The human mind is good at discoveringpatterns. One might argue that human intelligence and creativity reliesto a large extent on our ability to recognize patterns and match them withpreviously stored patterns. We easily see animal shapes and faces in clouds,or goblins and ghosts in tree trunks at night in the forest. We are intriguedby star constellations if they approximate regular triangles or quadrilaterals,or if they lie on a roughly circular arc. We also recognize and enjoy theregularity in tiling patterns. Often we try to explain the patterns inone domain with patterns from another domain. For example, Kepler tried to explain the relative sizes of the planetary orbits with the suitablynested circumspheres of the Platonic solids ( Fig.2), and there were attempts to draw the periodic table onto simple geometricalobjects such as cylinders. Often we use analogies to try to explain a newand unfamiliar domain with a model from a well-understood and intuitively plausible domain; as an example, the model of water flowing over a damof adjustable height has been used to explain the operation of an MOS fieldeffect transistor [8].

I believe that one's creativityand imagination can be improved by training one's ability to find suchanalogies. This is why I include such exercises in almost every courseI teach at U.C. Berkeley. I try to improve the students' design skills by raising to a conscious level sequences of thoughts and associationsthat appear in the search for a solution. We ponder questions such as:What is going on in the design process? Where do good ideas come from?What can we do to enhance the flow of good ideas? When do we know thatthe design is done?

In my graduate courses on geometricalmodeling and computer-aided design, these kinds of questions and the correspondingexercises that I discuss in this paper often involve an inductive stepgoing from a 2-dimensional form to a related 3-dimensional shape, or froma configuration of n elements in a highly symmetrical arrangement to aconstellation of n+1 and more such elements. Each problem was first solved to my own satisfaction to gain an idea of its possibilities and difficulty. I then clarified the design task and curtailed the solution space so that a structured exercise resulted. Typically, these tasks are attacked bythe students in two waves: First they are simply given the short, open-endedproblem statement and are asked to think about it and bring their ideasand questions to the next class. In a joint discussion we then agree onthe salient features that the solution should have and identify some criteriaby which we could judge the quality of the various designs. In spite ofthe stated constraints and of the focussing effect of our discussions,I normally have the pleasure to obtain new and unexpected solutions thatadd to the richness of the problem and transcend the previously known solutions.Often I find out later that others had pondered the same questions, andsometimes even wrote a whole book about related issues -- as in the caseof the delightful book "Orderly Tangles" by Alan Holden [6].Since the exercises often lead to artistically interesting and pleasingresults, they should be particularly well suited for this conference, bridgingthe gap between logic and the arts.

After the students have ponderedthe question for a while, we start discussing desirable criteria that onemight use to rank-order different designs. For instance, the higher-order Hilbert curves should maintain the symmetry of the starting frame, andthat symmetry should be as high as possible. The 3D solution may well havemore symmetry than the 2D curve. To achieve that goal, we might want tomodify the top level of the recursion in the original Hilbert curve sothat it forms a closed loop, thus introducing a second axis of mirror symmetryand leading to a prominent "H" in the second generation ( Fig.3a).Other aesthetic considerations may suggest that the extra-long line segmentof three unit lengths occurring near the center of the 3rd-order planarcurve should be avoided. Ideally one might want to avoid even two subsequent collinear line segments. Similarly, one might want to minimize the number of subsequent elements that lie in the same plane. While three subsequent coplanar segments cannot be avoided, if we want to visit all eight corners of a unit cube, sequences of more than three segments may be avoidable.Ideally we would like to find a simple recursive formulation for such astructure.

A plausible starting frame consistsof the closed loop shown in Figure3b. To use this frame as the basic corner element at the next level, one of its eight segments, i.e., the dark one in the lower part of Figure3b, must be opened up, and two new connections to adjacent, identical corner elements must be created by suitably twisting the L-elements thatwere formerly attached to it (Fig.3c ),we place eight such modules, reduced to half-size, into the eight octants of the original cube. They have to be properly oriented, and some unitshave to be mirrored (shown darker in Fig.4a) so that they can be connected readily with their neighbors. This should be done in such a way that a closed path results that basicallyfollows the path of the original frame ( Fig.3b).I have built a physical implementation of such a second generation structurefrom 64 plastic 3/4-inch pipe corner pieces. The problem with constructing larger physical structures lies in the fact that, regardless of the level of recursion, two halves of the sculpture are connected only with eithertwo or four pipe segments, which renders the physical structures ratherweak. On the other hand, in the virtual space of computer modeling, theprocess can be continued without limits using ever smaller copies of theoriginal module. The basic design can then be turned into an impressivevirtual sculpture with a suitable choice of pipe dimensions, texture, andcoloring ( Fig.4c).

For Hilbert curves of orderthree and higher, some interesting choices have to be made. I was ableto design a 512-segment 3D pipe in which there are never more than three coplanar line segments. However, I had to start with a different cornerelement for the 2nd-generation curve. Rather than removing the dark segmentin the base frame (Fig.3b),I chose to remove one of the elements adjacent to it (of which there arefour to start with) and to orient the connecting L-pieces so that theyturn away from the plane last visited inside the base-frame. These cornerelements now show C2 symmetry around the mid-point of their middle segment,and they can now be properly oriented in all eight corner positions toform a 2nd-generation Hilbert path (Fig.4b).To produce the corner element for the 3rd generation, we again break opena segment near one of the corners and suitably twist the adjacent L-elementsoutwards. This unit can then be assembled into a symmetrical closed loopwith carefully chosen orientations and mirroring operations. The drawbackof this solution is that the connection operation needs to modify one ofthe lowest level corner elements, thus a simple recursive composition ofeight identical corner elements is not possible.

The solution presented in Figures4a,c, has the same basic symmetry, but it exhibits sequences of foursubsequent coplanar pipe segments; on the other hand, I was able to describeit with a nice recursive formulation. If a closed curve is desired, thenone has to change the orientations of the first and last corner elementsat the top level of the recursive procedure. The approach generalizes tohigher dimensional cubes [ 3].The initial starting frame can always be seen as an n-bit reflected Graycode which runs through all permutations of an n-bit string in such a waythat each string differs from its predecessor in only a single bit. Scaled-down versions of this traversal of the starting frame are then placed -- withsuitable orientation -- into each "corner" of the original hyper-cube.

Next, we aim to cluster morethan three rings around the origin, without much concern whether individualpairs of rings mutually interlock. The task given to the students was tofind a constellation of four loops with the highest possible symmetry. Every loop should be in an identical position within the constellation,so that the basic symmetry operations transpose any one loop into any otherloop.

If one has not seen the solutionbeforehand, this problem turns out to be surprisingly difficult. Two approacheshave proven helpful to guide the students to finding a solution. The firstone is to ask what symmetry groups one might possibly expect. After somecontemplation, one finds that with four rings it has to be the tetrahedral or the octahedral group. The second approach starts with the notion thatone might want to interlock four triangles. The choice of triangles torepresent the loops seems natural, because each loop has to interact withthree other loops, and if we want to do this in a symmetrical manner, weshould choose a loop with 3-fold symmetry. Given that we want to placefour triangles symmetrically in 3D space, we need to define the positionsfor twelve vertices -- in a symmetrical manner. So the question then turnsto how one can place twelve vertices uniformly and symmetrically onto thesurface of a sphere. The insight to this secondary problem might come fromthinking about the densest sphere packing, or from contemplating the Platonicand Archimedean solids and looking for the occurrence of the number 12-- preferably in an object with tetrahedral or octahedral symmetry. WhenI initially contemplated this problem, I first thought of the twelve edgeson a cube. So I placed the vertices at the midpoints of these edges andconnected them into 4 triangles -- and the solution emerged almost immediately(Fig.6a).

I implemented this configurationas a physical sculpture from 4-inch cardboard tubes [ 9],spray-painted with copper enamel on the outside, and with a touch of fluorescentyellow near its center-- which made the sculpture glow on the inside whenhit with indirect sunlight ( Fig.6b).It should be pointed out, that in this arrangement every pair of trianglesmutually interlocks; cutting away one triangle would still leave the otherthree entangled. Also, the topology of Figure6b is the mirror image of that of Figure6a.

I continued my quest by lookingfor constellations of five and more intertwined loops. For five loops,an extension of the process that had helped me find the 4-triangle structurewas employed -- i.e., I tried an inductive approach. Each of the five loopswould have to interact with four others, thus using squares seemed likea reasonable start. This then required twenty vertices positioned symmetrically in space. The twenty vertices of the pentagon-dodecahedron offered themselves conveniently, and it did not take long to find a grouping of the vertices into four planar polygons -- which however were rectangular rather thansquare (Fig. 6c). Alsothe resulting constellation does not carry the full symmetry of the Platonicsolid from which it was derived, it just has one axis of 5-fold symmetry(C5) and five axes of C2 symmetry.

It is interesting to study theinterlocking pattern of this structure. No two rectangles interlock! Itlooks like a more complicated Borromean arrangement in which each loopsurrounds exactly two other ones in a cyclical relationship. Encouragedwith this result, we might try again to look for a Borromean arrangementwith four loops. However, a little conceptual reasoning will soon let ussee the difficulty of this quest. Let's use the notation A ---> B to indicatethat loop A encircles loop B on the outside. Thus the Borromean rings havethe cyclic relationship indicated in Figure7a. If we try to draw a similar diagram for the 4-ring constellation,then we run into the difficulty that in a complete graph with four vertices,there are three edges joining at each vertex; thus the number of incomingand outgoing arrows cannot be made the same everywhere. To draw a symmetricaldiagram in which all vertices are identical, we would have to use double-headedarrows, which we can interpret as an indication that the two rings (vertices)connected by such a double arrow are mutually interlocking ( Fig.7b).On the other hand, the complete graph with five vertices has four edgesjoining at every vertex, and we can readily draw such a graph with twoincoming and two outgoing arrows at each node ( Fig.7c).This corresponds to the configuration of five loops discussed above andshown in Figure 6c .

Before I had a chance to pushmy quest much beyond the constellation with five rings, I stumbled ontothe delightful book "Orderly Tangles" by Alan Holden [ 6].This is an invaluable resource containing dozens of such symmetrical, interlockingconstellations with as many as twenty loops. Figure8b shows a computer simulation of a tangle with ten triangles inspired by a model built by Holden. In this case, the vertices of the triangleslie on the midpoints of the thirty edges of the dodecahedron. The inductivereasoning outlined above for the steps from three to four and then to fiveloops can be continued. Once one understands the search procedure, it isnot too difficult to find the more complicated tangles.

Holden's book also containsa Borromean configuration formed by four triangles; its linking logic correspondsto Figure 7d , and it isrealized by a 3-ring Borromean configuration that rigidly holds in placea fourth, non-entangled triangle ( Fig.8a).Holden also shows how such symmetrical tangles can be carried beyond justsimple loops; he shows models of interlocking tubular tetrahedral frames.Another realization of the classical tangle of five tetrahedra with all20 vertices lying at the corners of a dodecahedron is shown in Figure8c. This part has been constructed with Selective Laser Sintering (SLS),one of the emerging layered Solid Free-Form (SFF) fabrication technologies [7].

The most often proposed solutionwas a sphere cut into two halves with a "band-saw" following the path ofthe 2D Yin-Yang dividing line. Some solutions were offered as clay models(Fig.10a), others as machinedparts (Fig.10b ), or assophisticated computer renderings ( Fig.10c).While I feel that this is not the best solution, since it is just an extrudedextension of the 2D figure, this is also a shape celebrated by great artistssuch as Max Bill ( Fig.10d).

Still, I was more intriguedwith attempts to cut the sphere with a surface that curves in both directions.A key question then arises: should the two halves be identical or mirrorimages? The most innovative proposal came from a couple of students whoreasoned, that the true analogy of going from 2D to 3D demanded that the sphere be cut into three identical parts -- possibly colored with the three primary colors red, green, blue.

The crucial characteristics to order and classify this plethora of shapes is the symmetry of the surface that divides the sphere. The following fundamental possibilities exist:The trivial solution cuts the sphere with a plane through its center; butthis does not exhibit any features of the Yin-Yang icon. A more interestingclass of dividing surfaces has C2 symmetry with respect to some axis throughthe sphere center; this results in two congruent halves ( Fig.9a,b).The "band-saw-cut" solutions ( Fig.10)also fall into this class. A generalization allows C3 symmetry around thisaxis and would thus cover the case of three identical subcomponents. Thethird and, to me, most interesting class, has glide symmetry, which brings the dividing surface back onto itself when it is rotated 180 degrees andthen mirrored on a plane perpendicular to that axis; this leads to complementary mirror-parts (Fig.9c).This shape is most defensible on philosophical grounds; we want to createtwo halves that are not identical but rather complements of one another.The most beautiful formulation of such a shape ( Fig.12a)is composed of three spherical surface pieces and two cyclides [2].This shape was also discovered by C. E. Peck in 1992.

Attempts at machining such ashape on a milling machine run into the problem, that the part shows aconcave groove that leads into a point with infinite curvature, which cannot be cut with any tool of finite dimension. The 2D Yin-Yang has constantcurvature and uses only circles of two radii that differ by a factor oftwo. I found a corresponding solution in 3D that replaces the two cyclidsurfaces with two tori in which the major radius R is twice the size ofits minor radius r. This shape can readily be described as a Boolean expressionof its five curved shapes and a few half-planes. The resulting shape isshown in Figure 12b, andan early attempt at machining it on a milling machine in Figure12c.

This problem has not been presentedto any student yet. The way I approached it myself was to make a wire skeletonfrom pipe cleaners ( Fig.13b)that contained the three coordinate axes (black) and three spirals in thethree main coordinate planes (white). Additional pipe cleaners (grey) producedsome connectivity among these spirals and formed partially spherical shells(Fig.13c). It became clearthat it was not possible to connect all spiral branches smoothly with oneanother; some jumps from one "onion shell" to another had to occur. Thus the surface cannot be totally closed. It turns out that this is a usefulfeature, since it would be rather dull, if we could not view the internalstructure of this surface. Later it became apparent that the edges thathad to be introduced to avoid the discontinuous jumps from one shell tothe next inner/outer one could themselves take on the shape of spirals.What serendipity!

One emerging solution exhibitedD3-symmetry along the {111} space diagonal in the coordinate system inwhich I had placed the original three spirals; and it showed six openings with helical edges leading inwards toward this axis. At this stage, I started to build paper models to get a better feeling for the surface topologyitself. I designed the spider pattern shown in Figure14a and made several suitably scaled copies. These were joined togetherin a nested manner, where the long arms of one spider join the short armsof the next larger spider. The result is shown in Figure14b.

 
Fig.15: a) Virtual model of Spiral Surface, b) half that surface showing spiralcurve in cross section.

The presented problems startwith the construction of a simple one-manifold, the 3D Hilbert curve, whereconnectivity, symmetry, and a recursive formulation are the dominant concerns.The subsequent tasks increase in complexity, adding topological considerationsof linking behavior, and evolving from one-manifolds (curves) to two-manifolds(surfaces).

The methods employed to tacklethese problems vary, but a dominant role is played by inductive reasoningand a judicious use of symmetry. The key challenge of the 3D Hilbert curveis to find a recursive formulation to build a generic corner element withthe desired connection properties at the corners so that the connectivity among the eight octant cuboids stays the same in each generation. In searching for orderly tangles of interlocking loops, it was most productive to determine the expected symmetry group, and then place a suitable number of vertices evenly onto the surface of a sphere so as to span the polygonal implementations of the loops. For the two surface-related problems, paper and pencil, oreven computer drawings did not seem adequate to explore the potentiallyvery large solution space. The use of clay, wire-mesh, and/or pipe-cleanersseemed to help a lot in the visualization of the problem and its possiblesolutions.

What all the solutions havein common is that the "best" solutions in terms of maximum analogy with the starting shape also have a high aesthetic appeal by themselves -- which adds a special bonus to these exercise tasks. This bonus becomes even greater with the advent of solid free-form (SFF) manufacturing technologies thatallow to turn these virtual design artifacts into nice physical sculptures.