. Playing with 3D Beziers for the recent boat articles
made me suddenly realise this was an effective solution, though not necessarily that used by the artist;
possibly a pure circle would have been easier for manufacture. Each edge of the strips runs between points on
each semicircle which provide the "anchors" for a three-point Bezier and the centre of the encompassing
sphere provides an obvious "handle" to draw the curves inwards. The strips seemed too concave, so the
"handles" were simply moved out a little from the centre.
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Figure 1.
Both figures may be viewed in 3D by placing a mirror edge on to the centreline and facing left, and nose on the other side of the mirror edge. Look at right half with right eye and at reflection of left half in mirror with the left eye. |
We need only calculate points on two strips starting
near the end of one semicircle, and curving off to opposite sides near the centre of the other semicircle.
After recording these, whether on the wireline plot or in the file to be used later for a raytracing, we get
the other six strips by the following steps:
* Rotate 180 deg. about the vertical axis. Record
* Rotate 90 deg. about the vertical axis. Invert. Record
* Rotate 180 deg. about the vertical axis. Record.
The recursive Bezier process conveniently divides the curves into equal segments, 17 in this case. (OK! The
end ones are half size.) With a Bezier also down the centre of each strip, we get 54 points and 34 (twisted)
rectangles in each of the eight strips. The rectangles in each strip must be divided into 68 triangles to
feed to the ray tracer, so the C++ program writes a 30 KB ASCII file including the 272 triangles for the 8
strips in
the form--Triangle { <n, n, n>, <n, n, n>, <n, n, n> } where "n" are <x, y, z>
co-ordinates of three points. A small utility is used to convert these to "smooth triangles", which include
the vectors at right angles to the surface at each point, to allow POVRay to render a smooth surface without
faceting.
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Figure 2 is the resulting ray tracing, where the
structure is laid over at 45 degrees to match the actual sculpture. The strips are actually 10 cm thick, but
this picture shows a single surface only as we are mainly interested in the overall geometry. The tracing
covers 900 x 1012 pixels for each eye view and for each pixel the calculation must find only those triangles
intersected by the line from the eye to the pixel, select the nearest, check the lighting (or shadow) on that
intersection point and its colour and intensity, and colour the pixel.
The fanned column in the background is not in its true relative position and this view shows only five of the
eight layers of granite blocks. The surfaces are twisted and the quadrilaterals are divided into plane
triangles using the upward/outward diagonals, which provides a fan of curved lines in harmony with the
overall shape.
Calculating co-ordinates of corner points is easy, but every block has a different shape so we can imagine
the challenge of cutting, polishing and erecting this structure.
Of course, it would not be put in the same league as Gaudi's Sagrada Familia cathedral in Barcelona, under
construction for 70-odd years with perhaps the same again to completion. It incorporates a fantastic variety
of structures of complex, organic shape with virtually every element a distinctive shape.
Its detail has mainly been planned by manual draughtsmen, in conjunction with modellers on a large model in
the basement keeping just ahead of the builders. Only in recent years has
Computer Aided Design taken a place in the process and one can appreciate how useful it will be. Gaudi left a
legacy of just plain hard work and they need all the help they can get.
Reprinted from the August 1999 issue of PC Update, the
magazine of Melbourne PC User Group, Australia
