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Derek Chan
CS294-10 Final Project: Polyhedron with Flowers
I have had a long-standing interest in the works of M.C.
Escher; thus, I looked towards him for inspiration in picking
my final project. One of his works (figure 1), simply entitled
Polyhedron with Flowers, particularly piqued my interest.
Figure 1
The original piece, created in 1958, was approximately
thirteen centimeters in diameter, and Escher carved in out of a block of maple. The nature
of the work as a sculpture, rather than as a two-dimensional print as most of Escher’s
works were, seemed to lend the piece towards a recreation on a three-dimensional
modeling machine.
Like most of Escher’s works, Polyhedron with Flowers is a fusion of both
aesthetics and mathematics. Its basis in regular polyhedra proved invaluable in my ability
to recreate the piece effectively without requiring the creation of each piece in some real
time three-dimensional modeling program. The sculpture can be considered to be based
off of two polyhedra: the small triambic icosahedron and the great stellated
dodecahedron. Both shapes share the same symmetry since they are both icosahedrons
with a triangular pyramid upon each face. The height of these pyramids is used to
separate the two shapes, as the pyramids on the small triambic icosahedron are
distinctively lower. When the apexes of these pyramids are connected, a dodecahedron is
formed. Thus the work can be seen as either of these shapes with a flower upon each
dodecahedral face and a leaf upon each icosahedral face, and this is how I chose to view
it when planning out my recreation.
I decided to make my recreation using SLIDE. SLIDE provided all the geometric
primitives that I ultimately needed, and SLIDE’s sweepmorph in particular proved
invaluable. Through its interface with tcl, SLIDE also readily allowed the creation of a
user interface, which could be used to tweak in real time various properties of the final
product towards the aesthetic judgment of a given user. In the final program, I have
provided multiple sliders to manipulate all the various parts of the model that would be
logical while still maintaining the basic form of the icosahedral flower. The sliders have
been subdivided into two windows: one primarily for modifying properties of the leaves
and the other to change the properties of the flower centers.
My process, in recreating the sculpture using SLIDE, can be subdivided into two
major tasks: the creation of the leaves and the creation of the flower centers. A flower
center consists of two parts: a stamen in the center surrounded by a cluster of petals. The
individual petals were created using sweepmorphs. I first created a single sweepmorph
that went along a path of a quadratic bezier curve. At each of the control points of the
curve is a circular cross-section of different radii. The first petal was modeled with
respect to the z-axis. I then instance eight of these petals and rotated them at equal angles
around the z-axis creating the petal cluster for my first flower. The positions of the
control points of the bezier path are changeable through the user interface. Sliders are
provided to modify the control points so that the height, spread (how much a petal curves
away from the center as it nears the tip), and radius of the petal cluster can be modified.
Sliders can also be used to modify the radii of each of the three cross-sections that the
sweepmorph passes. The stamen surrounded by this petal cluster was formed simply
using a sphere whose radius and distance from the center of the model are variable. Once
the entire flower center was created, it was instanced twelve times and each was
transformed according to the dodecahedral symmetries demonstrated in icosaball.slf (by
Prof. C. H. Sequin).
Like the petals, the leaves were similarly created using sweepmorphs. The faces
of a regular icosahedron are equilateral triangles; thus, the leaves originally consisted of
an equilateral triangle cross-section along a straight polyline path through the center of
the model. The size of the equilateral triangle at the base and tip of a leaf were both
changeable as were the distances of the base and tip from the center. The azimuth of a
leaf could also be modified rotating the leaf from its original position atop the faces of an
icosahedron. A single leaf was then instanced twenty times and transformed to each
icosahedral face. The transformations were derived using the aforementioned
dodecahedral transformations as a starting point however additional calculations, to
transform the leaves, had to be derived to match the
Figure 2
leaves up with icosahedral symmetry. In this first
attempt (figure 2) the leaves were basically variable
triangular pyramids, and this gave the leaves a very
artificial look. Quite obviously the model as a whole
left much to be desired aesthetically.
One of the challenges of my project was to simulate the organic feel of Escher’s
original work. Since the piece was handcrafted, it was imperfect in its symmetry and
geometry; however, this gave it a more natural and authentic feel. The sculpted work had
noticeably few hard edges and points. After my first attempt, I tried to simulate the
organic feeling of Escher’s original by varying the leaves more from the triangular basis
underlying each of them. I also tried to limit the linearity of the leaves and give them a
more rounded feel, as my original leaves were very pointy.
On my second pass with the leaves, I changed the
Figure 3
sweepmorph’s cross-section from an equilateral triangle to
a 28-point polyline (figure 3) centered on the origin and in
the xy plane. Although the polyline still went through the
vertices of an equilateral triangle, the new cross-section
better simulated Escher’s original piece and gave the model
the more organic feeling that I desired. The improved
cross-section gave the faces of the leaves a curved surface
as well as implemented the vein evident down the leaves in
Escher’s sculpture. My second attempt (figure 4), although
Figure 4
much improved, still had noticeably sharp edges between the faces of a given leaf and the
linear triangular pyramid shape that the leaves were based on was still somewhat evident.
To eliminate the sharp edges, I changed the straight polyline path of the sweep morph to
a straight-line path formed by a bezier curve. Although the shape of the path was the
same, SLIDE allows for the dividing of a bezier curve. The bezier curve also has the
advantage over a bspline curve in that it must end at its endpoints. The subdivision
property was important to allow the twist property on the leaves to give the edges
between the faces of a leaf an aesthetic curve. The number of subdivisions of the bezier
and the degree of twist in the leaves were both assigned to sliders. Cross-sections were
also added in between the tip and the base of the leaf so that the rate of change of the
width of the cross-section could at least be piece-wise linear. After adding these
properties I tweaked the part with the sliders I created and set the default values to create
the part I thought was most aesthetically pleasing resulting in my final model (figure 5).
The computer model was designed with
the intention to be created on the zcorp
machine. The recreation admittedly does not
attempt to remedy the overlapping amongst the
various parts of the flower when the parameters
controlled by the sliders are modified. The
zcorp machine generally handles internal walls
Figure 5
better, and the solidly filled nature of the model
lends itself towards the zcorp. The zcorp machine also allows for the use of color, an
often-prominent feature of flowers.
Although unlike Escher’s sculpture my recreation has identical copies for each of
its parts, I believe that I have been rather successful in my attempt to model Escher’s
work. Discontinuities in the smoothness of the surfaces are still noticeable; however my
attempts to give the model an organic feel have progressed far resulting in an ultimately
aesthetic piece.