Analogies from 2D to 3D 
Exercises in Disciplined Creativity

CarloH. Séquin

Computer Science Division, EECS Department
University of California, Berkeley, CA 94720


Human creativity reliesto a large part on our ability to recognize and match patterns, to transposethese patterns into different domains, and to find analogies in new domainsto known facts in old domains. In the realm of geometrical proofs and geometricalart, such analogies can carry concepts and methods from spaces that areeasy to deal with, e.g. drawings in a 2-dimensional plane, into higher dimensions where model making and visualization are much harder to carryout. Students in a graduate course on geometric modeling are challengedwith open-ended design exercises that introduce them to this analogicalreasoning and, hopefully, enhance their creative thinking abilities. Examplesinclude: constructing a Hilbert curve in 3D-space, finding an analogousconstellation to the Borromean rings with four or more loops, or developing3D shapes that capture the essence of the 2D Yin-Yang figure or of a logarithmicspiral. The proffered solutions lead to interesting discussions of fundamentalissues concerning acceptable analogies, the role of symmetry, degrees offreedom, and evaluation criteria to compare the relative merits of thedifferent proposals. In many cases, the solutions can also be developedinto attractive geometrical sculptures.


Fig. 1: a) Borromean Rings,b) Hilbert Curve, c) Yin-Yang.


Where does creativitycome from? What is "good design"? How do we know that we have a "solution"to a design problem? These are some questions behind the design exercisesthat I will discuss in this paper.

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].

Fig. 2: Kepler's attempt toexplain the ratios among planetary orbits

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.

Hilbert Curve

An exercise that datesback to 1983 [9 , 10]asks the students to develop a 3-dimensional, recursive, self-similar,space-filling, piece-wise linear path inspired by the 2D Hilbert curve[ 5] shown in Figure1b. This Peano curve, which in the limit fills the unit square, wasdiscovered in 1891. It can be nicely described by a recursive procedure.The problem statement makes it implicitly clear that we want the new curveto visit all the grid points of a cubic array with 2 n x 2 nx 2 n points so that no grid point has more than two line segmentsattached to it. The basic approach is fairly obvious: The overall cubeis split into eight equal octants which are visited in a particular sequence.These octants are split into octants again, which are then visited in thesame geometric order. All the interesting issues lie in the details!
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.


Fig. 3: a) Closed 2D Hilbertcurve, b) 3D Starting frame, c) corner element for 2nd generation curve.

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.


Fig. 4: (a), b) 2nd- and c)3rd-generation virtual 3D Hilbert pipes, with 64 and 512 pieces respectively.

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.

Symmetrical Constellations of Interlocking Loops

Two tightly intertwinedrings form a simple yet intriguing configuration that seems to have symbolicmeaning in several cultures ( Fig.5a).An attempt to place three loops in space as compactly and as symmetricallyas possible, leads to an arrangement known as the Borromean rings (Fig.1a, Fig.5b). Individual pairsof rings are not actually interlocked; the configuration only holds togetherwhen all three rings are present. However, when we try to place three perfectlytoroidal rings into a constellation of high symmetry, the result is a pairwiseinterlocking configuration with three-fold symmetry (Fig.5c).

Fig. 5: a) Two interlockingrings, b) tight Borromean configuration, c) three interlocking ring pairs.

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.


Fig. 6: a) Four symmetricallyclustered loops, b) physical realization, c) five "Borromean" rectangles.

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 .

Fig. 7: Interlocking schemes:a) Borromean rings, b) 4 rings, c) 5 rings, d) 4 "Borromean" rings

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.

Fig. 8: a) Four "Borromean"triangles, b) ten interlocking triangles, c) tangle of five tetrahedra(SLS).

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].


Yin-Yang symbolizesthe two complementary forces that comprise the Tao, the eternal dynamicway of the universe: Yin is the earthly, dark, passive, or female principle.Yang is the heavenly, light, active, or male force. Geometrically, theYin-Yang symbol divides a circle into two complementary halves that in some sense are "opposites" of one another. The task given to the students was to find a corresponding partitioning of a sphere in 3-space. The richness of the solutions proposed by the students in the Fall 1997 course CS 285, Solid Free-Form Modeling and Fabrication, was unusually rewarding ( Fig9,Fig.10).


Fig. 9: Various solutionsto constructing a 3D Yin-Yang.

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).


Fig. 10: The most pervasivesolution for a 3D Yin-Yang: A cut with a S-shaped developable surface.

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.

Fig. 11: Yin-Yang symmetry groups: a) mirroring,b) cyclic rotational, c) glide plane reflection.

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.


Fig. 12: 3D Yin-Yang modelswith mirror complements: a) cyclid-based, b,c) torus-based.

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.

Spiral Surface

The logarithmic spiral(Fig.13a ) is a fascinatingcurve, and may be considered the ultimate solution to self-similarity atarbitrary scales. As in the first problem dealing with the Hilbert curve, we might ask what an analogous curve through 3D-space might look like.One might argue that a 3D spiral curve should (eventually) pass throughall possible directions emerging from the origin of the coordinate system,and, at the same time, gradually move outwards at an exponential rate.The problem of visiting all points on a sphere with a continuous smoothpath has been addressed by Dan Asimov's "Grand Tour" [ 1],which, given a long enough time span, will approach every point on thesurface of a sphere with arbitrary closeness. All we have to do, is to let the radius grow exponentially as a function of time.
However, the task we want to focus on here, is to find a surface that captures the spiral properties.Ideally, we would like to obtain a spiral intersection curve whenever thesurface is cut with an arbitrary plane through the origin -- however, thiswould be asking for too much! But can we get spirals in at least threecutting planes that are mutually perpendicular to one another?

Fig. 13: a) Logarithmic Spiral,b, c) emerging pipe-cleaner skeleton.

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!


Fig. 14: a) Spider patternwith patch subdivision lines, b) paper model of Spiral Surface.

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.

Subsequently this surface was modeled on the computer. Using three Bézier patches to form theshape of one "L" with a 60-degree corner, it takes 18 patches to composeone complete spider (Fig.14a).The boundary constraints to guarantee smooth continuation of the patchesare not hard to derive; and the inner control points of the cubic patchesare adjusted to minimize any apparent bumps. Finally, Jane Yen added highlightsand shadows and rendered that surface (Fig.15a,b )with the Blue-Moon Rendering Tools [ 4].The next step is to make the surface thicker and to derive a solid descriptionfrom which a 3D model can then be built with one of the layered Solid Free-Form(SFF) fabrication techniques [ 7].


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

Discussion and Conclusions

The design exercisesdiscussed in this paper are a good example of an activity that bridgesthe realms of logical reasoning on the one hand, and of intuitive and evenartistic contemplation on the other. The region between art and mathematics is particularly suitable to study the creative process on (somewhat) open-ended design problems. The problem statements are loose enough to allow the mind to roam free and to come up with potentially wild and unorthodox solutions. At the same time, these geometrical puzzles possess enough structure andquantifiable properties so that one can apply an acceptable metric to theresults and rank-order different solutions. This process brings wild, far-flung,non-sequitur creativity into a more disciplined mode where there are designsolutions of defensible quality.
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.


Some aspects of thiswork were partially supported by the National Science Foundation undera research grant to study the designer/fabricator interface for rapid prototypingof mechanical parts. The help of Jane Yen and Jordan Smith with the renderingof some of the virtual sculptures is also gratefully acknowledged.


  1. D. Asimov, "The GrandTour: A Tool For Viewing Multidimensional Data." SIAM Journal of Science& Stat. Comp., Vol. 6, pp. 128-143, (1985).
  2. W. Boehm, "On Cyclidesin Geometric Modeling." CAGD, Vol 7, pp. 243-255, North Holland, (1990).
  3. E. N. Gilbert, "GrayCodes and the Paths on the N-Cube." Bell System Tech. Jour. 37, pp. 815-826,(1958).
  4. L. Gritz, "Blue MoonRendering Tools Home Page,"
  5. D. Hilbert and S.Cohn-Vossen, "Geometry and the Imagination." Chelsea, New York, 1952.
  6. A. Holden, "OrderlyTangles." Columbia University Press, New York 1983.
  7. D. Kochan, "SolidFreeform Manufacturing: Advanced Rapid Prototyping." Manufacturing Researchand Technology 19, Elsevier, Amsterdam, New York, (1993).
  8. C. Mead and L. Conway,"Introduction to VLSI Systems." Section 1.15, Addison Wesley, Reading MA,(1980).
  9. C.H. Séquin."Creative Geometric Modeling with UniGrafix." Technical Report UCB/CSD83/162, U.C. Berkeley, CA (Dec. 1983).
  10. C.H. Séquin."More... Creative Geometric Modeling." Technical Report UCB/CSD 86/278,U.C. Berkeley, CA (Dec. 1985).

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