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From cracks to catastrophes, “singularity theory” could shed light

June 5, 2008
Courtesy European Science Foundation 
and World Science staff

The­re’s of­ten more to eve­ry­day events than meets the eye. The fold­ing of pa­per, or drip­ping of wa­ter from a tap, are two ex­am­ples: they both in­volve the crea­t­ion of points known as sin­gu­lar­i­ties.

Sin­gu­lar­i­ties oc­cur at places of cut­off or of sud­den change, as in forma­t­ion of cracks, light­ning strikes, crea­t­ion of ink drops in print­ers, and the break­ing of a cup when it drops. These points re­quire soph­is­t­icated math­e­mat­i­cal tech­niques to de­scribe, an­a­lyse and pre­dict.

Lightning is one phe­no­me­non ex­hi­bit­ing sin­gu­lar­i­ties. Others in­clude crum­pled pa­per and drip­ping wa­ter. (Im­age cour­tesy NASA)
Sci­en­tists say many sin­gu­lar­i­ties have much in com­mon at all size scales—from mi­cro­scop­ic in­ter­ac­tions to the forma­t­ion of the uni­ver­se it­self dur­ing the so-called Big Bang. But these seem­ingly dis­par­ate events are usu­ally stud­ied by dif­fer­ent sci­en­tists in re­la­tive isola­t­ion.

A work­shop or­gan­ised by the Eu­ro­pe­an Sci­ence Founda­t­ion in Par­is in Jan­u­ary was one of the first at­tempts to un­ify the field of sin­gu­lar­i­ties by bring­ing to­geth­er ex­perts in the dif­fer­ent fields from as­tron­o­my to nano­sci­ence—the study of atom­ic-scale struc­tures. 

The meet­ing was aimed at de­vel­op­ing com­mon math­e­mat­i­cal ap­proaches to sin­gu­lar­i­ties. Im­proved un­der­stand­ing of the un­der­ly­ing math would have many ben­e­fits, for ex­am­ple in mak­ing ma­te­ri­als more re­sist­ant to break­ing, re­search­ers say. 

The event was a suc­cess and and paved the way for fur­ther re­search with great­er cross-pollina­t­ion of ideas, said the con­ven­or, Jens Eg­gers of the founda­t­ion.

The work­shop con­firmed, sci­en­tists said, that most or all sin­gu­lar­i­ties, from mi­cro­scop­ic cracks to the Big Bang, share a key prop­er­ty known as self-si­m­i­lar­ity. This means that un­der mag­nif­ica­t­ion the event looks al­most the same. For ex­am­ple a crack in a piece of plas­tic ex­hibits the same jag­ged struc­ture when mag­ni­fied, say, 100 times. This means com­mon math­e­mat­i­cal ap­proaches can be ap­plied.

But the dev­il is in the de­tails when it comes to com­par­ing dif­fer­ent types of sin­gu­lar­i­ties, work­shop par­t­i­ci­pants cau­tioned. Dif­fer­ent sys­tems might have some com­mon fea­tures such as self-si­m­i­lar­ity, but al­so un­ique as­pects that re­quire spe­cial­ised stu­dy. One aim of the work­shop was to iden­ti­fy the com­mon meth­ods that could be ap­plied as a founda­t­ion for more de­tailed spe­cif­ic study.

Jay Fine­berg of He­brew Uni­ver­s­ity in Je­ru­sa­lem, for ex­am­ple, pre­sented in­ves­ti­ga­t­ions of cracks in struc­tures or rock forma­t­ions. Fine­berg dis­cussed new ex­pe­ri­ments in­volv­ing gels, al­low­ing the crack’s struc­ture to be de­ter­mined in great de­tail down to mi­cro­scop­ic di­men­sions, yield­ing some un­ex­pected find­ings.

Cracks are of­ten sur­pris­ingly com­plex, Eg­gers not­ed, with “many small side branches, which ap­pear to have com­pli­cat­ed, if not frac­tal, struc­ture.” Frac­tal struc­ture here means much the same as self-si­m­i­lar­ity, in­volv­ing a ge­o­met­ric pat­tern that looks un­changed un­der mag­nif­ica­t­ion or re­duc­tion.

Anoth­er ex­am­ple con­cerned the sin­gu­lar­i­ties of crum­pling in pa­per, pre­sented by Tom Wit­ten of the James Franck In­sti­tute in Chi­ca­go. Crum­pled pa­per com­prises many ridges and tips that de­fy sim­ple anal­y­sis. There are many un­an­swered ques­tions even in de­scrib­ing each in­di­vid­ual cone-shaped tip, Eg­gers said; fig­ur­ing out the un­der­ly­ing math would not just help un­der­stand what hap­pens when we crum­ple pa­per, but al­so oth­er phys­i­cal sys­tems in­volv­ing ridges and tips, such as the way bi­o­log­i­cal mo­le­cules fold in­to their char­ac­ter­is­tic forms.

One branch of sin­gu­lar­ity the­o­ry is “catas­tro­phe the­o­ry,” which rose to prom­i­nence in the 1970s, in­i­tially de­vel­oped by French math­e­ma­ti­c René Thom and ex­pand­ed by U.K. math­e­ma­ti­c Er­ik Zee­man. Ca­tas­tro­phe the­o­ry deals with events with space-and-time com­po­nents, such as col­li­sions be­tween wave fronts, Eg­gers said. “In that case, a prob­lem that takes place in all of space can be re­duced to a prob­lem that takes place along cer­tain lines,” known as caus­tics, “which can be clas­si­fied ac­cord­ing to ca­tas­tro­phe the­o­ry.” But not all sin­gu­lar­ity prob­lems are ame­na­ble to this sim­plifica­t­ion.

The sub­ject “cuts across dis­ci­plines and spe­cial­iz­a­tions, such as ex­pe­ri­men­tal phys­ics, the­o­ret­i­cal phys­ics, and rig­or­ous math­e­mat­i­cal proofs,” Eg­gers said. “This work­shop very much re­flected this fact, as we had speak­ers from very dif­fer­ent back­grounds.”

* * *

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There’s often more to everyday events than meets the eye. The folding of paper, or drip of water from a tap, are two examples: they both involve the creation of “singularities.” Singularities occur at a point of cutoff or of sudden change, as in formation of cracks, lightning strikes, creation of ink drops in printers, and the breaking of a cup when it drops. These points require sophisticated mathematical techniques to describe, analyse and predict. Scientists say many singularities have much in common at all size scales—from microscopic interactions to the formation of the universe itself during the so-called Big Bang. But these seemingly disparate events are usually studied in isolation by different scientists with little interaction. A workshop organised by the European Science Foundation in Paris in January was one of the first attempts to unify the field of singularities by bringing together experts in the different fields of application from astronomy to nanoscience, the investigation of atomic-scale structures. The meeting was aimed at developing common mathematical approaches to singularities. Improved understanding of the underlying math would have many benefits, for example in making materials more resistant to breaking, researchers say. The event was a success and and paved the way for further research with greater cross-pollination of ideas, said the convenor, Jens Eggers of the foundation. The workshop confirmed, scientists said, that most or all singularities, from microscopic cracks to the Big Bang, share a key property known as self-similarity. This means that under magnification the event looks almost the same. For example a crack in a piece of plastic exhibits the same jagged structure when magnified, say, 100 times. This means common mathematical approaches can be applied. But “the devil is in the details” when it comes to comparing different types of singularities, organizers of the workshop cautioned. In other words different systems might have some common features such as self-similarity, but also unique aspects that require specialised study. One aim of the workshop was to identify the common methods that could be applied as a foundation for more detailed specific study of particular singularities. Jay Fineberg from the Hebrew University in Jerusalem, for example, presented investigations of cracks in structures or rock formations. Fineberg talked about new experiments involving gels, allowing the crack’s structure to be determined in great detail down to microscopic dimensions, yielding some unexpected findings. Cracks are often surprisingly complex, Eggers noted, with “many small side branches, which appear to have complicated, if not fractal, structure.” Fractal structure here means much the same as self-similarity, involving a geometrical pattern that looks unchanged under magnification or reduction. Another example concerned the singularities of crumpling in paper, presented by Tom Witten of the James Franck Institute in Chicago. Crumpled paper comprises many ridges and tips that defy simple analysis. There are many unanswered questions even in describing each individual cone-shaped tip, Eggers said; figuring out the underlying math would not just help understand what happens when we crumple paper, but also other physical systems involving ridges and tips, such as the way biological molecules fold into their characteristic forms. One branch of singularity theory is “catastrophe theory,” which rose to prominence in the 1970s, initially developed by French mathematician René Thom and expanded by U.K. mathematician Erik Zeeman. Catastrophe theory deals with events with space-and-time components, such as collisions between wave fronts, Eggers said. “In that case, a problem that takes place in all of space can be reduced to a problem that takes place along certain lines,” known as caustics, “which can be classified according to catastrophe theory.” But not all singularity problems are amenable to this simplification. The subject “cuts across disciplines and specializations, such as experimental physics, theoretical physics, and rigorous mathematical proofs,” Eggers said. “This workshop very much reflected this fact, as we had speakers from very different backgrounds.”