by Carl Pierer

Fig. 1

Braids are fairly simple to picture. A few interleaved strands of string, say, gives a complex and mesmerising object. They are aesthetically appealing, as their widespread use as ornament testifies. While most will be familiar with the standard braid used for braiding hair, there is basically no limit to complexity and beauty. Yet, braids are more than merely nice, artistic adornments for clothes and jewellery. The more and deeper you delve into braids and their complex interconnections, the more fascinating they become. Trying to look at them with a mathematical eye opens up pathways and connections to many deep and beautiful fields of pure mathematics.

Studied as mathematical objects, braids need to be defined rigorously. However, for present purposes it is enough to specify what we intuitively have in mind when thinking about braids. (Fig. 1) may well serve as an example. A braid consists of a certain number of strands n, say, together with a specification of how and where these strands cross each other. Furthermore, these strands (if they are not crossing) run parallel and we may adopt the convention that they are running from top to bottom. To avoid ambiguity, we require further that there are no two crossings at the same horizontal level. It is clear that for the braid to have any crossings at all, it must have at least two strands. If a braid does not have any crossings, it is called the trivial braid.

Fig. 2

Braids also have a close connection to (mathematical) knots. Mathematical knots are simply everyday knots where the loose ends have been joined together. If you take an overhand (or trefoil) knot (Fig. 2) and join the loose ends, you have a mathematical knot. Imagine an extension cords where one end has been plugged into the other. One important feature is that there is no way of undoing this mathematical knot by pulling or stretching or any other deformation that does not break the connection. If you reconsider our

Fig. 3

initial braid (Fig. 1), imagine that the ends have been joined up as in (Fig. 3). As all loose ends are now joined up, we can consider (Fig. 3) as a mathematical knot. Indeed, every mathematical knot can be expressed as a braid. Unfortunately, this is not a one to one correspondence. So, two different braids may end up as the same

Fig. 4

knot. Or one and the same braid may give you two different knots. One problem is that there is no particular reason to join up ends of the braid as we did in (Fig. 3). If we have an even number of strands, we can equally well join up the endings that are next to each other (as in Fig. 4), which would give a very different beast altogether. Indeed, this is not a knot anymore but a so-called link, because it consists of 4 different components. A knot is a one component link. However, this connection between braids and knots is not the main concern here.

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by Shadab Zeest Hashmi

The sun burnishes the walls every day for just over three quarters of an hour out of the fourteen I spend at my shop; the mannequin assumes a buttery glow then, her organza scarf liquefies in the golden light. The CD skips at mi amour every time but this hiccup is also golden and otherworldly. The sun lifts my surroundings, the merchandise, the credit card machine, the shelves, during this portion of time in a grandiose gesture; it’s our secret— the book I’ve been writing for years has a life somewhere and this is a furtive daily reminder. I’d rather not have customers at a moment so personal in a public place; I’m at a shopping mall, selling jewelry and shawls.

I’ve named my business “Moriama”— a variant of “Moraima” (or Mariam), the last empress of Al Andalus; my manuscript is a series of poems set in Al Andalus. No one knows or cares about this but I’m advised by friends to do away with the “World Gem Bazaar” part of the sign— too “middle-eastern” in 2003. The spectrum of emotions in response to the pressure to hide my identity for fear of hostility will permeate the poems describing the atmosphere of the inquisition as my book on Al Andalus progresses. Across from me is a mattress store. As I polish and arrange African garnets and checkerboard-cut citrines, nomad jewelry from Afghanistan and London Blue Topaz pendants, someone rolls and rolls dough next door, breathing in cinnamon powder and sugar I imagine, or mops spill after spill of coffee. We’re connected to each other by repetition, as if we were phrases of the same poem.

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by Mathangi Krishnamurthy

The bell clangs loudly and I shuffle back into class trying to avoid the boys running in at dangerous speeds. I find my desk and sidle into place. I almost bump into Solomon. Solomon never studies, so the teachers always ask him to sit by me. All he does, though, is copy my notes. The afternoon sun sends bright rays into class and I inch away from him to find a cool spot on the tiny bench. The room smells of heat and dust, and I see particles floating. I feel temporarily dizzy.

I often daydream through classes. Things come easily to me, and I both know and doubt this. I am deeply suspicious that this will someday be found out, and exposed as fraud. So I am most always simultaneously attentive, and anxious at school. But daydreams come easily, because school is boring.

Solomon is gazing out of the window in his sleepy manner. Some day, I want to be Solomon, who is so cool, so uncaring, and hardly ever worried about the teachers. Mostly, it just seems to be that the world passes him by, and that he is on some other mission; something dangerous, and adult-like. I often see him hanging out by the school stile with the older boys. They all must know something about him that I don't, because there he looks happy, instead of sullen, and quiet. Solomon is really, really, dark and his white shirt often soiled. My blue pinafore, in contrast, is always immaculately pressed, its pleats like the lines of a ruler. Solomon's knees are scruffier than mine, and mine a little, only because I fell down the colony hillock last week. He never says anything in class, so I am not even sure what his voice sounds like. I imagine it to be deep. I often turn my eyes away when he looks at me. It's easy, because we sit side by side, parallel to each other, like the eyes of a cow. The only time he looks towards me is English class, where I cover my notes with my left arm, even as I can feel his eyes boring into my flesh.

Mrs. D walks in, brisk and monochromatic. She is wearing a beige sari today, and I stare up into her almost double chin. She is so tall and so pale. Her severe light brown hair is capped close to her head, but falls at her nape into a wispy ponytail. Her mouth is set in a straight line, but two front teeth escape and soften the severity. Her name is Roda, or at least, that is how I think it is spelled. I found it by accident, when Mrs.R called out to her in the teachers' room. She is pretty when she smiles. She might smile any moment, and she always smiles at me. The noise drowns as she commands us to settle down and hand in our homework.

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by Hari Balasubramanian

A selection of facts, research and personal impressions.

In February this year, I traveled to the small town of Angangueo in Central Mexico. A 4-hour bus trip from Mexico City, Angangueo is in a rural part of the state of Michoacan, in the mountainous trans-volcanic belt. Here, in a few select high elevation forests with oyamel (fir) and pine trees, millions of monarch butterflies from east of the Rocky Mountains congregate each year, after an astonishingly long journey – over 3500 km – from Canada and the northern reaches of the US.

The monarchs stay in Mexico from November to March. When the sun is shining, the butterflies – which otherwise huddle together in tightly packed and well camouflaged clusters on the branches and bark of oyamel trees – take to the air, like a beehive that has been stirred. If the sun stays up, as it did the day I visited, the monarchs quickly fill the sky and everything around you; you can even hear the faint flutter of their wings. I was in the very thick of things when this photograph was taken [1]. Every speck in the picture below, however faint, is a butterfly.

Monarchs have fascinated me for years now. In Massachusetts, a few months prior to my Mexico visit, I'd seen the odd monarch or two flying unhurriedly, seemingly without a purpose. So languid was their flight – the classic flap, flap and glide – that there was no way to tell that each butterfly, following some mysterious signals – its reproductive system having been put on pause, allowing the organism to focus on the rigors of the coming journey – was leaving for a distant forest in Mexico.

It's worth reiterating this: each butterfly that migrates south starts the journey alone and has never made the journey before. When birds make long journeys, there are often older individuals that guide the young. In the case of monarchs there is no guide. The recent discovery that the thin, seemingly inconsequential antennae of monarchs house circadian clocks that help in orientation only deepens the wonder [ref. and figure]. Traveling by day and over land, a monarch, “with a mass 20% that of a penny” [2], covers thousands of kilometers in a two month period.

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by Charlie Huenemann

How wonderful it would be to be a systematic thinker! One marvels at the Aristotles, the Aquinases, the Descarteses, the Kants, and the Hegels and the Marxes (well, the Karl Marxes anyway), the Freuds – those who know how to approach anything, how to incorporate any material into a systematic empire, those who can see the universe as fulfillments of their own plans. It may sound like I am satirizing them, but I really do admire them: I admire their imagination, their enthusiasm, and their persistence. Chiefly I admire their ability to take their own thought so seriously, since every time I have tried to construct a system, it turns into fits of giggles.

What causes such a mindset? Let us first see if we can discern its preconditions – those elements necessary for the possibility of system-building, as it were. One must first be convinced that reality, or human experience, is coherent – a big assumption, granted, but absolutely required for a system. And the coherence must be intelligible to a finite human mind, and specifically the specific mind of the specific system-builder. One must further believe that the coherent, intelligible world order has a kind of hierarchy that allows for some parts of it to be more basic, more foundational, or more universal than others. For the system builder is not so deluded as to believe that all of the facts can be fit within a single head: only the organizing principles need be grasped and kept forever in one’s mental field of vision.

That the world is a coherent, intelligible hierarchy – this much at least must be believed by any would-be builder of a system. But no one is going to leave it at that! To harbor that belief is to have the ambition to explain the hierarchy, and propound it to oneself and to the world. I’d say the belief and the ambition go hand in hand – but then again, if anyone has ever had the belief without the ambition, we probably would not have heard of them. Oh yes, a final thing: the system has to be new, if we are dealing with a genuine builder, and not a worker bee.

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by Thomas R. Wells

The proponents of gun control in America are losing the argument and will continue to do so. Their complacency, typical of the left, that they are on the right side of history has blinded them to the fact that they have chosen to fight on the wrong ground. They keep harping on about guns killing people. As if guns were like cigarettes, and as if the numbers were big enough to matter.

I. The Public Health Argument Doesn't Work

Guns are indeed an excellent killing technology. They are really very good at transforming an intention to kill into its achievement. However, that doesn't mean that they are a particularly significant cause of death and it is rather ridiculous to imply that removing guns from citizens would change death rates much. America is not 42^nd in the world for life-expectancy because of guns, but because of much more significant effects like the social gradient in health.

Let's go into this a little more.

We hear a lot about the large number of deaths caused by guns in America – now up to 33,000 per year. This seems like a big number. It is nearly as big as the rate of death from car accidents (another area in which America is an international outlier, by the way). But 2/3 of gun deaths are suicides. Most of those deaths would still occur if people didn't have access to guns. Many murders committed with guns would also go ahead without them, albeit with a smaller chance of success.

Mass killings by individual loonies get far more attention than they deserve. It feels like there are a lot of them, and perhaps they are even increasing – 133 between 2000 and 2014. But in a country with 320 million people and poor funding of mental health services there are always going to be murderous loonies making the national news somewhere. These atrocities make for wonderful news stories, full of pathos and inspiring great moral indignation. But they are statistically irrelevant to Americans' public health. They are not an argument for gun control.

The overarching assumption that murders are caused by weak gun control laws is weak. The decline of gun control began in the 1980s, but the murder rate in America has actually fallen by half since then (back to what it was in 1950). The reason is that rates of violence have a lot more to do with social conditions and inequality than with particular technologies. Most of America is nearly as safe as Western Europe, but some areas of concentrated hopelessness have the murder rates of Central America. The real causes of violence are something America is particularly bad at addressing, among rich countries, perhaps because the left in America spends most of its time campaigning for things that have little to do to with social justice.

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