The four seasons

In various bits of the world, notably including much of Europe, it is usual to characterize the year in terms of four seasons:


The days are at their longest, the weather at its warmest, plants grow vigorously, many species of animal are busy raising their young while their food is plentiful.


Each day is shorter than the one before, the weather grows increasingly stormy, life is in transition – hoarding resources and preparing to endure hardship, or migrating to warmer climes.


The days are short, darkness abounds, the weather is cold and harsh, there is plenty of rain or even snow, life lies dormant; death stalks the land.


Each day is longer than the one before, the weather is capricious yet delivers ample warmth and rain, life wakes from its slumber and much of it has breeding as a high priority, animals that migrated to warmer climes for winter now return, all look forward to the imminent arrival of summer.

In practice, there are no crisp boundaries between these seasons; there are periods of transition from each to the next (one could even describe Autumn and Spring as periods of transition between the other two) and when those transitions happen is subjective or variable from year to year. Furthermore, the timing of the transitions depends significantly on latitude, as does the amount of difference between the extremes; (within the band of latitudes where this pattern faithfully describes climate, and modulo local variations due to other geographic variables) nearer the poles, winter is longer, summer shorter, spring and autumn more rapid and dramatic, compared to nearer the equator.

Foolish precision

Yet one occasionally sees otherwise sensible people speak of specific days on the calendar as marking the start or end of a season. Sillier yet, the Spring equinox is often described as the first day of Spring – I have heard that more often than the equivalent for Autumn and its equinox, or for the solstices as the starts of Summer and Winter, but those who say it do indeed seem to view these as boundary days. The equinoxes are when night and day are of equal length and the solstices when their lengths are shortest and longest; around equinox, the lengths of day and night vary most rapidly; around solstice, they hardly vary at all. In many cultures, the solstices, or some days close after each, are described as midsummer and midwinter; exponents of these quarter points as the days of transition between seasons thus propound the absurdity of Summer starting on midsummer day.

I can cope with the mid-point of a thirteen-week season not falling in the seventh (i.e. middle) week of it; but falling in the first week of it, or indeed on the very first day of it, makes a mockery of the prefix mid. To be fair, while midsummer is when the days are longest, it isn't when the weather is hottest or when conditions are at their most summery; that comes a month or so later, which is apt to be referred to, for example, as high Summer. The different features by which Summer is characterized reach their peaks at various points in the season, and some of these vary from year to year, so it would be rash to be too literal about centering the season on one of them; but the season should surely find them spread around its middle, rather than starting with one of them.

There is also a certain precision to the quarter points – they are defined by astronomical events and wobble about within the calendar, reminding us always that our calendar is based on approximations to the year's cycle – that goes beyond what one would get from assigning three months to each quarter. Saying that Spring begins in March (say) leaves a certain fuzziness as to exactly when in March, where saying that it begins at the equinox says that it begins on the 20th of March (most years – some years it's one of the days either side of that, but a specific moment in any case). Especially when considering the variations with latitude, the reality is better described by the fuzzier statement !

Charting the year

Now, as noted above, there are transitions between seasons, which means the seasons don't formally partition the year – each characterizes, to at least some degree, a portion of the year; and their respective portions overlap somewhat. In an overlap, the characters of the two overlapping seasons are seen intermixed, sometimes interleaving conditions quite typical of one or the other, sometimes exhibiting hybrid conditions. It is better, thus, to describe each season in terms of a period of time in which its character is typically felt, to a greater or lesser degree. Such periods, for each season, shall naturally overlap, reflecting the reality of the transitions between seasons.

Such a description has the same character as an atlas of the world; while each page has crisp borders, some parts of the world are depicted on several pages. The individual maps that comprise the atlas depict overlapping parts of the world. Each map is, typically, centered on a region that it represents reasonably faithfully, enduring some infelicity nearer the edges – the Earth is roughly spherical, so any representation of its curved geometry on a flat page is necessarily distorted; but a judicious choice of projection can make the distortions small at least within some modest area of each map. Just so, the periods of the year I would use to represent each season would, near their middles, reasonably faithfully match the caricature of their respective seasons; towards their edges, less so – but each period overlaps that of the adjoining seasons near its edges, offering the opportunity to view each season's period's edges from the perspective of another season, that places these dates more centrally and characterizes them more faithfully.

To be sure, a small atlas may neglect some parts of the world – the cartographer may slight some nation by showing it only ever near the edges of maps, where it is less well represented; then the atlas does a poor job of describing that nation, although it does technically describe it. Just so, my atlas of the year, with each season's period as a chart, is apt to describe the periods of transition between seasons poorly – they do not appear near the centres of charts, though each appears on (at least) two. This is, none the less, proper: these are the periods we are least able to describe, in any case, so it is proper that each of our charts recognize its inability to describe them well: no season-based chart pretends to offer a good description of them, when the characterization of seasons represents them poorly.

Smooth manifolds

Now, it so happens, there is a mathematical formalism for describing curved things (smooth manifolds) which is based on the idea of an atlas of charts, with overlaps between charts enabling one to work out how the pieces fit together. Books of maps, depicting the whole world, do effectively constitute atlases of charts in this formal sense, describing the (roughly spherical) surface of the Earth as a smooth manifold; and my proposed description of the year by four overlapping periods, each centred on a season, is formally an atlas of a circle, understood as a one-dimensional real smooth manifold, depicting the year, understood in terms of the various cycles that repeat in time with it.

Since the charts of an atlas can't, in general, faithfully preserve geometry – the shape of geography near the edges of a map is typically distorted to some greater or lesser degree – the theory of smooth manifolds doesn't oblige charts to worry about the geometry; it only requires them to provide co-ordinates that map smoothly from one chart to the next, where two charts describe the same region. Formally, each chart is a one-to-one mapping between part of the manifold and part of a vector space. Where two charts describe overlapping portions of the manifold, one can follow one chart from the vector space to the manifold, then the other chart back from the manifold to the vector space; the resulting composite is a mapping from on part of the vector space to another. For mappings between vector spaces we can define the notion of differentiability, hence of derivatives, differentiability of these and so on; so that we can define that a mapping is smooth if it is differentiable with smooth derivative (which implies, inductively, that we can differentiate arbitrarily many times and still get a smooth derivative). Two charts of an atlas are compatible if the composites – obtained by going from the vector space along one then back along the other, on their overlap – are both smooth. The only constraints the theory imposes on charts are that all charts in an atlas are mutually compatible and the union of the interiors of the parts of the manifold they cover should be the whole manifold.

To deal with the geometric distortions, the theory of smooth manifolds expresses the geometry on the manifold in terms of a tensor field, called the metric, on the manifold. The coordinates implied by each chart give their own representations to the metric, but distinct charts' representations say the same things about distances on the manifold, where they overlap, for all that they express those things differently. In a book of maps of the world, one generally sees lines of longitude and latitude, or some similar co-ordinate lines with given geometric properties on the Earths's surface; these effectively encode each chart's representation of the metric natural to the surface of the Earth. They can be used to work out distances, even in geometrically distorted portions of maps (although it gets more laborious there); and, where two maps overlap, the distances they imply along any given line (e.g. road, river or coast), between any two places (e.g. cities or mountain passes), should agree.

By pushing all geometric concerns off onto the metric, we liberate our choice of charts from these concerns, granting plenty of flexibility as to what charts we can use; we are no longer obliged to worry about distortions, as long as they are smooth. We still have to keep each chart one-to-one, so it can't represent the same place on the manifold by two places in the vector space, though. When the manifold's topology isn't that of the vector space, this means we can't represent the whole manifold by the interior of the part covered by any one chart, though we can often get quite close; we always need more than one chart.

On maps of the Earth, of course, we can violate the one-to-one rule if we want; and cartographers sometimes do – e.g. showing the international date line near both edges of the map, so that you get a useful overlap and can see a neighbourhood around each point in the world, aside from in the polar regions. In the theory of smooth manifolds, we don't do that.

The theory of smooth manifolds may seem a rather huge structure to use for the description of the year – describing the circle is so easy as to usually be deemed trivial. None the less, the richer and more fluid approach to description, that the theory provides, may give us a better way of thinking about the familiar cycle of the seasons. Descriptions of the circle usually involve cutting the circle somewhere, yet brush the cutting under the carpet – angles run from minus a half turn to plus a half turn, or from zero to a full turn, then magically flip back again to the start and we all understand how it works well enough to not think about how it distorts our descriptions – where an atlas provides the natural and elegant way to describe it cleanly.

Extending each chart

Consider a single map from an atlas: it uses a projection to map part of the world onto a flat surface. In most cases, one could extend that projection to cover a larger area of the world: it might become more distorted, so that the good cartographer would rather not do so, but we could do so. Depending on the projection, we could extend it to cover some large proportion of the Earth: simple perpendicular projection from a sphere onto a plane can cover most of a hemisphere, while many commonly-used projections can cover all but (a neighbourhood of) a cut between two opposite points (e.g. from pole to pole). Using such a projection, we can map the whole world with just two charts (e.g. one cutting from pole to pole, the other cutting half way round the equator, from quarter of the way round from the first cut to three quarters of the way round); even with a projection that can only cover most of a hemisphere at a time, four charts suffice (one centred on each vertex of a regular tetrahedron inscribed within the Earth).

Just so, with the circle representing the year, charted by the seasons, one can extend each season's chart outwards – so far in fact that it all but meets itself, so that each chart is almost the whole year. Indeed, the Western calendar, running from the start of January to the end of December, is a single chart that covers almost the whole year – omitting only the moment between the end of one year and the start of the next, where we jump from one end of the line to the other. This chart is centred on Summer and cut in Winter; Spring, Summer and Autumn are all fully covered by it and the passage of time, within them, is faithfully enough represented by it. With just one more chart – e.g. centred on Winter, running from the start of July to the end of June – we'd have a full atlas of the circle. We are, of course, at liberty to include other charts in our atlas, as long as they're compatible, so we can use four, if we like – one centred on each season.

Being liberated from geometric concerns means we don't have to give equal space on each chart to each day of the year; for example, we can make the chart's length of each day be proportional to the extent to which, in some given geographic region, our specific season's characteristics have been witnessed on that day, over a period of several years. Winter days all score low on Summer's days are long characteristic and on life's vigour, but each date in Winter has been warm and sunny in some years; rare though these are, a winter day thus has a non-zero (albeit tiny) length in Summer's chart. Days in a given season then have a full length in its chart, are quite short in the adjacent seasons' charts and almost vanish in the opposite season's chart. Days that normally fall in the transition between two seasons shall have less than full length in each of those seasons' charts and be fairly short in the other two seasons' charts.

One could, indeed, go further and be somewhat more expressive: for example, instead of looking at objective conditions, think back over all the summers of your life and give each day of the year a size according to how prominently it features in that subjective view, distorted by memory. If you're a Scandinavian, St. Hans aften (June 24th, culturally deemed mid-summer) may well feature prominently in Summer's picture of the year; if you routinely go to some music festival with a roughly fixed date each year, it'll do likewise.

Choosing where to cut

On a map of the whole world, one has to chose the bits to leave out, where the geometric distortion gets too extreme, and where to make a cut (to avoid excessive repetition). It's common for folk from each culture to put somewhere important to them near the middle of the map. Choosing what to put in the middle restricts where the cut can go – it has to be equally far away on each side – albeit with as much slack as one allows to how central the middle is.

One can, instead, start by choosing where to make the cut: for example, a cut from pole to pole through the Bering strait would lie in sea most of the way, which is a nice feature for a map of the world's land (this would place the middle ten-and-a-quarter to twelve degrees East of Greenwich, making Florence – city of art and culture, in contrast to the traditional home of an imperial navy – a good reference-point from which to define a central meridian). Choosing the cut implies which parts shall fall in the middle, of course.

There's also a cut through the Atlantic, just West of Iceland (hence also of Africa): it's not as good as the Bering one, through the Pacific, in that it cuts through Greenland (as well as Antarctica, which no pole-to-pole cut can avoid); but it gets ocean most of the way. This would give a map with the Bering strait a little to the East of centre, looking a little like hands of Asia and America reaching out to one another across the void of the Pacific, somewhat reminiscent of Michelangelo's Creation of Adam (albeit some might want to view the latter in a mirror to make the reminiscence feel more natural). But I digress.

Now, in our charts of the year, there's no hard-and-fast reason we should put each season exactly central in its chart, using a cut in the middle of the opposite season: for example, a cut at either end of the opposite season may be more apt, placing the chart's season a little off centre and shifting one's perspective on its relation to the year. For the telling of a story, a cut within the season depicted may even work better – notice how, at the start of this page, I began in Summer and ended in Spring; did that leave you more aware of these more joyous parts of the year ? The bit in the middle isn't always what sticks in the memory. Depending on how we want to use our charts, we can cut each wherever suits us, as long as each chart's cut appears in the interior of some other chart (so that our atlas has a full picture of the whole year).

Valid CSSValid HTML 4.01 Written by Eddy.