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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.