Gregory’s pillar

The article Gregory’s meridian line of 1673–1674: A St Andrews detective story by John Ceres Amson in the 2008 BSHM Bulletin tells how James Gregory, the first Regius Professor of Mathematics in the University of St Andrews, defined a meridian line in the 1670s with an accuracy of better than one part in 2000, an accuracy not surpassed for nearly a century.

This was the first secular meridian line in Europe. (Earlier meridian lines, mostly inside large churches, were used to establish the date of the Spring equinox so as to fix the date of Easter.)

Gregory was a very able mathematician, anticipating results of Newton and Leibniz on the calculus, and establishing the power series for sine, cosine and inverse tangent (though, as I told here, he was himself anticipated by the Keralan mathematician Madhavan (1340–1425)). Gregory’s time in St Andrews was not happy – as Dr Johnson noted somewhat later, the University was not a happy place at the time – and he left for Edinburgh after six years and died shortly afterwards.

Gregory’s pillar, marking the southern end of his meridian line, is marked on the Ordnance Survey map, and stands in the back garden of a farm cottage on Scoonie Hill, a couple of kilometres south of St Andrews. Last weekend we walked that way on a beautiful Sunday afternoon, when I took the picture above.

However, as Amson points out, there is a mystery here: the pillar is not visible from the Old Library where Gregory had his observatory. (This is the case even without the complication of new buildings in St Andrews and the growth of trees on Scoonie Hill in the intervening centuries.) Amson speculates that the original wooden pillar was on the side of the hill in the middle of an agricultural field, and was moved to a safer location in the cottage garden when it was re-erected by David Gregory, James’ grand-nephew and fifth Regius professor, in 1757.

Amson ends his article with a plea for greater recognition for this extraordinary mathematician and astronomer, and celebration of his achievement. Wouldn’t the University’s 600th anniversary be a suitable time to do this?

Regius chair at St Andrews

The University of St Andrews is advertising to fill the Regius Chair in Mathematics. As the name suggests, the appointment must technically be approved by the Crown; a rare distinction! (According to Wikipedia, there are a number of Regius chairs in ancient universities, and some in modern ones created by the present Queen; but Mathematics is the only Regius chair in St Andrews, and the first holder of the chair was James Gregory, about whom I shall hopefully say more later.)

All areas of mathematics are considered. It would obviously be good if the Regius professor could talk to existing members of the School of Mathematics and Statistics. (The advertisement speaks of “synergies”.) It is more important that the professor can start strong new directions of research in the School.

What are the advantages? Speaking from my own experience, I would say: a very congenial department with relatively low barriers between divisions, in an ancient university in a very attractive town; management unusually enlightened among British universities about what the purpose of a university is; strong and motivated students. For a golfer, there is no need for me to discuss the attractions; but if you think a walk is better without being interrupted by people swatting little white balls, it is on the Fife coastal path and close to the Highlands. Not much more than an hour’s travel brings you to Edinburgh.

The LMS and open access

The London Mathematical Society is launching a new open-access journal, to be called the Transactions of the London Mathematical Society, to stand alongside their Bulletin, Journal and Proceedings.

Apart from the obvious reason (that nobody knows what is going to happen in academic publishing and it is best to have all options covered), there are a couple of other reasons that emerge from reading their leaflet.

For example,

The new, purely open access journal will provid a place for authors whose funders, such as those institutions who have signed the “Compact for Open Access Equity”, insist that the papers they fund may only be published in purely open access journals, which would exclude being published in our hybrid journals.

I didn’t know about the Compact for Open Access Equity before, so I looked it up. A quick glance shows that it is a document to which universities, not research funders, sign up; it commits universities to

the timely establishment of durable mechanisms for underwriting reasonable publication charges for articles written by its faculty and published in fee-based open-access journals and for which other institutions would not be expected to provide funds.

This appears to address one of the major concerns about open-access publication funded by page charges, that the distribution of the money provided to fund this would be subject to bureaucratic decisions within individual universities.

But the LMS document suggests that something much more worrying is going on. Can anyone explain why an open-access publication in the LMS Bulletin is less valuable than one in the new LMS Transactions, simply because some other papers in this journal might be behind a subscription paywall? The LMS are good guys: they already offer a reverse moving wall (so that free access to articles is granted for the first six months); their publications are free or reduced price to “low income countries”; and they permit publication of a pre-publication version on the arXiv. So who is driving this?

Their own explanation is that they are not trying to encourage a move to open access, since they feel that existing journals already offer a good service. Indeed, after mentioning that the page charges are currently 1925 pounds per article (with a special introductory offer of 500 pounds for the Transactions), they say,

Of course, mathematicians who do not have access to funds to cover the APC are not obliged to publish in open access journals and they still have the other three journals, offering an identical peer review service. Authors will still be able to post pre-acceptance versions of their paper on the math arXiv.

I wish I found this reassuring.

The leaflets had been left on the table in the common room. In a teatime discussion, one of my colleagues pointed out that, although at present only UKRC-funded research is subject to the open-access requirement, there will be pressure to widen this to all research. The general principle is that all research receiving public funds should be open access, and all universities receive public funds.

Endomorphism monoids of graphs

A monoid is, for me, a set of mappings on a finite domain which is closed under composition and contains the identity mapping. The composition is, of course, associative. Thus, it is “a group without the inverses”.

A homomorphism from a graph X to a graph Y is a map from vertices of X to Y which maps edges to edges. We don’t care what it does not non-edges: they may map to non-edges, or to edges, or collapse to single vertices. An endomorphism of a graph X is a homomorphism from X to itself. As a special instance of a general principle, the endomorphisms of a given graph form a monoid: the identity map is certainly an endomorphism, and the composition of endomorphisms is an endomorphism.

Anyone who has heard me banging on about synchronization recently will know that I think that endomorphism monoids contain the key. A monoid fails to be synchronizing (that is, fails to contain a map which collapses everything to a single point) if and only if it is contained in the endomorphism monoid of a graph. Moreover, we can take the graph to have clique number equal to chromatic number; so the graphs which occur are rather special.

So it is worth asking, what do the endomorphism monoids of these special graphs look like? The answer can be a bit of a surprise sometimes.

Here I want to discuss just one example. The graph X is the n×n grid. (This is not in the sense of lattices and statistical mechanics. The vertices are the cells of an n×n array, and two vertices are joined if they lie in the same row or column. So any row or column is a complete graph of size n.)

There are just two kinds of endomorphisms of this graph:

• automorphisms, that is, endomorphisms which happen to be permutations, and map non-edges always to non-edges);
• endomorphisms which collapse the whole graph onto one row or column.

The first type are easy to understand. We can permute the rows of the grid, and permute the columns (independent of what we did to the rows); and we may if we wish transpose the grid, interchanging rows and columns. The group of automorphisms is what is known as the wreath product of the symmetric group of degree n with the cyclic group of order 2, in its product action. Its order is 2(n!)².

The second type are “essentially” Latin squares. Suppose we map to the first row, whose vertices are numbered from 1 to n. Since there are n² vertices altogether, and at most n can collapse onto any given vertex, each vertex in the row is the image of n vertices in the grid. We can fill in the array by putting into each cell the number of the vertex in the first row where it is mapped. Now two vertices mapping to the same place must be non-adjacent, so not in the same row or column. This means that our array has each of the numbers 1 to n exactly once in each row, and once in each column.

We call two Latin squares equivalent if one can be transformed into the other by permuting rows and columns, possibly transposing, and permuting the entries. Now if f is an endomorphism of the type just discussed, and g is an automorphism, then gf (meaning g first, then f) corresponds to the Latin square where we permute the rows and columns and possibly transpose. Also, if h is any endomorphism, then fh corresponds to the same Latin square with the symbols permuted (and the image may be a different row or column, but that doesn’t affect the Latin square.)

So the non-automorphisms in the monoid generated by f and G correspond to an equivalence class of Latin squares.

One’s first reaction to this should be delight. We have put an algebraic structure (a monoid) on the Latin squares; this should certainly tell us something interesting! Unfortunately, the structure of the monoid is rather boring. Suppose that L₁, L₂, …, L_r are representatives of all the equivalence classes of Latin squares of order n. For each i, we let S_i be the set of endomorphisms corresponding to squares equivalent to L_i. If G is the automorphism group, then each set S_i∪G is a submonoid. Also, the set S_i is closed under composition, so is itself a semigroup. Moreover, the product of S_i and S_j is S_i. So the composition, “coarsened” by thinking of each S_i as a single element, is the rather trivial one: the product of any sequence of elements is just the first one in the sequence.

Here is a picture of the monoid.

We can see more from this.

First, the number of inequivalent Latin squares of order n is huge, not far short of nⁿ². So the endomorphism monoid of the grid graph is vast, and the automorphisms form only a small part of it.

Second, the union of any collection of the S_i is itself a semigroup. So the number of subsemigroups is huge, exponentially large in terms of the order of the monoid itself. This is in contrast to what happens for groups, where a group of order n can have at most n^log n subgroups.

It is not quite clear what the existence of monsters like this has to say about synchronizing monoids …

By contrast, let Y be the complement of the graph X. We have the same two types of endomorphisms of Y. But those which are not automorphisms collapse the graph along rows or along columns onto a “transversal” set, a set containing one element from each row and one from each column. Since there is only one kind of transversal set, up to automorphisms of the graph, the picture of the endomorphism monoid differs from the picture for X in that there is only one semigroup S_i rather than a vast horde of them.

There are a lot of interesting questions inspired by all this!

Random synchronization

Mikhail Berlinkov posted a paper on the arXiv this week proving that two random transformations of an n-set generate a synchronizing semigroup with probability 1-o(1/n) for large n.

His approach was quite different from the one I’d been taking, using much more probability theory and less hands-on combinatorics. In particular, we wondered whether this is the first paper in finite semigroup theory in which a double integral appears!

But this is by no means the end of the story. The semigroup generated by two random transformations surely has many other properties; in proving some of these, I suspect that ideas about graph endomorphims will have a role to play.

So the story continues …

St Andrews Botanic Garden

St Andrews Botanic Garden describes itself, rightly, as a “hidden gem”. Tucked between Canongate and the Kinness Burn, on a 7-hectare site some distance from the old centre of the town (a number of signposts give the distance to the Botanic Garden in yards, with varying accuracy), it has a pond, rock garden, peat garden, glasshouses with both warm-climate and cold-climate plants, and space for educational activities, and is sheltered by trees on two sides, with rhododendrons and other flowering plants in the woods.

The pictures below are from a visit today, the first nice spring day we’ve had, so the flowers are not yet at their most spectacular.

The bad news is that the future of the botanic garden is in doubt.

The site is owned by the University. Since it no longer teaches or researches bootany, it has given over the management of the garden to Fife council. Because of funding cutbacks, the council can no longer afford the full cost of running the garden, and is cutting its funding by 50% (a six-figure sum). The University has proposed that part of the land be sold for development and the money invested to provide a fund to pay for running the rest of the site as a botanic garden. Accordingly, it has applied for planning permission for mixed development of the whole site. It states that it does not intend to develop the whole site but wants the planning process to consider all options.

In the meantime, a steering group made up of members of the Friends of the Garden and the Education Trust has been set up, to consider what to do. A successful fundraising drive to make up the shortfall would be the best solution, but this will not be easy in the present climate. It is easy to say that a single banker’s bonus would secure the future of the garden in perpetuity, but the world does not work that way. (This is a tragedy without villains, unless you count the bankers who brought us to this financial mess in the first place.)

The steering group will report next month.

St Andrews has already lost the famous Byre Theatre to the cuts. It would be a sad day if this lovely garden, or even half of it, were also to be lost. The Friends have a petition which you can sign; it closes on 5 May.

Beechwoods

The most beautiful tree in the world is the eucalyptus. If you know it only from backyard trees in Britain, or plantations around the Mediterranean, you will not agree; but if you have seen mountain ash in the Dandenongs, or ghost gums by an outback river, you may be convinced.

Anyway, it seems that we have a tendency to make emotional attachments to trees, especially those we grow up with; like baby birds, we are imprinted with our early experience.

All this is by way of introduction to a book about beech trees by the naturalist Richard Mabey, called Beechcombings, which I bought in a charity shop recently. (St Andrews is well supplied with charity shops!)

The book is a mixture of history of British woods, reflections on the author’s experience of growing up with beech trees and then acquiring a small wood (Hardings Wood) in the Chilterns, and more general philosophizing.

The beech tree is a relatively recent immigrant to Britain, having arrived here 8000 years ago, only 500 years before the Channel broke through and isolated Britain, and 1500 years before farmers arrived to clear the forests for agriculture. The natural spreading of the beech is very slow; it drops its seeds rather than letting the wind spread them, and does not have anything like the symbiotic relationship of oak trees with jays. (I didn’t know about this; but jays bury huge quantities of acorns in ideal places for new oaks to grow. A single bird, working a ten-hour day, can bury 65000 acorns in ten weeks. These provide food for the jays, but many of the acorns grow, and so the succession of oak trees and food for future generations of jays are both ensured.) So it is a mystery to scientists how the beech spread so far and so fast.

In Britain, its natural range is confined to the south-east, but it has been planted far beyond this. A week ago, we walked the Fife Coastal Path from the Tay Road Bridge to St Andrews, of which a large stretch was through Tentsmuir forest. This is a pine plantation, but in accord with recent forestry policy, beeches have been planted along the sides of the access road, and we noticed many young beech trees thriving in the midst of the pines.

In general the story of British woods, as Mabey tells it, is one dominated by the growing view that trees can’t look after themselves and need to be managed by humans. Then there is conflict about the purpose of the management: do we want oaks for warships, pines for paper, or beeches for fuel? Do we care what the forests look like, and the good they can do for our souls, or do we care only about economic imperatives? As one view or another prevails, large areas of forest are felled, for plantations or agriculture. The scariest story he tells is that of Stansted Great Wood, an ancient oak forest in Suffolk, being sprayed with 2,4,5-T (a component of Agent Orange) from a helicopter in 1967. The National Trust felled Frithsden Great Copse, a beautiful wood of hornbeam, maple and beech, and replanted it with conifers to commemorate Queen Elizabeth’s accession.

These attitudes are nowhere better expressed than in the reactions to the destruction of trees by the Great Storm of 1987. The Tree Council said “Trees are at great danger from nature”, while the National Trust opined “The Great Storm has desecrated the past and betrayed the future”. In fact, the gaps in the canopy caused by the losses of mature trees encouraged, and the rotting logs sheltered, the growth of young trees (except where well-meaning foresters had bulldozed the land clear ready for replanting).

What makes a beautiful tree? Some of the most impressive trees in places like Burnham Beeches are the result of many years of pollarding (harvesting branches from living trees). Beechwood was never in official demand to the extend that oakwood was. You cannot build a ship of beech. It was mainly used for fuel for country people, wagon wheels, and more recently for furniture (Windor chairs for dwellers in the rapidly growing suburbs). Pollarded trees are often more resilient and long-lived than virgin forest, although some people’s idea of “naturalness” may be offended. Looking at the way that different artists have depicted beech trees, from Paul Nash at one extreme to Arthur Rackham and E. H. Shepard at the other, shows how varied can be the features we appreciate in a tree.

Since I have lived in Britain, the beech has become my favourite local tree. (The only native relative in Australia, as far as I know, are the magnificent antarctic beeches of the Lamington plateau.) Partly this is for the vividness of their colouring: the grey trunks; in spring, the dazzling green of the new leaves and the shimmering carpet of bluebells; in autumn, the extraordinary shades of colour from rich yellow to rust brown as the trees shed leaves loaded with toxins.

The spread of beech is chancy: production of seed is very dependent on weather conditions, and typically they produce at intervals a heavy crop of mast with very little in intervening years. (This may also be a strategy to save mast from creatures that eat it, such as commoners’ pigs.) Climate change may have the effect of spreading the tree northward from its current natural range.