Abstract

The author's experience using "3D printing" technology for producing physical models of four-dimensional objects such as the 120-cell is summarized. For background and comparison, previous mathematical models of the 120-cell are reviewed first.

1. Introduction

Three-dimensional projections of four-dimensional polytopes are valuable for teaching and self-education about higher-dimensional geometry. Experience shows that physical modelsreal 3D objectsare especially useful for developing intuition and understanding about 4D polytopes. A variety of physical modeling techniques are possible, with three popular materials being paper (or cardboard), wire (or wire and string), and Zometool (a plastic construction set). This paper illustrates a new 3D-printing technique that I expect will supplant these traditional materials for a wide range of mathematical modeling applications.

3D-printing allows the creation of models that are very compact, intricate, accurate, and portable. This state-of-the-art technology involves the automated (robotic) assembly of physical models by assembling very thin cross-sections calculated from the designer's computer file describing the structure to be built. This new technology will have applications in the creation of all types of educational 3D models, e.g., polyhedra, topological or algebraic surfaces, molecular models, crystal structures, anatomical models, etc. However, this paper focuses on geometric models related to the 120-cell, and its family of truncations and expansions. This example is featured because the author considers it one of the most beautiful geometric objectsyet it is not widely enough known, in part because of the lack of models.

Software presentation of geometric structures, e.g., interactively rotating images on a computer screen, have a valuable and growing role in modern education. (Search the internet for "120-cell" to locate several fine examples.) However there has been and will always be a role for physical 3D models. In the heyday of physical mathematical models, from the 1870s to World War I, universities were proud to boast of their educational models. The German geometer Felix Klein was a strong advocate of models in mathematics education reform, and American universities that emulated the German system purchased German-made models such as surfaces and polyhedra. For an excellent overview of this period, see Kidwell [18], but she does not discuss models for illustrating higher-dimensional geometry. Fischer [6] illustrates a wide range of mathematical models from this period, including stick models of 4- and 5-dimensional hypercubes (in the collection of the Palais de la Decouverte, Paris), but nothing as complex as the 120-cell. Mueller [21] emphasizes the sense of wonder that such mathematical models can excite in the viewer.

To understand projection models, first consider the "Schlegel diagrams" of Figures 1-5. [26] These show the five Platonic solids, each projected from 3D to 2D. These can be understood as shadows, from a nearby point light source, of the edges of the polyhedra. The geometry of the 3D lengths and angles is distorted in the projection, e.g., a square may project into a trapezoid, but all the topological properties are preserved. In each case, the outer boundary must be considered as one of the faces of the polyhedron, divided into regions representing the remaining n-1 polygons. The outer polygon can be thought of either as overlaying all the other faces or as an inside-out infinite polygon, filling the remainder of the plane. For example, in the cube's shadow at lower left, one must see six tetragons, not five, to understand that there are three squares meeting at each vertex. Figures 1-5 show "perspective projections," meaning the shadow-producing light source is nearby. From a light source infinitely distant, "Orthographic projections" result, in which the shadows of some edges will cross.

By "dimensional analogy" one can understand the structure of 4D polytopes from their perspective and orthographic projections into 3D space. Section 2 of this paper presents a history of 3D projection models of 4D polytopes, illustrating a range of examples known to this author. Then in Section 3, new 3D printing technology is described and several models are presented starting with my first geometric experiments and working up to intricate 4D models.

2. Some physical models of the 120-cell and other 4D polytopes:

The history of 4D Euclidean geometry is documented in a number of references. [4, 2, 28, 30] In 1852 Ludwig Schläfli discovered the six regular convex 4D polytopes: the 5-cell, 8-cell, 16-cell, 24-cell, 120-cell, and 600-cell. He also discovered four of the ten non-convex ones. But higher dimensional geometry was not a popular field of mathematics then, and his results were not appreciated. Also, Schläfli worked algebraically; the paper has no images of any sort which might have helped readers appreciate it. In 1858 and 1860 his paper was partly published in English translation by Arthur Cayley. However, it was only published in its entirety posthumously, in 1901. [24] In the 1880s, the topic became popular and many mathematicians, unaware of Schläfli's work, were competing to discover the higher-dimensional regular polyhedra.

Stringham [32] was the first to independently rediscover the six regular convex 4D polytopes. His 1880 paper includes images of what appear to be paper models. Figure 6 shows one of his figures that illustrate the structure of the 120-cell. Its 3D projection can be built by choosing one dodecahedron to be at the center and surrounding it by layers of 12, 20, 12, 30, 12, 20, 12, and 1 dodecahedra, totaling to 120. Stringham illustrates four stages in this process. Figure 6 is the result after three layers, i.e., an assemblage of 45 dodecahedra. There are also analogous images for the structure of the other regular 4D polytopes.
 

Fig. 6. From Stringham [32]

In 1883, Victor Schlegel constructed accurate projection models of the six regular 4D polytopes, made from brass wire and silk thread. [25] These were first exhibited at the 1884 meeting of the Society of German Naturalists. This was the height of the educational mathematical model period, and 4D polytopes became mainstream enough that physical models of the six were soon commercially available through mathematical supply catalogs. Schlegel's designs were sold first by Ludwig Brill of Darmstadt, and then by his firm's successor, Martin Schilling of Leipzig. [23; 28, p. 191] Schlegel's five-page instructional brochure uses Figures 1-5 to explain the models by means of dimensional analogy. [26] Examples of these wire and string models of the 5-, 8-, 16-, and 24-cell are on display at the University of Göttingen Mathematics Department. The only surviving examples I have been able to locate of the German 120-cell and 600-cell models are in the mathematics collection of the Smithsonian Museum in Washington, DC. However at this time they are in storage, missing their strings, and in need of conservation.

Sommerville [29] gives a mathematical description of these models:

Fig. 7. Cardboard models showing layers of the 120-cell, from Dyck [5]

Horsburgh's 1914 Handbook of the Napier Tercentenary Celebration records that projections of the six regular 4D solids by J.E.A. Steggall were on display. Described only as "a representation, with the dissection of certain parts," their material is not specified. [17] It also records the projection model of the 600-cell made of brass wire and silk by D.M.Y. Sommerville. In contrast to Schlegel's cell-centered projections, this projection is centered on a vertex, as detailed in [29]. There is no image, but the description in [17] concisely explains the construction:

Figure 8. Donchian's 120-cell model [2]

Various individuals, captivated by the geometric beauty of these forms, have independently made 120-cell models for their own gratification or edification. Several 5-foot welded stainless steel models constructed in the 1990s by Marc Pelletier are most impressive. Yuro Otobe has constructed a series of soldered and painted steel wire models (up to 20 cm in diameter) since about 1980. They have been presented in seven previous issues of Hyperspace and are on display at the museum of the Mathematical Institute of the University of Tokyo. Paper (or cardboard) is a popular, inexpensive modeling material, so it is likely that many people have used it for polytope projection models in addition to Figures 6 and 7 above. A notable example is that Koji Miyazaki constructed beautiful color paper models of the 120-cell illustrated in [20, p. 74, 95].

In 1970 Steve Baer designed and produced a geometric modeling set called "Zometool" with plastic nodes and wooden struts [1]. Marc Pelletier discovered (when 17 years old) that the lengths and directions allowed by this kit permit the construction of accurate models of the 120-cell and related polytopes. The kit went out of production however, until 1992, when it was redesigned in a plastic version that can be used to make an extremely wide range of mathematical models. [35, 15] In particular it is capable of making models of the 120-cell and 600-cell, and their family of truncations and expansions.
 

Figure 9. Zometool model of truncated 120-cell

Figure 9 shows a Zometool model of the truncated 120-cell, a 4D polytope consisting of 120 truncated dodecahedra and 600 regular tetrahedra. Instructions for its construction are given in [15, p. 211]. This is one of many "Archimedean" polytopes discovered by Alicia Boole Stott and first described in her 1910 paper [31]. She was a daughter of George Boole, untrained in mathematics, and has a fascinating story outlined in [4, p. 258]. This 1.5 meter model was constructed in a student workshop led by the author at Mathcamp 2000 in Vancouver. [19] (The students had made a Zome 120-cell model on the previous day, in preparation for this larger project.) In this projection, it produces a stunningly beautiful 3D form with 10-sided passageways piercing it in six different directions. In the several hours required for its construction, the students gained a significant appreciation of the geometric issues. This is also an excellent social, engineering, and visual experience. However, making large Zome models of several related polytopes would require excessive time.

3. Models produced by 3D Printing

In this context, I set out to make new models of 4D polytopes using state-of-the-art rapid prototyping technologies. There are various automated 3D assembly methods currently under development by different research groups. [3] What they have in common is that the 3D design specified by a computer file is fabricated in a machine which sinters, laminates, or solidifies hundreds of very thin cross-sectional layers. One early technique ("laminated object manufacturing") uses a computer-controlled knife or laser to cut sheets of paper, which are automatically stacked and glued to each other to make a wood-like material. The "stereolithography" technique directs a laser that catalyzes a polymerization reaction for each cross section at the surface of a liquid that gradually rises, resulting in a plastic model after the fluid is drained. The Fused Deposition Modeling (FDM) technique is effectively like a "hot melt glue gun" at the end of a robotic arm, which builds a plastic model in the air. Selective laser sintering (SLS) uses a laser to sinter wax or metal dust in a chamber to which is added a new (thin) layer of dust after each cross section. When complete, the loose dust is vacuumed away and the hardened model remains. The technology I have used the most is "3D printing", like this last, but the dust is plaster and the mechanism of an ink-jet printer is used to print the cross sections with water. Developed by the Zcorp corporation, this technique is promoted as relatively inexpensive. [34] Unless otherwise stated, the models below were produced by Zcorp and are each about 8 cm in diameter. Typically, several hours are required to build each.
 

Figure 10. Uniform compound of five truncated tetrahedra.

 
 

Figure 11. Uniform compound of six pentagonal prisms.

To familiarize myself with this technology, the first two models that I made were solid polyhedral compounds with icosahedral symmetry. Figure 10 shows the compound of five truncated tetrahedra, derived by truncating all the tetrahedra of the standard compound of five regular tetrahedra inscribed in a dodecahedron. Figure 11 shows the uniform compound of six pentagonal prisms. Each prism has its 5-fold axis aligned with one of the 5-fold axes of the icosahedral symmetry group, and is rotated on its axis to a position intermediate between that of the two opposite faces of a dodecahedron. Described in a 1976 paper by John Skilling, [27] as far as I know these two compounds had not been made previously in paper or any other media. However, they could straightforwardly be made in paper, and instructions for Zometool models of both are available on my web pages [9].

After gaining experience with the technology, file formats, and other production issues, I decided to try hollow models with a thin shell, which would be more difficult to produce by any other technique. Figure 12 shows an original form, which I call the snub truncated icosahedron. Like the snub Archimedean solids, i.e., the snub cube and snub dodecahedron, this form is derived by inserting a chiral structure of triangles between the faces of a simpler polyhedron. In this case, the simpler polyhedron is the truncated icosahedron, consisting of 12 regular pentagons and 20 regular hexagons. However, the result is not an Archimedean solid and the snub triangles must differ slightly from equilateral. I chose an open-faced edge model, in the style of Leonardo da Vinci. The result is a very light structure, like an eggshell.
 

Figure 12. Snub Truncated Icosahedron

Having gained better insight into the technology, and a sense of how thin the struts can be, I felt ready to make open models with internal structure. Figure 13 shows a 10-cm form with five nested concentric spheres. Artistically, I was paying homage to the long tradition of "Chinese Balls" in which ivory spheres are turned on a lathe and then carved, to produce intricate concentric forms. [16] For my structure, I chose a form which is the 31-zone zonohedron based on the directions of the rotation axes of the icosahedral symmetry group. Each of the five layers has the same structure as the outer, most visible, orb: there are 30 large 12-sided oval openings, 12 smaller 10-sided openings, 80 irregular hexagonal openings, and 120 small rectangular openings. Oval "corkscrew spirals" in the 12-sided openings connect the layers with each other. I consider this a sculpture rather than a mathematical model, although that distinction can be difficult to draw. It is titled Deep Structure.
 

Figure 13. Deep Structure

In the underlying mathematical structure of Figure 13, there are 4080 line segments positioned in space. However, the computer format for specifying 3D printing requires a volume bounded by a list of triangles. So a significant part of my work was to develop an efficient general purpose algorithm that creates a triangulated surface surrounding an arbitrary set of line segments. The algorithm is described in [13]. For producing the result in Figure 13, there are 27214 triangles altogether, a very low number for such a complex, high-genus structure.

The same algorithm can produce suitable triangulated structures when given a list of edges for a projected polytope. The necessary vertex coordinates and edge lists for polytope projections are relatively straightforward to produce. [4, 33] Figure 14 shows a virtual model of the 120-cell. This is a standard orthogonal projection along a line that connects the centers of opposite cells. However, as a sculptor, I am interested in visually interesting effects, so for my first 3D model, I made a small variation to the algorithm, producing the chiseled sculptural form of Figure 15 rather than the mathematical model in Figure 14. Its blue color results from a dye added to the plaster. Again, as I consider this a sculpture, I give it a name: the Five-Legged-Bee Hive. It was exhibited at the art display of the International Colloquium on Mathematics and Art, at Maubeuge, France, in September, 2000. I more recently gained access to a FDM machine, and produced the 8cm plastic model shown in Figure 16.
 

Figure 14. 120-cell edge model image

 
 

Figure 15. Five-Legged-Bee Hive, based on 120-cell

Figure 16. 120-cell model of ABS plastic. (8 cm FDM model)

 

Next, Figure 17 shows a perspective projection of the 120-cell, a 3D analog to a 2D "Schlegel Diagram". For visual interest, the strut thickness was made to vary smoothly from thin at the center to thick at the exterior. Because of its intricacy, a different technology was employed when building it. Figure 18 shows a wax model, produced by a Sanders machine [22]. This design was worked out in discussions with Carlo Sequin, who also paid for its production cost. The 3D printing is 5 cm across, executed and photographed by Bathsheba Grossman. [8]
 

Figure 17. 120-cell perspective projection image.

 
 

Figure 18. Wax realization of Figure 17, 5 cm diameter.

Figures 19 and 20 show models of two 4D Archimedean polytopes discovered by Alicia Boole Stott. [31] Both are orthogonal projections, built from a translucent epoxy by the stereolithography process using a 3D Systems rapid prototyping machine. [36] The first shows the "truncated 120-cell," which consists of 120 truncated dodecahedra and 600 tetrahedra. It is derived by truncating the 600 vertices of the 120 cell. A Zome model of it is shown in Figure 9. The second is the "rectified 120-cell," which consists of 120 icosidodecahedra and 600 tetrahedra. It is derived by truncating the 120 cell further, to the edge midpoints.
 

Figure 19. Truncated 120-cell. 14 cm stereolithography model.

Figure 20. Rectified 120-cell. 10 cm stereolithography model.

4. Conclusions

As a sculptor and applied mathematician, I am interested in understanding and creating beautiful geometric forms and then presenting them to the public. See [9-13] for a range of my creations. For each object, I must choose materials that are visually and structurally appropriate. When the underlying structures are of an intricate four-dimensional nature, this offers a difficult challenge. Following on my study of previous techniques, I have come to appreciate the impressive capabilities of rapid prototyping techniques for realizing forms that would otherwise remain trapped in the imagination.

Beyond sculpture, models of 4D structures are of established educational value. In the past, paper, wire, and Zometool have allowed the construction of useful teaching models. To this list I now recommend that rapid prototyping techniques be considered for such geometric education purposes. A 120-cell model such as Figure 16 is conveniently passed around in a classroom, held in the hand, rotated, studied, and compared to a different model in the other hand. Although relatively expensive at present, the costs are decreasing. In the future, I expect, all schools will have 3D printing capability. Teachers will easily be able to produce a variety of stimulating models to be used for lessons and to stock a "cabinet of curiosities" which can excite a sense of wonder about mathematics.
 
 

References

I thank Dr. Peggy Kidwell of the Smithsonian Institution for research assistance and access to the Smithsonian's models in storage, Angela Vierling and John Sharp for providing valuable information and leads on the history of mathematical models, Imin Kao of SUNY Stony Brook for providing access to the machine on which Fig 16 was built, Carlo Sequin for funding the model of Figure 18, and Dr. Manfred Hofmann of RPC, Switzerland for the model illustrated in Figure 19.

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