This is the abstract for my
invited talk at CCCG
Rapid Prototyping of Geometric Models
George W. Hart
Rapid prototyping refers to a wide range of state-of-the-art techniques
in which a 3D design specified by a computer file is fabricated in a machine
which calculates the object's cross sections, then sinters, laminates,
or solidifies hundreds of very thin cross-sectional layers. I am a sculptor
creating works that celebrate the beauty of geometry, but I also use rapid
prototyping technologies to create interesting mathematical models. The
presentation will illustrate a range of recent examples, and demonstrate
some software I have written and used in this process.
There are various automated 3D assembly methods currently under development
by different research groups. The three models shown in Figures 1d, 2d,
and 3d are fabricated by the technique called stereolithography. A file
listing a set of bounding triangles is presented to a computer-controlled
laser which catalyzes a polymer-ization reaction (for each calculated cross
section) at the surface of a liquid that gradually rises. After several
hours of building, the fluid is drained and a plastic model is the result.
While accurate and durable, the models are currently quite expensive. In
the future, the cost of rapid prototyping should come down to the point
where such models can be made at low cost, and they will be widely used
for educational purposes.
Figure 1 shows drawings by Leonardo da Vinci, and a stereolithography
model that recreates one of the forms he illustrated. Figure 2 shows an
original design based on a set of topological operations: it is the expansion
of the expansion of the "propellorization" of the tetrahedron, rendered
in a hollow open-face manner. Figure 3 shows another novel design, this
one based on six nested polyhedra in geometric progression; a general-purpose
algorithm was used to wrap the edges with bounding triangles in a very
References - provide some further details about the algorithms
used in each step, illustrate additional models, and discuss my related
Acknowledgment: The illustrated models were fabricated by Dr.
Manfred Hofmann of RPC, Marly, Switzerland.
|Figure 1a. Rhombicuboctahedron drawn with "open faces" by Leonardo
da Vinci (circa 1496). This and Figure 1b are two of the sixty illustrations
Leonardo prepared for Luca Pacioli's 1509 book De Divine Proportione.
|Figure 1b. "Elevated rhombicuboctahedron." Leonardo and Pacioli
explored nonconvex polyhedra which result from simpler forms by "elevating"
them, i.e., erecting pyramids with equilateral triangles on each of their
|Figure 1c. Computer model of "elevated rhombicuboctahedron"
provided for fabrication.
|Figure 1d. Final stereolithography model, 8 cm diameter, made
of epoxy resin. Small (0.01 inch) stair-steps indicate the layers.
|Figure 2a. "Propello-tetrahedron". This self-dual polyhedron
has 16 vertices, 30 edges, and 16 faces (4 triangles and 12 kite-shaped
tetragons). It is the simplest polyhedron with chiral tetrahedral symmetry,
it serves as this model's "seed." In the next step is "expanded" by separating
the faces and inserting a tetragon for each edge and an n-gon at each n-fold
vertex. Then this expansion operation is applied again.
|Figure 2b. "Expanded expanded propello-tetrahedron". This novel
polyhedron has 240 vertices, 480 edges, and 242 faces (8 triangles and
234 tetragons). It is realized here in canonical form (meaning its edges
are tangent to the unit sphere and the center of gravity of the contact
points is at the origin) and has octahedral symmetry. It is fun to visually
follow the meandering edge paths.
|Figure 2c. Puncturing all the faces and creating ribs to a depth
of one half of the radius produces this computer model for fabrication.
It is actually in the same "Leonardo style" as Figures 1a, 1b, and 2a,
but the depth parameter has been set much larger.
|Figure 2d. Final stereolithography model, 4 cm diameter, made
of epoxy resin. It contains a hollow space half its diameter. Close inspection
reveals that it is assembled from parallel cross-sectional layers (100
|Figure 3a. Truncated icosidodecahedron (TID) rendered in the
open-face style of Leonardo. This Archimedean polyhedron consists of 12
decagons, 20 triangles, and 30 squares.
|Figure 3b. Edge structure of six nested copies of the TID, each
0.8 the size of the next larger, and connected to each other by five conical
helixes in each decagonal opening. 180 edges in each TID and 12 groups
of 5 5-segment spirals, make 1380 segments total.
|Figure 3c. By efficiently wrapping the 1380 edges in triangles,
we obtain a computer model of a solid TID^6. Ready for fabrication, it
bounded by 9600 triangles.
|Figure 3d. Final stereolithography model, 8 cm diameter, made
of epoxy resin. No other technology can produce such an exquisite crystalline
G.W. Hart, web pages, http://www.georgehart.com/
G.W. Hart, "Computational Geometry for Sculpture," Proc. of the Seventh
Annual Symposium on Computational Geometry, ACM, Tufts University, 2001,
G.W. Hart, "4D Polytope Projection Models by 3D Printing," Hyperspace,
G.W. Hart, "In the Palm of Leonardo's Hand," Nexus Network Journal,
Craig Kaplan and G.W. Hart, "Symmetrohedra," Proc. of Bridges 2001:
Mathematical Connections in Art, Music and Science, Southwestern College,
Kansas, July 2001.
Douglas Zongker and G.W. Hart, "Blending Polyhedra with Overlays," Proceedings
of Bridges 2001: Mathematical Connections in Art, Music and Science,
Southwestern College, Kansas, July 2001.
G.W. Hart, "Loopy," Humanistic Mathematics, (to appear) 2001.
G.W. Hart, "Reticulated Geodesic Constructions," Computers and Graphics
G.W. Hart, "Solid-Segment Sculptures," Proceedings of Colloquium on
Math and Arts, Maubeuge, France, Sept. 2000. (to appear)
G.W. Hart, "Sculpture based on Propellorized Polyhedra," in Proceedings
of MOSAIC 2000, University of Washington, Seattle, August 2000.
G.W. Hart, "Millennium Bookball," Visual Mathematics 2(3), and in
Proceedings of Bridges 2000: Mathematical Connections in Art, Music
and Science, Southwestern College, Kansas, July 2000.
G.W. Hart, "Zonohedrification," The Mathematica Journal, 7(3), 1999.
G.W. Hart, "Icosahedral Constructions," Proceedings of Bridges: Mathematical
Connections in Art, Music and Science, Southwestern College, Kansas,
G.W. Hart, "Zonish Polyhedra," Proc. of Mathematics and Design '98,
San Sebastian, Spain, June 1998.
G.W. Hart, "Calculating Canonical Polyhedra," Mathematica in Research
and Education, vol. 6-3, 1997.