minimal surfaces, crystals, shortest networks, and undergraduate research

Minimal Surfaces, Crystals, Shortest Networks, and Undergraduate Research

Frank Morgan

S e c t i o n 1. M i n i m a l S u r f a c e s

What is the least-area surface on a tetrahedral frame? (See Fig. I.) Because soap film tends to minimize surface energy or area, a good candidate may be obtained by dipping a tetrahedral wire frame in soap solution. Six triangular pieces of film stretch from the six edges to the singular point at the center. They meet in threes along four seams running from the four vertices of the tetrahedron to the center. This surface is called the cone

over the tetrahedral frame. Only fairly recently have mathematicians vindicated the soap film's behavior:

T H E O R E M 1.1. (Taylor [19, W.6] , Lawlor, Morgan [10, 1.1]). The surface of the cone over the regular tetrahedron in R 3 (R ") is area-minimizing.

Admissible surfaces are assumed to have the given boundary and separate the four faces of the tetrahedron. Taylor's original proof was only by the process of elimination. More recent work of Lawlor and me has produced the following simple direct proof, which generalizes to R n (the hypercone over the regular simplex in R" is area-minimizing). So let us start right off with a proof and enjoy greater leisure thereafter.

Figure 1. This soap film on a tetrahedral frame is area- minimizing. Admissible surfaces are assumed to have the given boundary and separate the four faces of the tetrahedron. Photograph by F. Goro.

THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 3 �9 1992 Springer-Verlag New York 37

Figure 2. The proof uses the four equidistant points Pi at the centers of the faces of the tetrahedron. Note that Pl - Pj is normal to the triangular piece of cone surface it pierces. The proof considers the flux of the constant vector fields Pi and Pi - Pj.

Proof: Let Pi be the point at the center of face F i. See Figure 2. By scaling, we may assume the Pi are unit distance apart. The vector Pi - P / f r o m Pi to pj is the unit normal to the triangular piece of the cone surface which it pierces.

Now consider any competing surface S dividing space into regions labeled R i according to which face F i they contain. Denote by Sq the part of S separating R~ from Rj. Now we will consider the constant vector fields Pi and argue that

area S = ~ area Sij

~ (flux (pi - pj) through Sij) ij

= ~ (flux pi through Fi). i

The inequality follows from the fact that Pi - Pj is a uni t vector field. Equality holds only if the vector field is normal to the surface, as it is for the triangular pieces of the cone. To obtain the last equality, focus on the occurrences of Pi in the second-last expression, which contribute the flux of the constant vector field Pi out of R i through all the bounding surfaces Sq. Because it is a constant vector field, this contribution equals the flux of Pi into R i through the face F i, yielding the last equality.

Note that the final expression does not mention the surface S; it is just a constant. Because the area of any admissible surface S is greater than this constant, with equality for the cone surface, the cone surface is area- minimizing.

3 8 THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 3, 1992

Figure 3. It is an open question whether there is a smaller surface of higher topological type which does not separate space into four regions. (Computer graphics by Jean Taylor, Rutgers University and The Geometry Center, after an idea by R. Hardt.)

Remarks. Note how the proof makes essential use of the four (pairwise) equidistant points Pi at the vertices of a regular tetrahedron dual to the given tetrahedron. The similar proof that the cone hypersurface over the regular simplex in R n is area-minimizing uses n + 1 points Pi at the vertices of a dual regular simplex.

This write-up deliberately suppresses concern about -+ signs; rest assured, they work out OK.

O p e n Quest ion . Theorem 1.1 admits only surfaces that separate the four faces of the tetrahedron. It is an open question whether there might be a smaller surface of higher topological type that does not separate the faces. Figure 3 imagines such a surface with tun- nels connecting the faces in pairs, front to back and right to bottom.

Actually, for general theoretical reasons any other soap film (without bubbles) on the whole tetrahedral frame would have to have less area than the cone surface. Therefore, this open question is one that any child might solve--by dipping a tetrahedral frame into soap solution and discovering another surface.

S e c t i o n 2. Crystals and N o r m s on R"

Crystals. Crystals try to minimize a surface energy which depends on the direction of the unit normal n with respect to the underlying crystal lattice. For many crystals, certain directions are very cheap, and these are the only directions occurring in the crystal surface: The crystal is a polyhedron. For example, if you look closely, you can tell that a salt crystal is a tiny cube. See

It follows immediately that

x . y ~ ~(x) cI)*(y). (1)

If equali ty holds, one says that x and y are dual. If �9 is just Euclidean length, then cI)* is also jus t

E u c l i d e a n l eng th . The i n e q u a l i t y is the f a m o u s Schwarz inequality that the dot product of two vectors is less than or equal to the p roduc t of their lengths, with equali ty if the two vectors are parallel. Thus, dual is just a generalization to other norms of parallel.

Theorem 1 on area-minimizing surfaces generalizes to cI)-minimizing surfaces, whe re (I) is a norm on, say, R 3 and the energy ~(S) of a surface S is given by

r = fs r

with n the unit normal to the surface.

Figure 4. Many crystals are polyhedralmhave finitely many faces---because the surface energy for those orientations happens to be very cheap. (Photographs of S. Smale [20] and E. Breiskorn, from Mathematics and Optimal Form by S. Hildebrandt and A. Tromba [9].)

Figure 4. For other materials, such as solid helium, the energy is nearly isotropic and the crystal is smooth and almost round. In the borderl ine case, as for pivalic acid, the crystal is smooth but distinctly non round (see Fig. 5). Compare also [18].

Norms. Consider a general surface energy given by a norm cI)(n) on vectors n in R 3 or R n. A norm c~: R ~ -~ R is homogeneous [cI)()~x) = )~cI)(x)] and convex [(I)(x + y) ~< ~(x) + (I)(y)]. A norm is characterized by its unit ball, the set of all vectors of norm at mos t 1. For the unit norm ball of Figure 6, the vertical direction is cheap because you can go far with unit cost. The hor- izontal direction is expensive. The convexity of a norm is equivalent to the convexi ty of its uni t ball. The s m o o t h n e s s of a n o r m (always ignor ing the nonsmooth point 0) is equivalent to the smoothness of its unit ball.

Norms come in pairs. Associated to a n y n o r m �9 is a dual norm ~* def ined by

(I)*(y) = sup{ x �9 y : cI)(x) ~< 1 }.

Figure 5. A pivalic acid crystal is smooth but not round, a kind of interpolation between round and diamond shapes. (Photograph from Glicksman and Singh [7].)

Figure 6. A norm ~ is characterized by its unit ball, the set of all vectors of norm at most 1. For this unit ball, the vertical direction is cheap because you can go far with unit cost.

THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 3, 1992 39

THEOREM 2.1. Sample theorem on ~-Minimizing Surfaces. Given a norm �9 on R 3, if there exist four ~-equidistant points ("an equilateral tetrahedron"), then the cone over the dual tetrahedron is ~*-minimizing.

Remark. As before we admit only surfaces which separate the regions containing the four faces of the tetrahedron. The proof is a minor variation on the proof of theorem 1.1 (see Fig. 7).

Proof: We may assume the Pi are ~-unit distance apart. Consider a "dual" tetrahedron such that a triangular piece of the cone surface has normal dual to Pi - Pj.

Now consider any competing surface S dividing space into regions labeled Ri according to which face Fi they contain. Denote by Sly the part of S separating R i from Rj. Now we will consider the constant vector fields Pi and argue that

�9 *(s) = 0

=

= f s

X fs,,(Pi-Pl)'n = ~ (flUX (pi -- pj) through Sij)

/j

= ~ (flux pi through Fi). i

The first two equalities are immediate from the defini- tions. The third equality follows from the fact that Pi - pj is a D-unit vector field. The inequality is the basic norm inequality (1) earlier in this section. Equality holds only if the vector field is dual to the normal to the surface, as it is for the triangular pieces of the cone. The second-last equality is the definition of flux. To obtain the last equality, focus on the occurrences of Pi in the second-last expression, which contribute the flux of the constant vector field Pi out of Ri through all the bounding surfaces Sq. Because it is a constant vector field, this contribution equals the flux of Pi into R; through the face F i, yielding the last equality.

Note that the final expression does not mention the surface S; it is just a constant. Because the D-energy of any admissible surface S is greater than this constant, with equality for the cone surface, the cone surface is ~-minimizing.

A natural question about Theorem 2.1 asks how many norms �9 satisfy the hypothesis that four cI)-equidistant points exist. The following theorem due to C. M. Petty says that all norms do.

THEOREM 2.2. [16, Theorem 4]. For any norm �9 on R 3, there are four C~-equidistant points.

Interestingly enough, the result breaks down in higher dimensions:

Open Question. Are there n + 1 equidistant points for any smooth norm on R"?

The question is open even for R 4. There are some more surprises:

Figure 7. For this unit @-ball, these five points are equidistant. Consequently, the cone over the dual triangular prism is ~*-minimizing---a new kind of singularity.

Proposition [10, 3.4]. For some smooth norm c~ on R 3, there are five equidistant points. Consequently, the cone over a dual triangular prism is ~*-minimizing.

The five points and the unit D-ball can be as in Figure 7. At the vertex of the minimizing cone, nine triangular surfaces and six curves meet at a point. This is a new kind of singularity that does not occur in soap films. It seems to have been recognized as a minimizer in mathematics before being discovered in nature!

Open Questions. Can there be six equidistant points for a smooth norm on R3? How many in Rn?

For the nonsmooth norm on R 3 with cubical unit ball, the eight vertices of the cube are equidistant. In general, for arbitrary norms, 2 n gives a sharp bound on the number of equidistant points in R" [16, Theo- rem 4].

Sect ion 3. M i n i m a l N e t w o r k s

Warm-up Question. What is the shortest network connecting the four corners of a square?


/

%

I %, %%

I J

t"

f f

\ /

p %

, / ~ ' / [

Figure 8. The shortest network connecting the four corners of a unit square is the double Y. In general, length- minimizing networks meet in threes.

One possibility is the U, of length 3 (see Fig. 8). A very good solution has four segments meeting at the center in an X, of length 2V2 ~ 2.82. The best solution of all is a double Y, of length V~ + 1 ~ 2.73. Indeed, it is a classical fact that at interior junctions

length-minimizing networks meet only in threes

(cf. [6, pp. 354-361, 3921). The interstate highway system probably should look more like Figure 9. A recent article in The New York Times described a new theorem on the value of using such triple junctions (see Fig. 10).

This principle can be well illustrated by soap films. Etch a map of the United States on parallel plexiglass plates, joined by plastic pins at the locations of cities. Dip it in soap solution. The soap film, in its attempt to

Solution to Old Puzzle: How Short a Shortcut?

By GINA KOLATA

T WO mathematicians have solved an old problem in the design of networks thai has enormous practical impor-

tance but has baffled some of the sharpest minds in the business.

Dr. Frank Hwang of A.T.&T. Bell Laboratories in Murray q i l l N.J., and Dr. Ding Xhu Du, a pc._,doctoral student at Princeton University, an- nounced at a meeting of theoretical computer scientists .last week that they had found a precise limit to the design of paths connecting three or more points.

Designers of things like computer circuits, long-distance telephone lines, or mail reutings place great stake in finding the shortest path to connect a set of points, like the cities an airl ine will t ravel to or the switch- ing stations for long-distance telephone lines. Dr. Hwang and Dr. Du proved, without using any calcula- tions, that an old trick of adding extra points to a network to make the path shorter can reduce the length of the path by no more than 13.4 percent.

"This problem has been open for22 years," said Dr. Ronald L Graham, a research director at A.T.&T. Bell Laboratories who spent years trying in vain to solve iL ':The problem is of t remendous interest at Bell Labora- tories, for obvious reasons."

In 1968. two other mathematicians guessed the proper answer, but until now no one had proved or disproved Illeir conjecture.

In the tradition of Dr. Paul Erdos, an eccentric /-Iunganan mathematician who offers prizes for solutions to hard problems that interest him, Dr. Graham offered $500 for the solution to th|s one. He said he is about to mail his check.

The Shorter Route

Thelength of string needed to join the three comers of a triangle is shorter if a point is added in the middle, as in the figure at right.

Figure 10. This article from The New York Times, October 30, 1990, described a new theorem of Du and Hwang on the value of triple junctions. Copyright �9 1990 by The N e w York Times Company. Reprinted by permission.

"I

Figure 9. The shortest highway system connecting these cities meets only in threes (From The Shortest-Network Prob- lem, Marshall W. Bern and Ronald L. Graham. Copyright �9 1990 by Scientific American, Inc. All rights reserved. [3]).

Figure 11. A soap film between plexiglass plates exhibits a shortest network, with roads meeting in threes. Photograph from Schecter [17]. (Courtesy of Gordon Graham/Prism.)

THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 3, 1992 41

minimize length or energy, connects the cities by a network of roads meeting in threes (see Fig. 11).

Now replace length by a norm ~(x) and try to minimize the ~-length of a curve C

�9 (C) = f c $ ( T ) ,

where T is the unit tangent vector. We will assume that is uniformly convex, that is, that the boundary of the

unit ~-ball has positive inward curvature, bounded away from 0. This assumption guarantees that a straight line is the unique cheapest path between two points, so that ~-minimizing networks consist of straight line segments, in general, meeting at auxiliary nodes. The following theorem says they still meet in threes, at least for smooth norms in the plane.

THEOREM 3.1. (Adam Levy, Williams '88 undergraduate thesis [11], [2]). If ~ is smooth and uniformly convex, ~-minimizing networks in R 2 meet only in threes (at interior nodes).

His proof showed that junctions of four or more segments are unstable. Only recently has Levy's theorem been generalized to higher dimensions. For length, the bound of 3 holds in all dimensions, but for the class of differentiable norms cI), n + 1 is the sharp bound on the number of segments meeting at a node in a cI)-minimizing network in R" [10, Theorem 4.4].

The summer after his thesis, Levy led a group in the Williams SMALL undergraduate research project to ex- amine the smoothness hypothesis in his theorem. They came up with the following surprise.

THEOREM 3.2. (SMALL Geometry Group 1988: Al- faro, Conger, Hodges, Kochar, Kuklinski, Levy, Mah- mood, and yon Haam [1,2].) For some piecewise smooth, uniformly convex norms �9 in R 2, rb-minimizing networks sometimes meet in fours (but never in fives or more).

Indeed, they showed that an X is minimizing for the unit norm ball of Figure 12. Note that the ball is uni-

8 t-t.

Figure 12. The X is ~-minimizing for this unit ball, which is uniformly convex, and smooth except at the four comers, which stick out to make the diagonal directions, which occur in the X, cheap. Thus, for some uniformly convex norms, minimizing planar networks can meet in fours.

formly convex, and smooth except at the four comers, which stick out to make the diagonal directions, which occur in the X, cheap. The proof uses some elegant symmetry arguments to reduce the analysis to a calcu- lus problem.

These results have just appeared in an article [2] in the Pacific Journal of Mathematics. The authors are dis- tinguished both by being undergraduates and by num- bering eight! Another interesting feature of the article is that of the 15 references, 6 are earlier work by undergraduates.

E. J. Cockayne [4] had earlier considered uniformly convex norms (I), but did not discuss the dependence on the smoothness of ~. M. Hanan [8] had observed that for the nonuniformly convex "rectilinear" norm (with a square unit ball), minimizing networks in R 2 can meet in fours. (With a cubical unit ball in R", 2" segments can meet at a point. It is an open question whether there ever could be more.)

The following year, one member of the group, Mark Conger '89, went on in his thesis [5] to generalize The- orem 3.2 to six segments meeting along the axes in R 3. His conjecture that the sharp bound for uniformly convex norms in R" is 2n remains an open question.

I must say that working with these students has helped me in my own work on minimizing surfaces in general dimension and codimension. They do read my manuscripts, check details, and edit, but their main contributions have gone beyond that. The principles and heuristics they develop working on minimizing networks have given me new ideas and corrected my prejudices on the higher-dimensional problems.

A survey of some of these results appears in [13, Chapter 10].

S ec t io n 4. Undergraduate Research

In ever-increasing numbers, undergraduates are doing mathematics, proving theorems, writing papers for publication, and giving talks at mathematics meetings. An active special session at the annual mathematics meetings in San Francisco, January 1991, was dedi- cated to undergraduate research. In the first two years of our SMALL undergraduate summer research project at Williams, the number of majors per year rose from 11 to 28. We think this increase is partly due to SMALL.

The SMALL project involves about 20 students, in- cluding a few from outside Williams, awarded about $2500 for 10 summer weeks' work in mathematics and computer science. Each s tudent belongs to two of seven research groups, each with a student leader and a faculty advisor.

The student groups work largely independently, although the faculty advisor usually provides the problem and some guidance. The students generally are


quite busy and do not like to be interrupted by the faculty! The less-experienced students learn from the others, and the most advanced like to bounce ideas off the rest.

The students range from seniors to freshman, in- cluding men and women, blacks and Hispanics, and sometimes students not even intending to major in mathematics or computer science.

One woman, who had no intentions of attending graduate school, changed her mind during the summer project and is now a graduate student in computer science at Wisconsin. A nonmajor, who used to consider mathematics just an avocation, is starting graduate school in mathematics at UC Berkeley this fall. A Hispanic student, who never seemed to settle down in his coursework, succeeded ou ts tand ing ly in the SMALL project his freshman year and has written an honors thesis in mathematics.

Although focused on 10 summer weeks, SMALL ac- tivities continue throughout the school year. At one of the first colloquia of the fall, SMALL students describe the project to faculty, friends, and potential new SMALL participants. SMALL alumni often participate in colloquia, pursue independent work, or write theses. This year, s tudent colloquium audiences num-

bered 50. Likewise, 50 Williams students applied for SMALL, in addition to the 40 applicants from outside Williams.

The demands of the summer project on the faculty advisors are alleviated by the large number of faculty involved (six or seven) and the major role played by the s tudent group leaders. These s tudent leaders, sometimes seniors who have already written honors theses, are the ones who really run the groups. As a result, faculty already immersed in research need only spend about an additional 5 hours a week on the project and can be away for up to 2 weeks at a time. But faculty participation does assume a summer-long com- mitment to research.

Faculty have received modest stipends of about $2500 for the 10 weeks, occasionally supplemented for major administrative tasks.

The SMALL project is now an official NSF Research Experiences for Undergraduates (REU) site, with additional funding from the Ford Foundation, NECUSE (New England Consortium for Undergraduate Science Education), NSF grants, G.T.E., Shell, and Williams College.

The name SMALL, in case you are wondering, is an acronym for the faculty on the original proposal.

|[Z.4 hW P.O. Bmcao

i-za-~a, lk'~-"~

:I'm arTct 2~,~ ,~ old.

c, ab~. c.~l;,raer, , f f ' , , '~m ~-~1[ i :~ r , , ; d .

9~mcw',~ "{.

la-o'l" l~glzr ?

i,',r or~ bobble.~ ~r ~n~

. . . . ( a ~ q , .~ ' -~ .d- ~ ~ ~ , , ~ t . ~

- i / ....

"Ch nlk.=- 0u

Figure 13. Cindy Cero began her mathematical correspondence at age 12.

THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 3, 1992 4 3

In addition to REU sites, the NSF is now supporting summer geometry institutes involving researchers, graduate students, undergraduates, and high school teachers and students. Write to the Mathematical Sci- ences Division at the NSF for information.

It probably is good to start research early on. I got one letter that began "I 'm Cindy Cero and I'm 12 years old." Cindy described her experiments with soap films that led her to ask the fundamental questions about structure and singularities: "Can you help me under- stand why they formed what they did in the middle" (see Fig. 13). By the end, she sounded exactly like any other mathematician: "I 'm having a hard time finding information on bubbles so any other resources you could give would be very helpful." Soon she surpassed us all: "P.S. Enclosed is a se l f -addressed enve- l o p e . . . ;" but then relapsed into the common clos- ing: "Need information before March."

I got feeling sorry that it was not until graduate school that my advisor, Fred Almgren, got me inter- ested in minimal surfaces . . . until an old document reminded me that I, myself, started research at an early age, with an exceptional teacher. See Figure 14.

A c k n o w l e d g m e n t s

This article is based on an AMS-MAA address [15]. The work was partially supported by the National Sci- ence Foundat ion and the Institute for Advanced Study.

References

1. Manuel Alfaro, Mark Conger, Kenneth Hodges, Adam Levy, Rajiv Kochar, Lisa Kuklinski, Zia Mahmood, and Karen von Haam, Segments can meet in fours in energy- minimizing networks, J. Undergrad. Math. 22 (1990), 9-20.

2. Manuel Alfaro, Mark Conger, Kenneth Hodges, Adam Levy, Rajiv Kochar, Lisa Kuklinski, Zia Mahmood, and Karen von Haam, The structure of singularities in cI)-minimizing networks in R 2, Pacific J. Math. 149 (1991), 201-210.

3. Marshall W. Bern and Ronald L. Graham, The shortest- network problem, Scientific American (January 1989), 84- 89.

4. E. J. Cockayne, On the Steiner problem, Can. Math. Bull. 10 (1967), 431-450.

5. Mark A. Conger, Energy-minimizing networks in R", Honors thesis, Williams College, 1989, expanded 1989.

6. R. Courant and H. Robbins, What is Mathematics?, Ox- ford: Oxford Univ. Press (1941).

7. Martin E. Glicksman and Narsingh B. Singh, Microstruc- tural scaling laws for dentritically solidified aluminum alloys, Special Technical Pub. 890, Philadelphia: American So- ciety for Testing and Materials (1986), 4~ 61.

8. M. Hanan, On Steiner's problem with rectilinear distance, J. SIAM Appl. Math. 14 (1966), 255-265.

9. Stefan Hildebrandt and Anthony Tromba, Mathematics and Optimal Form, New York: Scientific American Books, Inc. (1985).

10. Gary Lawlor and Frank Morgan, Paired calibrations applied to soapfilms, immiscible fluids, and surfaces or networks minimizing other norms, preprint (1991).

11. Adam Levy, Energy-minimizing networks meet only in threes, J. Undergrad. Math. 22 (1990), 53-59.

12. Frank Morgan, Geometric Measure Theory: a Beginner's Guide, Boston: Academic Press (1988).

13. Frank Morgan, Riemannian Geometry: a Beginner's Guide, Boston: Jones and Bartlett (1992).

14. Frank Morgan, Soap bubbles and soap films, in Mathe- matical Vistas: New and Recent Publications in Mathematics from the New York Academy of Sciences (Joseph Malkevitch and Donald McCarthy, eds.), New York: New York Academy of Sciences (1990), Vol. 607.

15. Frank Morgan, Compound soap bubbles, shortest networks, and minimal surfaces, video of AMS-MAA address, Win- ter Mathematics Meetings, San Francisco, 1991.

16. C. M. Petty, Equilateral sets in Minkowski spaces, Proc. AMS 29 (1971), 369-374.

17. Bruce Schecter, Bubbles that bend the mind, Science 84 (March 1984).

18. Jean E. Taylor, Crystalline variational problems, Bull. AMS 84 (1987), 568-588.

19. Jean E. Taylor, The structure of singularities in soap- bubble-like and soap-film-like minimal surfaces, Ann. Math. 103 (1976), 489-539.

20. Steve Smale, Beautiful crystals calendar, 69 Highgale Road, Kensington, CA 94707.

Figure 14. Here, at age 2, I watched my morn blow soap bubbles [12, p. ii].

Department of Mathematics Williams College Williamstown, MA 01267 USA E-mail: [email protected]


minimal surfaces, crystals, shortest networks, and undergraduate research

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