George W. Hart's

Rapid Prototyping

Web Page

I am active in using rapid prototyping (RP) technology for a range of purposes, including art, math, and education. This page collects in one place some models I have designed, with links to papers that have further information about the algorithms, etc. Files for most of these models are provided, which you are welcome to download and replicate on your own RP machine, as long as you give me design credit when displaying them.

If you have comments or suggestions about these models, please send me email.

This image shows me with a Sierpinski Tetrahedron model, described below.

Background

Rapid Prototyping or Solid Freeform Fabrication refers to a range of new technologies which construct physical three-dimensional objects by assembling thin layers of material under computer control. Objects can be made which are extremely accurate, complex, and beautiful, and which no other technology can produce. For basic information on this rapidly evolving technology, I recommend this site or this site or search Google for new developments.

Presently this is a somewhat expensive technology used mainly in high-end product design, and in research universities. But in the future, the cost will certainly come down and everyone will be able to create amazing physical objects with these machines. Consider how laser printers cost hundreds of thousands of dollars in the 1970's but now are commonplace in every school and business; I similarly expect rapid prototyping machines to be ubiquitous and inexpensive in ten years, available in all universities, at most high schools, at the copy shop down the street, etc. There will be many applications to art, mathematics, and education which I am beginning to explore now.

Mathematics

As a sculptor I am necessarily interested in three-dimensional geometry. As a hobby I am also interested in the mathematics of four-dimensional geometry. From a 4D object, one can calculate 3D "shadows" which are often beautiful but very complex objects. RP machines can easily produce these structures, which are stunning to look at even if one doesn't understand the underlying higher-dimensional ideas behind them. Below are two examples of such models; I hope in the future to be able to provide models for some algebraic surfaces and other interesting mathematical forms. This paper describes some of the mathematical models:

George W. Hart, "Creating a Mathematical Museum on your Desk", Mathematical Intelligencer, 27, No. 4, Winter, 2005

The 120 Cell is a 4D structure made of 120 regular dodecahedra. This "shadow" of it has the form of one large dodecahedron filled in with 119 smaller dodecahedra. In 4D all the dodecahedra are regular, but in this 3D shadow, angles are necessarily distorted, so only the innermost and outermost dodecahedra appear regular.

At right is shown an SLS model, about 3 inches in diameter, made on a DTM 2500Plus machine. It is a beautiful object to hold in your hand and rotate slowly. To make your own, just download this STL file (0.35MB) and send it to your local RP machine.

Below is a gorgeous 4-inch model made by the Extrude Hone "ProMetal" process. Made of stainless steel powder which is infiltrated with bronze to braze it together, it should be quite long lasting. So I am happy to think that in 2000 years, someone could be holding this same model in their hands and spinning it around as I am now. These are available for purchase through Bathsheba Grossman.

Here is a popular fractal, often called the Sierpinski Tetrahedron, because it is a three-dimensional generalization of the two-dimensional Sierpinski triangle. This is a "fifth-order" version, as there are five different sizes of octahedral holes. This model is scaled so the tetrahedron edge is about 8.5 inches long. You can get a better sense of scale from the image at the top of this page.

It is easy to write a program which generates a computer-rendered image of this form, but it usually has the little solid tetrahedral parts just barely meeting at mathematical points. If you designed a physical model like that, it would fall apart into dust. An RP machine requires a well connected solid interior (with a triangulated manifold boundary) so the software to produce this file has a few tricky aspects.

Here is the stl file (1.0 MB) for you to download and feed to your own RP machine, to make your own model.

Even more beautiful and intricate is the truncated 120-cell, a 4D object made of 120 truncated dodecahedra and 600 tetrahedra. At right is shown an SLA model of an orthogonal shadow of it, about six inches in diameter. It is quite stunning to view the tunnels which penetrate it in six different directions. To make your own, just download this STL file (0.81 MB) to send to your local RP machine.

The paper cited above also describes this structure.

I used these and many other 4D shadow models in a seminar on The Fourth Dimension which I taught at Stony Brook University in the spring of 2003.

You may also be interested in reading a paper describing the mathematics behind this and other models of 4D objects I have made.

This is a model of a polyhedron first described by the mathematician Michael Goldberg in a 1937 paper. (It is "8,3" in his series of such polyhedra. For this model, I chose the largest one with under 1000 faces; it has 972 faces---12 pentagons and 960 hexagons.) But as far as I know, neither he nor anyone else in the intervening years had previously worked out how to calculate the lengths and angles to make these polyhedra with planar faces. I'll describe more about these polyhedra and ways to make them in a future paper.

This is a five-inch diameter model made on a 3D Systems InVision machine. If you want to make your own, here is the STL file (1.7MB).

Here is a nested series of seven spheres which can rotate freely, independently of each other. This construction is an homage to the long tradition of turning concentric ivory spheres on a lathe. As outlined on this page, this artistic tradition started in Nuremburg in the seventeenth century and is still carried out in parts of Asia.

Each sphere is based on the edges of a different Goldberg polyhedron. From inside to inside, they are:

2, 0 (42 faces);
2, 1 (72 faces);
3, 0 (92 faces);
2, 2 (122 faces);
3, 1 (132 faces);
4, 0 (162 faces);
3, 2 (192 faces);

This is a three-inch diameter model made on a Stratasys 3000 FDM machine. If you want to make your own, here is the STL file (1.4 MB).

When built, the twelve pentagons are aligned in all the spheres, so you can look right through them in to the center and out the other side. After randomizing the orientations, it is something of a puzzle to restore this property.

This is a tangle of ten equilateral triangles, just one of many interesting polygon tangles first described by Alan Holden. The stl file is here (0.01 MB).

For the history of these forms and many more examples, see this paper. The bottom of that page also has a link for you to get the java software which I wrote that allows you to generate the stl files for all of Holden's tangles plus infinitely many new ones.

Here are models of two uniform polyhedral compounds with icosahedral symmetry. These were first described in the mathematics literature by John Skilling in 1976. I don't know of anyone making a physical model of either of them before I made these in 1999.

At top is the compound of five concentric truncated tetrahedra. Below is the compound of six concentric pentagonal prisms. To understand these, you must see each as several interpenetrating solids. In the compound of five truncated tetrahedra illustrated at top, you can see a large equilateral triangle facing you. Its edge length is roughly the same as the overall radius of the object. There are also large hexagons of the same edge length. One truncated tetrahedron consists of four of these triangles and four of the hexagons. The overall form is five of these interlocked. Similarly, in the lower figure find large squares and pentagons of the same edge length. Five squares and two pentagons make one pentagonal prism, and there are six interlocked prisms. It may be easier to see the structure if you read the stl file into a 3D viewing program and slowly rotate the view.

The models are made of plaster on a Zcorp machine, which uses inkjet printer technology to squirt water selectively in the places where the plaster dust is to be hardened, leaving the unmoistened plaster dust to be vacuumed away. The stl files are here (0.03 MB) and here (0.03) MB.

Here is another popular fractal, the Menger sponge. The FDM model shown at right is third-order, i.e., there are three sizes of holes. The stl file to make your own is here (1.8 MB).

The surface area (and therefore file size) grows exponentially with the order. If you would like me to email you the 26 MB stl file for a fourth-order Menger sponge, let me know. It is easier for you to generate it on your own computer. A simple algorithm can be based on the idea that a voxel is empty if there exists a trit position (in base three representation) where two or more of its X, Y, and Z coordinates have the value one. If you have access to Mathematica, you can use my software in this paper to generate stl files for Menger sponges of any order.

The lower image at right shows two halves of a third-order Menger sponge, with a hexagon of paper between them. You probably know how to cut a cube in half with a planar slice to create a hexagonal cross section: you cut on a plane which is a perpendicular bisector of any of the cube's four long diagonals. (If you don't understand this, go cut a cube of cheese or potato, then come back.) A great visualization exercise is to figure out what the cross section is when you cut a Menger sponge along that hexagonal slicing plane. In other words, what is shape of the area where the half-sponge is in contact with the paper?

The stl file for you to build your own halves is here (1.1 MB). The model shown at right has 5.5 cm edge length, made by selective laser sintering. It is a fun puzzle to give someone the two halves together and ask them to try to visualize and draw the boundary surface before separating the halves. Then they can open it up and see how close they came to the correct cross section.

To give you a chance to think about it before seeing the answer, I put a photo of the separated halves on a separate page, here.

These are tetrahedra and an octahedron built from a small inventory of block shapes that are Voroni cells around face-centered cubic lattice points in the polyhedra. A more detailed explanation (and the stl files to make your own set) are available here.

Here is a 4-inch model of a two-layer geodesic sphere. There are 260 triangles in the outer layer. The inner layer has 12 pentagons and 120 hexagons. As far as I know, it is the world's only chiral two-layer geodesic sphere, as explained here.

The stl file, if you want to make your own copy on your own RP machine, is 20MB.

This is an old favorite of mine, the snub truncated icosahedron. The pentagons and hexagons of a soccer ball are separatged by a chiral boundary of triangles. Read more about it here.

Here is a fairly cool object, if you like music and geometry: It is a model of the orbifold representing three-note chord types, as described by Clifton Callender, Ian Quinn, and Dmitri Tymoczko here. Each sphere represents a type of three-note chord, but abstracting away any particular transposition. The struts connect pairs of chords that differ by one voice changing one semitone. The top left is the augmented chord, sitting at the cone point. It is adjacent to the major triad and the minor triad. At the bottom right is the "unison chord," meaning all three voices sound the same pitch. Most chords have six neighbors, because any of the three pitches can be raised or lowered a half step. Along the orbifold boundary at the bottom are all the chords which have two identical pitches. These generally have four neighbors, except the unison chord, which has only two neighbors. The major and minor triads have five different triads as neighbors, but six connections, counting the two ways they can lead into each other.

The physical model shown is 5.5 inches long, made of nylon by SLS. The stl file is here. A high-res photo is here, with enough detail that you can look closely to see the tiny stairsteps of layered fabrication.

Fractal polyhedra clusters can be derived by putting copies of one polyhedron at the vertices of another. Scale the small polyhedra so they just touch their neighbors. Now take these clusters and position copies of them at the vertices of a larger polyhedron. Repeat to any depth level.

At right is an icosahedron of icosahedra of stellated dodecahedra. More info and the stl file to make your own are available here.

Here is a set of reconstructions of geometric sculpture designs by Morton Bradley.

More information about these forms and Bradley's original work, and the stl files to make your own copies of these, are available here.

Historical Models

In the future, small museums, schools, and interested individuals will be able to download and fabricate files to create an "instant exhibit" that replicates important original objects stored in museums. For example, to teach evolution a teacher might download and reproduce a set of skull models showing the cranial evolution from primate ancestors to modern man. These would be passed around in the classroom or displayed with appropriate signage. I have recreated a series of models of polyhedra which Leonardo da Vinci drew when he illustrated a mathematics book for Luca Pacioli. For historical details and additional information about these models, see this paper:
George W. Hart, "In the Palm of Leonardo's Hand: Modeling Polyhedra", Nexus Network Journal, vol. 4, no. 2 (Spring 2002).

This first is an "elevated icosidodecahedron" composed of 120 open equilateral triangles.

The model shown in my hand at right is made on an FDM machine. I have also made a wooden model of this and other Leonardo constructions, shown here.

The STL file is available here (0.43 MB) for you to produce on your RP machine.

This second is a spherical form with 72 open faces (24 triangles and 48 trapezoids). The form was originally a Renaissance teaching model to illustrate one of the constructions in Euclid's Elements (Book 12, proposition 7).

The model shown is about 3 inches in diameter, made on a Zcorp machine.

The STL file is available here (0.19 MB) for you to make your own.

Art

I am a sculptor and work with a wide range of materials, including RP designs. Here are some examples of my sculpture. Generally I do not release the files for a work I consider a sculpture as I want to have control over its reproduction. But below are a couple of interesting forms which I am happy to have you reproduce.

This is six nested truncated icosidodecahedra in an open structure. Be sure to notice the conical corkscrew spirals which connect the different layers together.

The model shown is about three inches in diameter, made in SLA. You can download the STL file (0.47 MB) if you want to make your own.

A short paper which explains the structure is available here.

Shown at right is an SLS model about 3 inches in diameter of a ball made of approximate rhombi. It is a good study model for understanding spatial symmetry, as it has the rotational symmetry of an icosahedron but no planes of mirror reflection. It is also something of an homage to the mathematician/astronomer/instrument maker Abraham Sharp.

You can download the STL file (0.99 MB) and make a copy on your own RP machine. Then search for the twelve points where five rhombi meet. (In the image one is at the very bottom, and two are on a horizontal line halfway up.) Notice that they do not point directly towards each other. The idea behind the geometry is described in this paper.

Here is one I call Tangled Reindeer. This is an SLS model, about 3 inches in diameter. The idea is to imagine a pile of reindeer with their bodies at the center. You can have a copy of your own by building this stl file (0.8 MB).

Here is a paper describing the methods underlying this and some other recent RP sculptures of mine.

This is a woven assemblage of Salamanders, in homage to M.C. Escher. It is a prototype for a large sculpture assembled in a group sculpture barn raising I led when I was artist-in-residence at MIT. Shown at right is a small 2.5 inch diameter model made on a new Stratasys/Objet Eden 333 machine. This stl file (0.54 MB) is available if you want to make your own copy.

Here is a paper describing the methods underlying this and some other recent RP sculptures of mine.

This is a small SLS model of a Snarl, which is both a puzzle and a sculpture. This 2-inch diameter scale is a bit small for seeing all the details of the design. To fully understand it, you can tape together a larger paper model of it, using the template in this short paper that I presented at the G4G6 conference in March, 2004. To make your own rapid prototype model of the solution use this stl file (1 MB).

Here is a synthesized form reminiscent of a "sand dollar" (stl file, 1.7MB) and another which is a kind of spiral toroid creature (stl file, 6MB). Both are from a project I call Echinodermania I, II, and III. A paper describing the generating techniques is available here.

I have added some images of additional forms from that paper on a page of More Echinodermania.

This sculptural form is based on the (10, 3)-a network, described in A.F. Wells' 1956 book The Third Dimension in Chemistry. Recently the same underlying crystal structure has been popularized by Toshikazu Sunada, who called it the K4 crystal. Its projections in various directions are very different, so see more photos of this fascinating 3.5 inch nylon model here.

The .stl file for it is here if you want to make your own physical copy with solid freeform fabrication equipment.

And now for something completely different...

This is an organic-looking form produced by an algorithm for "growing" shapes as cellular structures. It is actually just one moment in a 3D "movie" which produces a series of related forms, each with one more cell than the previous form. You can understand the algorithm by watching it grow on this page, which also has a range of additional examples.

Here is the stl file (2.6MB) for this particular moment in its development. As shown at right, it was made in ABS by FDM, with a height of six inches. The white material and the small size of the photo makes it hard to see the cellular structure. This larger photo of the model after I painted it makes very clear how it is constructed of many individual cells.

Puzzles

Here is a puzzle made of twenty identical parts that are fabricated on an FDM machine. More information is available here. The .stl file is available here.

Algorithms

The algorithm described in the paper listed below may be of interest to people who wish to convert sets of line segments into 3D models for RP production. The method it describes for efficiently wrapping segments with triangles is one of the steps in several of the structures shown above. At right is a 5 inch diameter form with five concentric spheres produced by this method.

George W. Hart, "Solid-Segment Sculptures," presented at Colloquium on Math and Arts, Maubeuge, France, 20-22 Sept. 2000, and published in Mathematics and Art, Claude Brute ed., Springer-Verlag, 2002.

I later improved the algorithm to be more robust and have more adjustable parameters. A paper describing the later version is here:

G. Hart, "Sculptural Forms from Hyperbolic Tessellations," Proceedings of IEEE Shape Modeling International 2008, pp. 155-161 (online pdf version)

Again, you are welcome to reproduce these models on your own RP machines as long as you give me design credit when displaying them.

I taught a special topics course at Stony Brook in Fall 2003 and again in Fall 2004 on these topics. You may enjoy looking at some of my students' projects in 2003 and 2004.