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Progress Report: The Computational and Scientific Graphics

Laboratory at MSRI DE-FG03-95ER25250

David Hoffman

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Progress Report: The Computation and Scientific Graphics Laboratory at MSRI

David Hoffman

1 Staff and Software Development

Hoffman continued his research in differential geometry with applications to the study of the

microstructure and singularities in compound polymers. This work takes place at MSN, where he has set up a computation and graphics laboratory.

James T. Hoffman is the full-time, senior graphics programmer for the laboratory

project. Here is an outline of J. Hoffman’s software-related work from January 15, 1996 to

Jan 15 1997. IMPROVEMENTS TO EXISTING APPLICATIONS

0 A) Mesh Server

- 1) Fixed long-standing bug in shearing compensation - a capability allowing Mesh to

- 2) Improved reporting on detected lines an planar geodesics.

- 3) Fixed several bugs which had resulted in the omission of triangles near cut-ends in the output data (but not in the mesh viewer).

- 4) Created a new output file that contains labeled coordinates of-cut endpoints and distinguished points, as well as other data, to facilitate the generation of translation vectors by VPS/render (See item under VPS/render.)

grow through regions where the metric changes rapidly.

0 B) Mesh Clients

- 1) Converted several old Mesh clients (written by others) to C++, and edited to conform

- 2) Re-created missing VPS construction scripts for various Mesh clients. - 3) Updated several clients, including modifications for recording translation vectors, and

to coding standard.

wrote VPS scripts for construction.

0 C) VPS/Render

- 1) Added network processing operators to make the results of constructions of net- works suitable for subsequent processing by the Brakke Surface Evolver including: a) rotation/reflection replication operators which seam the pieces along lines/planes. b) general seaming operators to be used with arbitrary types of construction. c) torus wrapping operators.

- 2) redesigned the network display operators, providing the ability to selectively draw and distinguish different network components.

- 3) Created a module to write network data in a VRML format.

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- 4) Upgraded the image module (to enhance encapsulation and improve interfaces).

- 5 ) Added an operator for reading items in the new output file of mesh, to allow automatic creation of translation vectors based on the displacements between cut endpoint and/or distinguished points in certain mesh clients.

- 6) Added a Schwarz reflection network construction operator, and a conjugation attribute function. Used together these capabilities allow a single output file created by Mesh, having both real and imaginary components, to be constructed into an arbitrarily large piece of the surface, then rotated in conjugate space to give any member of the associate family of surfaces.

- 7) Added several real and vector operators for simplifying coding of arithmetic.

0 D) SolSurf

- 1) Added the ability to use criteria other than zero crossing to define surfaces, supporting the approximation of surfaces, other than zero sets which partition space into two regions.

- 2) Fixed long-standing bug which resulted in occasional dropped facets.

- 3) Added a command line option for adding offsets to functions, simplifying scripting animations.

0 E) TEMsim

- 1) Fixed bugs which resulted in incorrect projections in the cases of three volumes.

- 2) Created library of TEMsim data for D and P surfaces generated using seaming con- struction operators in VPS/Render and then evolved to CMC surfaces at various volume fractions using the Brakke Surface Evolver.

0 F) K/Kvew (application to visualize Kobayashi Metric)

- 1) Added elements to aid in visualization of the domain. - 2) Improved file handling and the interface between the computation and viewing pro-

grams.

0 G) Algorithms/Data Structures

- 1) Added 4 bytes of attribute flags to objects in networks (nodes, edges, and faces) in run-time data structures (not affecting file format) and modified algorithms which use attribute flags to avoid clashes, and increase reliability and maintainability.

- 2) Added diagnostic functions such as one to histogram face sizes and edge lengths in networks.

IMAGES & ANIMATIONS

0 Created an animation of a CMC family of D surfaces varying in volume fraction.

0 Elser’s Surface * Created a series of images and an animation of Elser’s surface. (Available at http:// www.msri.org/Computing/jim/geom/zero/elser/)

0 The P-G-D Associate Family * Created an animation showing the relationship of the triply pe- riodic P, G, and D surfaces, which are all members of the same associate famiIy of minimal sur- faces. (Available at http:// www.msri.org/Computing/jim/geom/minimal/assoc~b/PGD/)

0 Costa Surface Deformation * Created deformation animations for the Costa-Hoffman-Meeks minimal surface families, serving as a prototype for other surface families (Available at http:// www.msri.org/Computing/jim/geom/minimal/library/costa/)

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Above: A screen from TEMsim, a suite of programs developed to help determine the mathematical structure of ni;itcrials interfaces by simulating electron microscope experiments. In color i s the candidate surface. related to the ggroid. I n monochrome is the simulated electron micrograph.

Z & ! K , . ~ ~ W . W w z Illustrations from the work of D. Hoffman. E. Tbomnc: pi a1

using TEMsioi and its precursors. Left and belot "double diamond" OBDD interface. an actual micrograph i s shown above its simulation. Before recent mathematical work. only simple models (spheres. cylinders, planes) were

Scherk's minimal surface. a model fo r grain boundaries of materials i n the lamellar phase. The ima$e was used again

Chemistry at the Frontier." by Phill ip Ball.

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k , r F ' ] considered as possible interfaces. Below right: Some of this .i

research was published in Nature---the cover illustration is

i n 1996 for the cover of "Designing the Molecular World:

c .

EXPERIMENTAL MATHEMATICS voiuhtE 4, NUMBER 1 14

A K PETERS, LTD

Top: 19th Century model of Ennepefs surface with its 201h Century computer realization. Right column: Enneper‘s surface with two of its moden generalizaiions below. The first has more complicated end-behavior. The second, found b r Chen and Gack- statter, has more com licated to o 2 ogy (genus 1).

E. Thayer, a former doctoral student of Hoffman. The paper concerns the asymptotic behavior of these surfact as thq genus increases and the desingularization of families of intersecting planes.

Bottom Ieft: Cover 11 P ustration $r a research article by

d

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color computer graphics by 3.T.Hoffinan. h4SRI 1996.1997

Images for papers, including final versions of “Complete embedded minimal surfaces of finite total curvature,” (D. Hoffman and H. Karcher) “A new turn for Archimedes,” (D. Hoffman) (D. Hoffman) ‘‘WE1 Paper

DOCUMENTATION/WEB PAGES

Mesh Client Web Pages 1) Created framework for extensive Web pages on Mesh clients. 2) Completed web pages for several clients.

Mesh Server Web Pages 1) Converted the Mesh User Manual to HTML (starting with output of latex2html).

Scripting 1) Created a Perl script acting as a final filter for all pages, assuring consistent formatting. 2) Created several special-purpose Perl scripts for automatically creating page content (input to the final filter) from concise, easily-maintained files consisting of entrys of keywords with one or more values.

Content 1) Added several Mesh client and minimal surface pages. 2) Created a new format for the Mesh clients index page. 3)* Created a framework and a consistent set of scripts for the generation of the images and VRML models for each of the Mesh clients. 4) Created the TEMsim Volume Projections page, showing standard projections of several common triply periodic dividing surfaces. (Available at http:// www.msri.org/Computing/jim/software/temsim/proj/) 5 ) Created web page to illustrate several periodic level surfaces. 6) Created web page to illustrate several of the possible tri- partitions of the unit cube by a soap-film.

J. Hoffman and L. Radzilowski are collaborating on a user manual for TemSim.l We

have developed an interactive WEB site and repository for MESH, a computational tool

for computing and drawing minimal surfaces.2Hoffman also prepared the.illustration pages

for D. Hoffman’s dossier for an invited submission to the LVMH competition: “La genkse

des formes, prix de la science: mathematique, physique et sciences de la terre,” . Two pages

are included here. The first concerns generalized Enneper’s surfaces, the second illustrates

some of the uses of the software TEMsim.

2 Interaction at MSRI and LBNL

0 Hoffman has been cooperating with Horst Simon, the Director of NERSC, to enhance

the collaboration between NERSC and MSRI, which was outlined in the original

proposal for relocation of NERSC to LBNL. A series of concrete steps are being

taken at this time. These include: planning a proposed conference on computational

problems in chemistry; a joint visitor program; planning efficient ways for use of

NERSC computational facilities and training courses by MSRI members.

‘For this and other softwaxe developed by this project, see http://www.msri.org/Computing/JTHapps/

*See http://www.msri.org/Computing/JTHapps/m~h/m~h-clients/

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0 Following a suggestion of H. Simon, we have done some numerical computation con-

nected to a partitioning problem for cubes, which is motivated by problems of load

balancing in multi-processors. Specifically, what is the (generalized) surface of mini- mum area that partitions a solid cube into three regions of equal area? Some results of

this can be seen on the Web.3 as well as on the illustrations on the previous page. A

solution surface is necessarily one with constant-mean-curvature. In addition, it meets

the boundary orthogonally and its singularities, if any, consist of three sheets meeting

at equal angles along a curve. We used the Brakke Surface Evolver to compute these

examples, which are local -but not necessarily global-minima. We must specify in

advance both the way the surface meets the boundary of the cube and its singularity

graph. With these constraints fixed, we are able to compute the minimizing config-

uration. These computations suggest that the area-minimizing configuration consists

of two cylindrical surfaces meeting a planar surface in a line segment. This surface

projects down to the solution for the tri-partition problem for a square. At present,

this is an open problem.

0 J. Hoffman prepared illustrations of critical surfaces, so-called double bubbles, for Joel

Hass (UC Davis and MSRI) and Roger Schlafiy. These two mathematicians solved a

longstanding open problem: what is the surface (bubble) of least area that encloses

two regions of equal area. Their work uses an interval-arithmetic computation to show

that toroidal bubbles cannot occur in a minimizer. However, they can arise in critical

bubbles. Hass and Schlafly show that these are not area-minimizing. The minimizer

consists of two spherical regions divided by a planar disk. J. Hoffman’s computer

graphic images of the critical bubbles appeared in an article by Schlafly and Hass in

American Scientist (September/October 1996). These illustrations can also be found

on the Web.4 Some of them are reproduced on the next page.

0 Hoffman aided in the conception and coordination of the MSFU Multicast Backbone

project. The Multicast Backbone software has been developed in large part at LBNL

in the networking group. In September 1996, MSFU was awarded an NSF grant for

two years to support of the project. Its goal is to promote the use of the Multi-cast

Backbone (MBone) in the mathematical community as a means of collaboration and

communication. It would allow mathematicians everywhere to participate in MSRI

activities via the Internet.

The original proposal to the NSF listed 15 institutions who had agreed to purchase

the necessary equipment and assign the requisite personnel to the project. Since then

three more institutions have signed on, and we expect gradually to add more, after the

initial ones are properly set up. Joe Christy, David Hoffman, and Bob Osserman are ~~

3http://www.msri.org/staff/bio/jim/imag/s~f/cmc/cube/ind~.html . 4http://www.msri.org/staff/bio/jim/imag/s~/cmc/revol/~d~.html

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in charge of the project, with Dave Wright providing general oversight and technical

support, both internally and to our collaborating institutions.

We view this project as an experimental one, to see how best to utilize the new

technology to the benefit of the mathematical community. We have learned several

valuable lessons already, and will be sharing them via the MSFU MBone information

page: http://www.msri.org/mbone/. One thing we have done is to link lecture notes

and overheads to prerecorded lectures (see http://www.msri.org/lecturenotes/.) We

have automated the production of these web pages to minimize the labor involved.

Both web pages are accessible from MSFU’s home page http://www.msri.org/.

In January, MSFU presented the MBone Project to a wide mathematical audience at the joint AMS/MAA meetings in San Diego. We set up an actual internet with two

nodes, one for broadcast the other for reception. We would have preferred broad-

casting directly from MSRI but there was no internet access provided to the exhibit

floor.

During the Spring of 1996, Hoffman was one of the organizers (together with Amir

Assadi (U. Wisconsin) of a weekly seminar on human vision, held at MSRI. The

seminar had between 15 and 30 participants and met for about 14 weeks. One of the

participants and speakers was the physicist Donald Glaser, a Nobel Prize winner from

UCB and LBL, whose current research focuses on the physiology of human vision and

its replication in computing hardware.

0 David Hoffman arranged the visit and lecture of Edwin Thomas, from the Department

of Materials Science at MIT to MSFU and the Center for Electron Microscopy at

LBNL, where Michael O’Keefe was the host. Thomas spend three more days at MSFU

working with D. Hoffman, J. and L. Radzilowski, a post-doc from MIT in materials

science.

3 Geometry

D. Hoffman continued his research in the theory of complete embedded minimal surfaces.

F. Wei (Virginia Tech) and Hoffman continue their collaboration on complete embedded

minimal surfaces of genus one with screw-motion symmetry.

John MCuan is an NSF post-doctoral fellow working since September, 1996 under the

direction of David Hoffman, Robert Osserman and Craig Evans. In the 1995-1996 academic

year, McCuan, was an MSFU Post Doctoral Associate who worked with Hoffman on prob-

lems involving constant-mean-curvature (CMC) surfaces bounded by convex plane curves.

In the Fall of 1996, he spent half of his time teaching at UCB. At MSFU, he is working with

Hoffman on problems involving minimal and CMC surfaces. In particular we are making

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some progress in the direction of classifying embedded annular minimal surfaces of infinite

total curvature. We have shown explicitly that there an infinite number of different ends of

this type, which are asymptotic to the helicoid.

McCuan is currently working on the following problem. Given a convex planar domain

and a function, f, with zero boundary values, whose graph has constant mean curvature, are the level curves of the f necessarily convex? Several years ago, Hoffman carried out

a simulation that showed that the graph itself is not necessarily convex. (For example, if

the domain has an acute corner on its boundary) then the graph can have negative Gauss

curvature near the corner.) This is now known to be true by analytical arguments. However,

the known maximum principles do not show that the level curves of f have to be convex.

McCuan's construction is modeled on the idea of viscosity solutions of PDE. It already

shows that under some additional assumptions the level curves are convex. There is a good

chance he will get the general case.

Related to the research Hoffman and McCuan are doing on minimal surfaces is the recent

work of A. Bobenko (TU, Berlin), which is not yet published. Bobenko visited MSRI for

several days in January. Using the theory of theta functions and a spinor representation of

minimal surfaces, he has been able to replicate the Weierstrass representation for the genus

one helicoid 5 . His point of view may allow generalizations to higher genus examples with

a single embedded end of infinite total curvature. If one is looking for embedded examples,

the machinery of spinors gives a criteria that rules out certain Weierstrass representations.

We are also interested in exploration of the use of classical theta functions in the attempt

to construct of a complete, simply connected embedded minimal surface. The only known

examples are the plane and the helicoid.

When Bobenko was at MSN, he expressed some doubt about a result of Hoffman

and McCuan: we claimed that not only the ends we constructed but also the ends he was

discussing were all asymptotic to a helicoid. Bobenko believed that there was a translational

shift, allowing a gap and making it impossible for the end to be asymptotic to a single

helicoid. We were able to simulate a few cases and draw them out far enough to show

that the experimental evidence contradicted his expectation and supported the truth of our claim. We also rechecked carefully; there is not a mistake in our proof.

Michael Callahan was supported for half an academic year as a graduate student and

research associate. Callahan works in several areas, including differential topology of four

manifolds and minimal surface theory. He is also an expert programmer. He worked with

D. Hoffman, J, McCuan and J. Hoffman to analyze the construction by Veit Elser, of a family of triply periodic surfaces, which had some properties in common with the helicoid.

This surface is constructed as the set where a complicated function, Q : R3 + C has

5D. Hoffman and H. Karcher and F. Wei, The genus one helicoid and the minimal surfaces that led to its discovery, Global Analysis and Modern Mathematics, Publish or Perish Press, 1993, K. Uhlenbeck, editor, p. 119-170

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constant phase, modulo 2 ~ / 3 . The function 3P can vanish and therefore the surface can

have singularities. To compute such a surface, our routines that implemented zero-crossing

algorithms would not work. Elser himself had to compute the level surfaces pi3 -Re(q3) = 6 for small S. This gives two surfaces that coalesce to the desired surface as 6 3 0.

The method worked out by Callahan and implemented by J. Hoffman allows a robust

computation of the surface, which in turn resulted in the clear images. Our analysis of the

surface is summarized in the short article “A New Turn for Archimedes,” published in the

November 7,1996 issue of Nature. A preprint version of the article is appended to this report

(under its original name: Mixing Mathematics and Materials) and the illustrations are

reproduced here. The article is a commentary on the issue of the Philosophical Transactions

of the Royal Society entitled “Curvature and Chemical Structure.”6 Briefly, here is what it is about:

The understanding of the interface formed in copolymers in key to prediction of their

material properties. At present, there is only a partial understanding of the relationship

between the geometry of the range of possible periodic surfaces that might occur as inter-

faces. The class of periodic constant-mean-curvature surfaces are reasonable candidates,

but this is being hotly debated in the current l i t e r a t ~ e . ~ . There are some very sugges-

tive and as-yet-unexplained mathematical phenomena having to do with the coincidence of

triply periodic minimal surfaces and the zero sets of simple trigonometric polynomials. We

would like to understand why this is the case. From a computational point of view, the

zero sets are much easier todeal with than the minimal surfaces that they shadow. We will

do a further study of this surface in coming months.

4 Equipment

Our two existing SGI Indigo2s have 150 and 250 Mhz R4400 processors. The 250 Mhz

system was upgraded with an IMPACT Compression board for live screen capture and

real-time playback of animations stored as JPEGs. This system is to be upgraded with an

RlOOOO processor board, and its R4400 board will be swapped out to replace the slower

processor in the other Indigo2. The RlOOOO system will receive 256 MB of memory, a

4GB Seagate Cheetah drive in a double external fastkwide SCSI enclosure, and an external

Iomega Jaz drive for backup and archival storage. The Macintosh 8500 is upgraded with 64

additional megabytes of memory, an internal 1GB Iomega Jaz drive for backup. An Epson

Color Stylus Pro XL color ink jet printer has been installed as a network device reachable

from both Macintosh and Indigo2. An additional workstation will be added in the form of

an SGI 0 2 with a 180 R5000 processor and built in compression/video support.

6Phil. Trans. R. SOC. London A (1996) 354, J.Klinowski and A. L. MacKay, editors. The article Level Surfaces for Cubic Tricontinuous Block Copolymer Morphologies, by C. Lambert,L. Radzilowski, E. Thomas, which we cited last year for its essential use of the software TEMsim, also is published in this volume

‘See, e.& Hajduk et al. Macromolecules, 27, 4063-4075 (1994)

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Figure 1.a (upper left) A unit cell of Elser's surface. Figure 1.b (upper right) A unit cell of one of the surfaces in the family. This one is in the middle of the family. An animation of the entire family is viewable at : http: / /www.msri .org / Computing/ jim /geom/zero /periodic/ elser. Figure 1.c (lower right) One of the three congruent surfaces that meet a t 120 degree angles along the line singularities to form the surface in 1.b. Figure 1.d (lower left) Line singularities: rod packing with octahedral s y m m et r y.

Figure 2 The Helicoid.

~, ... . .: . . .. . .. .. . ~ . ...

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Figure 3a (left) The gyroid, a triply periodic, space-dividing minimal surface, discovered by A. Schoen in the late ‘60s. It contains no lines and has no reflective symmetries. Figure 3b (right) The solution set to sin x cos y + sin y cos z + sin z cos x = 0.