on the origins of numbers

When I saw the news story about the fraction signs of Linear A, the cuneiform writing system of ancient Minoans, I felt inspired to go looking for the origins of mathematics. I.e., I was asksing myself: where was the transition when natural abilities that you might also find in other animals led to calculating, book keeping, and the whole culture of mathematics that we have today.? I didn't really find that magic spark moment, but I still learned lots of interesting things about different cultures deal with numbers, and how scientists are debating the numerical (or just quantical) skills found in some animals.

The feature is out now:

Are numbers in our nature?

Current Biology Volume 30, Issue 21, 2 November 2020, Pages R1283-R1285

FREE access to full text and PDF download

Animals from insects to primates can distinguish quantities, but their processing is different from the arithmetic we learn in school. (Photo: Luca Ambrosi/Unsplash.)

can Pythagoras take a shower?

During a recent stay at a cosy little hotel in Germany, we had a shower cabin with a square footprint (ABCD) and a sliding door (ab) that operated such that its endpoints a and b slid along the sides of the square, so point a moved along AB, while point b moved along BC.

We wondered if a well-rounded hotel guest – let’s call him Pythagoras (hint, hint) - could fit into the shower cabin and close the door behind him. Specifically, if the side length of the square is 1 meter, what is the maximal radius of a rigid rotund guest that could fit in and close the door?

Answers on a postcard. (This really happened, and the young mathematician in the family worked it out. However, I suspect that somebody, eg Martin Gardner, must have come up with this riddle before.)

paradox lost

My review of the book Paradox: the nine greatest enigmas in science, by Jim Al-Khalili is out in Chemistry World online:

FREE ACCESS

It also appears in the print issue on page 69 (together with lots of other book reviews, worth checking if you're hunting for presents ...).

the world according to Royal Mail

in attempt to get my brain round the staggering price increases, I had a good look at Royal Mail's new price leaflet and found that according to them, Greenland is larger than South America:

Now I do know that applications like Google maps use the Mercator projection because it is easiest to calculate when you want to zoom in and out of maps. However, in my opinion, if you display a static map like this there is simply no excuse for using it. It distorts the relative importance of various parts of the world and is thus offensive to anybody who cares about such "small" places like Africa and South America.

The full leaflet with the offending picture is also online as a PDF file.

what the world doesn't look like

I find the use of the Mercator projection to portray the world offensive in any context (as it hugely distorts the sizes of countries and continents, Greenland isn't really larger than South America, it's a lot smaller!), but in the context of trying to help developing countries (which all appear small and insignificant in this projection) it really drives me up the wall.

Apparently, applications that allow you to zoom in and out of maps, like Google maps routinely use the Mercator projection, as this is the easiest one to calculate (it’s all in squares), but that’s no excuse for displaying the world in a blatantly misleading way, as, for instance, Oxfam does here:

Food price volatility map

I am sure the clever people working at Google and other internet companies can come up with a way to program interactive maps without making tropical countries shrink to insignificance?

If one has to have a cylindrical projection (where all the longitudes and latitudes are straight lines), one could use the Gall-Peters projection:



(Wikipedia)

which with its very unfamiliar look reminds us how wrong the maps are that we see more often. Personally, I prefer the round or oval projections, as they also remind us that our planet isn't rectangular, e.g. the Lambert azimuthal equal-area projection.

PS checking up on Google I realised that even their satellite view is in Mercator projection. Surely the satellites don't observe a cylindrical planet from space?

a very useful formula

I am puzzled that my children's secondary school doesn't seem to teach the most elementary form of the binomial theorem, i.e.

(a+b)² = a² + 2ab + b²
(a-b)² = a² - 2ab + b²
(a+b)(a-b) = a² - b²

In German selective secondary schools (Gymnasium) you get these 3 equations drilled into your head until you know them in your sleep, backward and forward (well at least this was still the case when I went to school), and rightly so, as they are incredibly useful for everything from quadratic equations through to mental arithmetics (allowing you to calculate things like 53 * 47 in a flash). They are known as "binomische Formeln" and have jokingly been attributed to a mathematician called Binomi, though of course the name refers to the fact that they are about algebraic expressions based on two terms.

Looking them up on Wiki, I found that the German entry Binomische Formel, which explains their usefulness in great detail, is linked to the English entry Binomial theorem, which is about the more general version [(a+b)ⁿ], applying to powers of all sorts. The latter would of course be too difficult for most pupils, so I am wondering whether it's only the German system that has come up with the the idea of rebranding the simplest case to make it accessible and useful. (A quick check of the wiki entries in Spanish, French, and Dutch reveals they also focus on the general formula.)

Any clues to this mystery appreciated.

the maths behind the Sarrazin affair

Former SPD politician and Bundesbank manager Thilo Sarrazin has unleashed a torrent of outrage (but also support from some quarters) with his hypothesis that muslims are going to swamp Germany and make up the majority of the population within a century, simply by having more children. In spite of acres of coverage going over the subject matter back and forth, I haven't seen a mathematical treatment of the claim to check whether it could actually happen, so I did my own back of the envelope calculation.

What would it take to make Sarrazin's fears turn into reality? For a 10 fold population increase in a century, we need roughly a doubling per generation. Let's assume that traditionalist muslim families (henceforth: "traditionalists") have an average of 8 children, of whom 4 will follow the traditionalist way, have 8 children again, etc. With this model (key parameters: 50% uptake of traditionalist lifestyle, 8 children per tradionalist couple) the number of traditionalists would double from each generation to the next, so these (or higher) parameters would support Sarrazin's claims. Note that we can safely ignore the 50% of children who don't follow the traditionalist way, as they will gradually adjust their fertility to the prevailing level of the country which is sub-replacement level.

I haven't seen scientific data regarding these two numbers, but my gut feeling would be that both numbers will in fact be lower, so the traditionalist population will _not_ double in a generation, and may not even grow at all. And seeing that net immigration has vanished in recent years, we can ignore that part as well.

So here is my stationary model: Assuming that 1/3 of children from traditionalist households stick with the traditionalist ways and have an average of 6 children, that would leave us with just 2 traditionalists in the new generation to replace the 2 parents we started with, i.e. we have exact replacement.

Of course if the parameters are smaller than 1/3 and 6, the traditionalists will gradually disappear (though possibly not as quickly as the rest of the population).

... and the winner is ...

seeing I posted on the shortlist for the Diagram Prize for the oddest book title of 2009, I have to report the final result as well.

So the winner, revealed in Horace Bent's blog this week is:

Daina Taimina: Crocheting Adventures with Hyperbolic Planes (A K Peters)



I actually voted for this one, if only on the grounds that it represents an interesting way to popularise mathematics. Although I was also tempted by Governing Lethal Behavior in Autonomous Robots, which didn't reach the medal ranks.

PS: if you do want to take your needlework into non-euclidian realms, amazon.co.uk has the book, but you'll need some patience: Usually dispatched within 1 to 3 weeks. Looks like they're not monitoring this prize.

venn diagrams

Car design is of course one of the creative activities we're most exposed to, as all our streets are clogged with the things, so I do tend to have strong opinions on what I am confronted with, even though I'm not going to buy a car any time soon. One of the designs I really like is the outline of the backside of the original Ford Ka, which is like a Venn diagram with the lights defined by the overlap between two sets (best appreciated from straight behind, as the kink will disappear).

I was reluctant to put this on here, as I didn't want to do advertising for cars, but I noticed recently that the new version of the model has switched to a completely different (more Japanese-looking) design, so assuming that this one isn't being built any more, I can praise it without fear of boosting car sales (happy to support the vintage car trade, though).



and of course it looks even better with the reflection of the OU Museum of Natural History ...

numerically confused

I'm getting the impression that the English language is getting increasingly mixed up at its interface with maths, i.e. the use of numbers, singular, plural, etc.

Examples:

* Nobody agrees what speciation means (quote from a news feature highlighted in a read box, Nature last week, page 867). "Nobody" means: Not a single person, mathematically: 0 Surely, to disagree, you need at least two people, so every single person on his or her own would be in agreement with the rest of the set, i.e. him or herself. So the smallest possible level of agreement, if you have as many opinions as people, would be: No two people agree ...

* often heard in speech and also seen in print: She's one of those people who ... If you deconvolute that, it means: "She's one people out of the number of people who ..." As there is no singular to the word people as used here (as opposed to people meaning population), I find this usage completely unacceptable.

* One thing that is common usage and seems to be insisted upon by editors, though I find it highly illogical, and it is not found in other languages I know: "He's one of the writers who doesn't ..." The verb in the relative clause is, of course, attached to the "who", and who refers to "writers," and is thus plural. I have no idea how this got into the rule books, but it's just stupid and wrong.

* Why does the media hate me? asks Martin Amis in the Guardian. I would argue that media is the plural of medium, so it should be used as a plural. Although I can see what Amis is trying to do - he wants to present "the media" as a sinister power that has decided to ruin his career by writing bad things about him. The alternative view, which I find factually and grammatically more satisfying, is that "the media" are many different institutions and people who have separately come to the conclusion that he is a ...

* Often heard and read: "The amount of items ... " When I learned English many years ago, I believe I was told that countable items should be referred to as "number of items", while bulk materials that aren't countable may be referred to as e.g. "the amount of water". There is of course a further complication in that we tend to speak of amounts when talking about money, which is countable, but if a specific unit is used, we could still say the number of dollars, or the amount (as measured) in dollars.

a mathematician's tree

If mathematicians were to design trees, the result would probably look a bit like this sculpture:



which sits in the parks of Magdalen college, Oxford, and which I snapped from the River Cherwell on one of our canoe trips.

silly questions

Review of
Can cows walk down stairs? By Paul Heiney (ed.) Sutton Paperback 2006
and Do cats have bellybuttons? By Paul Heiney (ed.) Paperback 2008

These two books came out of a remarkable institution that lived only for a few years. London-based ScienceLine (not related to the current project scienceline.org) offered scientific answers to any question that ordinary citizens chose to fire at it via phone or email. After a few years and 16,000 questions answered by 8 gurus (teachers? scientists?), government funding for the project ceased and it closed down in September 2003.

These two books are its legacy – two colourful bouquets of vaguely science-related questions plucked from the ScienceLine archives and arranged by non-science author Paul Heiney.

I came across them because the youngest member of my family picked the first of these books and enjoyed it. With the style of the questions very much along the lines of children’s questions (we are told nothing about the age or demographic profile of the people who asked them! They may be mostly children, for all I know.), the short(ish) answers and the funny cartoons, it’s no wonder the books appeal to children, and probably also to grown-ups whose scientific education got stuck at that level.

For the scientist, they are a mixed blessing. Reading most of the questions but only a selection of the answers, I found equal measures of interesting and trivial stuff on both sides. Some of the answers contained silly errors that should have been eliminated at proof reading, e.g. mix-ups of the effect of convex mirrors (in book 2, the answer suggests wrongly that it makes things appear closer). Some answers didn’t match the question (one question was about the limits of the galaxy, and the answer was about the limits of the universe), and some left me unsatisfied, including the answer to the title question regarding cows and stairs. Now I know that they can walk upstairs but not downstairs and that it has something to do with their knees, but I still don’t understand why they can’t walk down. I’m sure cows in the Swiss Alps can walk down the mountainside!

The questions are a reminder of how difficult it is to ask a good scientific question. If these are the 500 best ones, I feel for the people who had to answer the 15,500 questions that were not selected.

Still, the kind of dialogue established by ScienceLine is a good idea in principle and it’s a shame it was discontinued. If I were to establish something like this, I would probably avoid promising to answer every question. If, say, only the 5 best questions get answered every day, and one of them is syndicated out to be printed in a few newspapers for a fee, maybe ScienceLine could even be commercially viable without that government grant that proved so short-lived. At least the number of questions sent in and the popularity of the books suggests that there is a demand for this kind of dialogue.


PS: A nitpicker’s guide to some of the other errors:
Asked whether one is more likely to win the lottery twice in a row or twice in a lifetime, the gurus say “it would be easier if” and go on to answer a completely different question. In fact the question is very straightforward, as the outcomes representing two wins in a row are a subset of those for two in a lifetime. Thus, anybody playing the same lottery more than twice in their lifetime is more likely to achieve the latter result than the former.
Asked how many prime numbers exist, the gurus give the right answer (infinitely many) and proceed with a wrong explanation, saying it must be infinite because the whole numbers are infinite as well. In fact, there is a very simple proof that would have enlightened the readership:
Imagine the opposite were true, and there were only a certain number of primes. Write them all down, multiply all, and add 1. Call the result P. Try dividing P by all known primes. It is not divisible by 2, as there will be a residue of 1. Same situation for 3, 5, 7, and all primes on our original list. Hence P is a prime that was not on our original list, proving our initial assumption wrong.

why music doesn't add up

review of

How equal temperament ruined harmony (and why you should care)
By Ross W. Duffin

Norton paperback 2008

Music, I was led to believe, is a supremely elegant manifestation of pure mathematics. The intervals we know as fifths (think “Twinkle - twinkle”), fourths, and octaves correspond to simple fractional relations between the sound frequencies, of 2/3, 3/4, and 1/2, respectively. And as an illustration, one gets shown the corresponding keys on a piano keyboard.

What nobody told me in the first 44 years of my life is that the intervals you play on the piano do not correspond to the simple fractions cited above. I only started hitting on this problem when my daughter’s cello teacher dropped a hint that the piano was out of tune by default. The piano is actually tuned not in pure intervals but in a system called “equal temperament” for the simple reason that the fractions don’t add up. If you add up 12 fifths, all around the circle of fifths (C - G - D - A - E - B - F+ - C+ - Ab - Eb - Bb - F - C) you get 3/2 ^12 = 129.746. Theoretically, the first and the last C in this series should be seven octaves apart, so their frequency relation should be 2^7 = 128. And not 129.746. And there are even worse clashes with other intervals. So in fact it would be impossible to tune a piano according to the pure intervals defined by simple fractions. This is why we as a civilisation have settled for equal temperament, which means the octave is split into 12 equal semitones.

Equal temperament (ET) is so widespread today that knowledge of the alternatives has gone missing, and even many musicians are unaware of the problems that this compromise solution causes. Duffin argues that some of the “unequal” solutions favoured in renaissance music and through to the end of the 19th century (he dates the total victory of ET to 1917) would still be useful today and that the question of temperament should be considered afresh for each piece of music, taking into consideration the likely intentions of the composer, the context of its creation, and what’s best for its harmonies. This will all be self-evident for practitioners of early music who use historic instruments and temperaments already, but it may be new to many people dealing with the classical repertoire from Bach to Beethoven (who, the author argues, cannot have become used to hearing ET by the time he went deaf).

This argument is all very well and convincing, but it would fill only around 30 pages, so to bulk his pamphlet up to a marketable 196 pages, the author has included lots of repetition (as you tend to do in music!) and biographical profiles of everybody who has ever voiced an opinion on temperament, from Mozart’s father, via the flautist Quantz, through to the cellist Pablo Casals. And cartoons. And diagrams. But all this is redundant in principle, so if you’re just after the meat of the matter, you can probably read the relevant pages within an hour, at a bookshop cafe.

What remains is the impression that music is in fact a lot less mathematically elegant than it is often claimed to be, and that it is a rather messy compromise between pure mathematical beauty and practicability. The good news is that a messy system leaves you free to mess with it, giving performers more freedom. When I play out of tune, I can always claim I am experimenting with different temperaments.

the drunkard's walk

I've reviewed

The Drunkard’s Walk: How randomness rules our lives
By Leonard Mlodinow
Allen Lane 2008
ISBN 978-0-713-99922-8

for Chemistry & Industry, and my full-page review is in issue 22, page 31, which is out this week.

Here's a snippet:

For me, the highlight of his book is the short biography of the gambler, medic, inventor, and arguably father of statistics, Gerolamo Cardano (1501-1576). He published 131 books, invented a part that you will find in every car on the roads today, and made a living from applying statistical analysis to gambling at a time when everybody else considered the outcome of chance events as determined by the will of God. With the spellbinding lives of Cardano, and subsequent luminaries including Pascal (with his famous triangle), the Bernoulli clan, and Thomas Bayes, Mlodinow manages to make the normally boring science of statistics an interesting read. Not to become bogged down in history, he intersperses these parts with many examples of misuse of statistics and probability in modern life.

PS: To avoid confusion in navigating this blog, I've now separated blog entries referring to reviews of my books from those referring to reviews of other people's books:
booksreviewed: my books (passive indicates I'm at the receiving end of reviewing)
bookreview: reviews that I have written (or occasionally, reviews that I've read and appreciated)

Caotica Ana

Julio Medem's 6th dramatic film is a feminist travelogue in time and space. As the heroine travels away from the cave on an island (features reminiscent of Lucia) where she grew up, she becomes aware of a connection to other women who lived before her and died a violent death.

A review in epd-film this month calls the movie "a dream as described by a mathematician". As usual Medem brings his very own geometry of time and space, with multiple layers of philosophy that one has to peel apart by repeated viewing.

What's new here is the strong presence of art. Ana's paintings in the film are in reality those of the director's sister, Ana Medem, who died in an accident in 2000. Over 50 other artists have contributed work that can be seen in the film.

Another revelation is the lead actress Manuela Velles, with her debut performance. Celebrated by the spanish press as "Medem's new muse", she does have the quiet radiance of a typical Medem heroine and reminded me very much of Emma Suarez who appeared in his first three films.

Links related to the movie can be found in its Wikipedia entry

martin gardner

When I was about 30 years younger than today, I enjoyed reading the popular mathematics books by Scientific American columnist Martin Gardner, so I was pleased to rediscover him in this recent magazine feature about him and his legacy. I was surprised to learn that he's still alive (he's only 93), and there were a few other things I didn't know, such as the fact that there are regular gatherings in his honor, and that he wasn't even a mathematician.

There's also a very impressive Wikipedia entry.

random walks

I was shocked to find out from the paper featured on the cover of Nature this week that we (humans) don't just walk around randomly. Analysis of the movements of 100,000 mobile phone users revealed that there is some order in their movement. Presumably they go to work in the morning and come home in the evening, if I may have an educated guess.

They should have restricted the analysis to people who don't have a regular job (like e.g. home-based freelancers like myself, or the unemployed, homeless,pensioners, etc.) I'd reckon that those of us who live life off the commuter's treadmill are closer to a random walk model than the research suggest. And I, for one, really do walk randomly when I get the chance to explore an unfamiliar city ...

maths challenge

I've been moaning about the maths education in secondary schools here (England & Wales) for a while, which essentially teaches children to avoid thinking at all cost and replicate solution schemes just with different numbers.

Now here's Oxford mathematician and pop science writer Marcus du Sautoy with a comment on this problem. As a true mathematician, he puts numbers to the economic problem created by the lack of a proper maths education. While for me it's mainly a cultural thing.

And here's the news story about the report which he refers to. Very depressing stuff all that.

think logarithmically

The natural way of thinking about numbers, according to a report out in today's issue of Science magazine, is to think logarithmically, i.e. to plot them onto a line leaving equal distances between 1, 10, 100 ... The researchers came to this conclusion after studying the maths skills of indigenous people in the Amazonas area.

Now this suggests to me that schools go against nature by teaching children to think of numbers linearly (where 1,2,3 get equal spacing), an effort which has to be reversed as soon as the same children grow up to be scientists, because they have to re-learn the logarithmic way of thinking, where orders of magnitude are more important than individual numbers, as in the progression from metre, to millimetre, micrometre, nanometre, picometre, etc.

What a waste. They should start teaching logs in the first year at school ...

measuring the world

Finally, I got round to reading "Die Vermessung der Welt" (Measuring the World), by Daniel Kehlmann. This is a somewhat fictionalised double biography of Carl Friedrich Gauss and Alexander von Humboldt. In their parallel lives, both did indeed measure parts of the world, one as a mathematician and surveyor on home ground, the other as an explorer and naturalist.

The book is very readable and follows in the tradition of books like "longitude" -- this "mid-brow" terrain is still relatively new to German literature; until recently there was a wide canyon separating the U and the E literature, i.e. the popular and the literary fiction. Novels bridging this gap, possibly starting with Patrick Suesskind's "Perfume" have been rewarded with successful translation deals, and Kehlmann's book is now available in English too.

While it was very entertaining to read, and useful in terms of organising a number of household names of the 19th century into a network of who knew whom (they didn't have facebook back then!), I felt slightly unsure about the fine line between biography and fiction, I mean did Gauss really make that promise to learn Russian as a favour to a Russian prostitute? I'll need to read a proper biography at some point.