Skip to content

Stephenson:Neal:Quicksilver:61:…a certain natural wave(Alan Sinder)

From the Quicksilver Metaweb.

This is a digression concerning grooming featuring Algebraic topology

Stephensonia

Daniel Waterhouse finds himself the object of social grooming by Isaac Newton.

Authored entries

Grooming As Seduction

Avoiding stereotypes, consider Shampoo — the farcical satire of 1960s sexual practices, a hairdresser George Roundy (Warren Beatty) who finds he can't resist the women he glamorizes. They can't resist him either, and unfortunately they are all linked somehow. Juxtaposing tropes from Restoration comedy with Southern California dialogue and a healthy, hilarious dash of running commentary from election returns, an awareness cuts through the film like a scalpel. Beatty's freewheeling and funny escapades with three of his lovers culminate in an election night party as Nixon wins which all three attend, along with another older, richer man who is married to one of Beatty's conspirators, and whom Beatty happens to be soliciting for startup funds for his own beauty business. At the party, all three women desperately want Beatty's attention, and the resulting debacle causes him to reconsider his lifestyle.

Claire Tomalin notes in her Samuel Pepys: The Unequalled Self that 'brushing hair was considered an erotic act. Newton for all his prissiness was a money lending farm boy. He'd certainly seen the returning Royals and returning courtiers entertain their whores or sluts at school. Like Daniel, he must have had the cast aside wantons alight on his bed. And somehow Daniel knew of the speciality prostitutes at the Stourbridge Fair when they went to buy prisms. Daniel has enough sense of self to avoid the seductive nature of being groomed, and seems to tolerate the activity to avoid upsetting Isaac. Young Newton appears to be one you don't want to alienate. One wonders if the grooming was Isaac assigning status to young Waterhouse.

Grooming Among Chimpanzees

Social grooming is clearly a form of conflict resolution. Conflict resolution is the process of resolving a dispute or a conflict permanently, by providing each side's needs, and adequately addressing their interests so that they are satisfied with the outcome. Conflict resolution aims to end conflicts before they start and end bloodshed.Bonobo.jpeg
A Bonobo Chimpanzee

Grooming is also used to relax tension from threats and aggression. It helps to maintain friendly ties among family and community members and to lessen the stress of infants during weaning. During times of relaxation, a wild chimpanzee may often be found grooming another chimpanzee or its own hair. A chimpanzee may request or solicit grooming by approaching another chimpanzee and getting their attention by presenting a part of its body for grooming. It may scratch itself or start to groom itself. Grooming is a very important social and skin care behavior. The most obvious function is the removal of pieces of debris from soil, vegetation, and dried skin from hair. It may present the groomer a tasty snack. A grooming session may include several individuals of varying ages and continue for a few seconds, minutes, or hours. The chimpanzee uses one hand to hold the hair back while the other hand, lips, or teeth are used to pick out and remove the small pieces of debris.

Tim Friend reports: “... [l]ike pop culture that arises suddenly and spreads quickly through a group of teens, new behaviors also appear in chimpanzees. In the mid- 1990s Gombe chimps suddenly began using leaves to squish bugs called ectoparasites, which they pull from each other's coats while grooming. Maybe the practice arose because one chimp didn't like getting its hands messy, or maybe a chimp discovered that the leaves made it easier to kill the hard-shelled bugs. Ta chimps, on the other hand, remove the parasites from a partner, place them on a forearm, smack them with their hand and then eat them. Boesch writes, "Here again, we have two different solutions to the same problem."

Grooming, which strengthens social bonds, provides some of the richest examples of chimp cultural behavior. Take, for example, the "handclasp," first observed among chimps in the Mahale Mountains. Recalls primatologist William McGrew of Miami University in Oxford, Ohio, "Each of the participants simultaneously extended an arm overhead and then either or both grasped the other's wrist or hand, or each grasped the other's hand. Meanwhile, the opposite hand was used to groom the other individual's underarm area revealed by the upraised limb."

Only adults and adolescents perform the behavior, and the pairs usually are male and female. They perform the grooming handclasp more often, every 2.4 hours on average, than any other type of behavior including food-sharing, tool-using and sexual and aggressive displays.

At nearby Gombe, scientists have taken special interest in the fact that the chimps there use a similar but significantly different technique. There, chimpanzees raise their arms overhead and sometimes grasp an overhead limb for support. Since the sites are only about 100 miles apart, the differences cannot be due to genetic isolation.

More recently, the Mahale chimpanzees have adopted a new grooming technique called the "social-scratch," which has caught on like another pop culture phenomenon. Here, the grooming chimpanzee rakes the hand up and down his subject's back at the end of a grooming session. The social scratch is not used at Gombe, nor has it been observed at any other site.

Commenting in the journal Science, McGrew wrote, "We have enough data in enough populations that we can start doing the sorts of comparisons that cultural anthropologists do across human populations." And already there is enough evidence of behavioral differences among chimp communities that anthropologist Frans de Waal can boldly conclude, "Biologically speaking, humans have never been alone; now the same can be said of culture."... ” [1]

Captive Chimps Need To Groom

Captive chimpanzees groom themselves and each other in zoos in the same manner as wild chimpanzees. The biggest difference is that they often perform this comforting and soothing behavior while sitting or lying high among the tall support structures of their indoor habitats or lying precariously on rock formations or in the grass out doors.

Mothers groom fussy temperamental infants being weaned from nursing against their wishes. Females relaxing outside on a hot day find a shady spot to spend long periods of time grooming. On occasion, it appears the entire social group is engaged in a friendly, comforting social encounter grooming each other.

Splitting Hairs

Meredith F. Small goes further: “… When researchers compared the chimps' kinship to their behavior, they found that males who sit together and groom each other are not closely related. … Kinship doesn't underlie male cooperation, in other words. Chimpanzees get to know each other and keep track of the political intrigue that goes with making, breaking, and manipulating relationships. Although kinship is important, chimps often also rely on the fragile ties of friendship. … One of the best examples of chimp culture can be seen in the Mahale Mountains of Tanzania and in Gombe, 90 miles to the north. Though the two areas lie on the same side of Lake Tanganyika, the chimpanzees have more elaborate grooming habits in Mahale. In Gombe, when a male chimp lumbers up to a friend and sprawls out on the ground, the friend will usually groom him by gently passing a hand through the fur on his back, chest, face, or leg. In Mahale, chimps prefer to face each other, lock hands, and raise their arms in a mutual salute. The same style is seen at several other sites across Africa, and in captive populations, but not in Gombe. Is the Mahale style simply the most efficient way to groom an armpit? Or is it the chimpanzee version of a secret handshake?

Anthropologist William McGrew has studied the Mahale chimps, and several other groups, for 20 years. He not only believes that their grooming is cultural, but also thinks there are grooming subcultures as well. Recently, when McGrew showed his students at Miami University in Ohio some old photographs of chimps grooming, he noticed something: One group at Mahale groomed the usual way while another group at the same site had a slightly different technique. "This is like the difference between the three-fingered salute by the Boy Scouts and the two-fingered salute by the Cub Scouts," McGrew says. "We are really dealing with nuances. But they're there.”[2]

Math Inspired by Homoerotic Grooming

For Isaac, Daniel's hair and its care may have been an experience in topology; Which may explain his pleasure in finding the natural wave that answered a scientific query. Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. He may have been working on what is called the “hairy dog theorem” (or variants — “hairy ball”) and knots.

Topological spaces show up naturally in mathematical analysis, abstract algebra and geometry. This has made topology one of the great unifying ideas of mathematics. General topology, or point-set topology , defines and studies some useful properties of spaces and maps, such as connectedness, compactness and continuity. Algebraic topology is a powerful tool to study topological spaces, and the maps between them. It associates "discrete", more computable invariants to maps and spaces, often in a functorial way. Ideas from algebraic topology have had strong influence on algebra and algebraic geometry.

The motivating insight behind topology is that some geometric problems depend not on the exact shape of the objects involved, but rather on the "way they are connected together". One of the first papers in topology was the demonstration, by Leonhard Euler, that it was impossible to find a route through the town of Königsberg (now Kaliningrad ) that would cross each of its seven bridges exactly once. This result did not depend on the lengths of the bridges, nor on their distance from one another, but only on connectivity properties: which bridges are connected to which islands or riverbanks. This problem, the Seven Bridges of Königsberg, is now a famous problem in introductory mathematics.

Similarly, the hairy ball theorem of algebraic topology says that "one cannot comb the hair on a ball smooth". This fact is immediately convincing to most people, even though they might not recognize the more formal statement of the theorem, that there is no nonvanishing continuous tangent vector field on the sphere . As with the Bridges of Königsberg , the result does not depend on the exact shape of the sphere; it applies to pear shapes and in fact any kind of blob, as long as it has no holes.

In order to deal with these problems that do not rely on the exact shape of the objects, one must be clear about just what properties these problems do rely on. From this need arises the notion of topological equivalence . The impossibility of crossing each bridge just once applies to any arrangement of bridges topologically equivalent to those in Königsberg, and the hairy ball theorem applies to any space topologically equivalent to a sphere. Formally, two spaces are topologically equivalent if there is a homeomorphism between them. In that case the spaces are said to be homeomorphic , and they are considered to be essentially the same for the purposes of topology.

Formally, a homeomorphism is defined as a continuous bijection with a continuous inverse , which is not terribly intuitive even to one who knows what the words in the definition mean. A more informal criterion gives a better visual sense: two spaces are topologically equivalent if one can be deformed into the other without cutting it apart or gluing pieces of it together. The traditional joke is that the topologist can't tell the coffee cup she is drinking out of from the donut she is eating, since a sufficiently pliable donut could be reshaped to the form of a coffee cup by creating a dimple and progressively enlarging it, while shrinking the hole into a handle.

One simple introductory exercise is to classify the letters of the English alphabet according to topological equivalence. To be simple, it is assumed that the lines of the letters have nonzero width. Then in most fonts, there is a class {a,b,d,e,g,o,p,q} of letters with a hole, a class {c,f,h,k,l,m,n,r,s,t,u,v,w,x,y,z} of letters without a hole, and a class {i,j} of letters consisting of two pieces. For a more complicated exercise, it may be assumed that the lines have zero width; one can get several different classifications depending on which font is used.

The method of algebraic invariants

The goal is to take topological spaces, and further categorize or classify them. An older name for the subject was combinatorial topology, implying an emphasis on how a space X was contructed from simpler ones. The basic method now applied in algebraic topology is to investigate spaces via algebraic invariants: for example by mapping them to groups, which have a great deal of manageable structure, in a way that respects the relation of homeomorphism of spaces.

Two major ways in which this can be done are through fundamental groups, or more general homotopy theory, and through homology and cohomology groups. The fundamental groups give us basic information about the structure of a topological space; but they are often nonabelian and can be difficult to work with. The fundamental group of a (finite) simplicial complex does have a finite presentation.

Homology and cohomology groups, on the other hand, are abelian, and in many important cases finitely generated. Finitely generated abelian groups can be completely classified and are particularly easy to work with.

Results on homology

Several useful results follow immediately from working with finitely generated abelian groups. The free rank of the n-th homology group of a simplicial complex is equal to the n-th Betti number, so one can use the homology groups of a simplicial complex to calculate its Euler-Poincaré characteristic. If an n-th homology group of a simplicial complex has torsion, then the complex is considered nonorientable. Thus, a great deal of topological information is encoded in the homology of a given topological space.

Beyond simplicial homology, one can use the differential structure of smooth manifolds via de Rham cohomology, or Cech or sheaf cohomology to investigate the solvability of differential equations defined on the manifold in question. De Rham showed that all of these approaches were interrelated and that the Betti numbers derived through simplicial homology were the same Betti numbers as those derived through De Rham cohomology.

Setting in category theory

In general, all constructions of algebraic topology are functorial: the notions of category, functor and natural transformation originated here. Fundamental groups, homology and cohomology groups are not only invariants of the underlying topological space, in the sense that two topological spaces which are homeomorphic have the same associated groups; a continuous mapping of spaces induces a group homomorphism on the associated groups, and these homomorphisms can be used to show non-existence (or, much more deeply, existence) of mappings.

The problems of algebraic topology

The most celebrated geometric open problem in algebraic topology is the Poincaré conjecture. The field of homotopy theory contains many mysteries, in particular the right way to describe the homotopy groups of spheres.

Poincaré conjecture

The Poincaré conjecture is widely considered the most important unsolved problem in topology. It was first formulated by Henri Poincaré in 1904. In 2000 the Clay Mathematics Institute selected the Poincaré conjecture as one of seven Millennium Prize Problems and offered a $1,000,000 prize for its solution. The conjecture states:

Every simply connected compact 3- manifold without boundary is homeomorphic to a 3-sphere.

Loosely speaking, this means that if a given "three-dimensional object" has a set of sphere-like properties (most notably that all loops in it can be shrunken to points), then it is really just a "deformed version" of a 3-sphere. (Note that "three-dimensional", in this context, refers to the intrinsic topological dimension of the object, the one the object has regardless of where it may live.)

The conjecture has induced a long list of false proofs, and some of them have led to a better understanding of low-dimensional topology.

Analogues of the Poincaré conjecture in dimensions other than 3 can also be formulated:

Every compact n-manifold which is homotopy equivalent to the n-sphere is homeomorphic to the n-sphere.

The Poincaré conjecture as given above is equivalent to the case n=3. The difficulty of low-dimensional topology is highlighted by the fact that these analogues have now all been proven (with dimension n=4 being the hardest one by far), while the original 3-dimensional version of Poincaré's conjecture remains unsolved.

Its solution is related to the problem of classifying 3-manifolds. A classification of 3-manifolds is generally accepted to mean that one can generate a list of all 3-manifolds up to homeomorphism with no repetitions. Such a classification is equivalent to a recognition algorithm, which would be able to check if two 3-manifolds were homeomorphic or not.

One can regard the Poincaré Conjecture as a special case of Thurston's 25-year-old Geometrization Conjecture. The latter conjecture, if proven, would finish off the quest for a classification of 3-manifolds. The only parts of the Geometrization Conjecture left to be proven are called the Hyperbolization Conjecture and the Elliptization Conjecture.

The Elliptization Conjecture states that every closed 3-manifold with finite fundamental group has a spherical geometry, i.e. it is covered by the 3-sphere. The Poincaré Conjecture is exactly the subcase when the fundamental group is trivial.

In late 2002, reports surfaced that Grigori Perelman of Steklov Mathematical Institute, Saint Petersburg might have found a proof of the geometrization conjecture, carrying out a program outlined earlier by Richard Hamilton. In 2003, he posted a second preprint and gave a series of lectures in the United States. His proof is still being checked.

Conclusion

Isaac's hairstyling could have been his way to release steam, and think about math. Or maybe not. Some people are able to perceive polarization of light. It is said to be visible as a yellowish horizontal line (with "fuzzy" ends, hence the name "brush") visible against the blue sky viewed while facing away from the sun.