Bad drawings but wonderful prose

It’s not all that long since I last (and indeed first) referred to the Math With Bad Drawings blog within these pages.

However, a post that appeared there the other day is such a peach of a short story that I couldn’t refrain from bringing it to your attention.  It is, as befits the nature of that blog, quite mathematical in character (and, more specifically, about differential calculus – though it doesn’t really go into the gory details) so if that sort of thing scares you too much, feel free to run away and hide behind the sofa until my next post (which almost certainly will be about something completely different).

Assuming you’re still here, the story is called The Differentiation: A Survivor’s Tale and is, uniquely among all the mathematical fiction I have ever read (which is a fair amount, over the years), told from the perspective of the exponential function.  The whole thing is firmly based on the behaviour of different classes of functions under the operation of differentiation and I suspect it would be fairly incomprehensible to anyone without a reasonable grounding in calculus, though it could be quite a useful way of helping to remember the general principles, without getting bogged down in the technical details, for somebody who is just learning the subject.  Given the pedagogical nature of the blog as a whole, I suspect that may have been at least partially the author’s intent.

The story also has a nice twist in the tail that makes it almost work as a social commentary on something or other (though I can’t say more without spoiling the punchline for anyone with sufficient mathematical background to follow the story in the first place).

Perhaps the best categorisation of it is as a mathematical horror story, which is one of the ways it’s been tagged on the original blog.  That works on at least two levels as for the mathematically inclined it is quite a chilling tale and for anyone else the very fact that it is mathematical is probably sufficient to induce a cold sweat.

Anyway, I should probably refrain from further analysis and let the story speak for itself, to those who have ears to hear.

(And, yes, there was a stealth pun in that last sentence, since differential calculus is one of the major subdivisions of the branch of mathematics known as real analysis.)

A gift from Wales to the World

One of several blogs I keep an eye on is the aptly-named Math With Bad Drawings (though, actually, I think the drawings do have a certain charm and they are in any case done with pedagogical rather than aesthetic intent).  This blog is by an American mathematician (hence the mis-spelling of maths 🙂 ) and consists of illustrated essays on a variety of mathematical topics.

I was recently flicking back through the archives of this blog and came across an interesting post that I didn’t notice when it first appeared, last December, even though I was following the blog by then (I guess it was pretty close to Christmas, which is generally a pretty busy time when it’s easy to skip over blog posts). It is a post that describes itself as a brief biography of the equals sign (=).

You may be thinking that this isn’t the most enthralling of subjects and, although a mathematician myself (with a fairly keen interest in mathematical notation and history to boot), I’d be inclined to agree with you.  However, here’s the exciting thing I learned from the post: the equals sign was invented in Wales (*).

The article doesn’t actually contain all that much information about the early history of the sign, though it has some fascinating stuff about its meaning and usage, as well as related symbols like > and <.  There was just enough detail to enable me to hit Wikipedia and do a quick Google search for other sites to cross-check the facts (not very extensive research, I know, but probably sufficient to establish that Ben, the author of the MWBD blog, wasn’t just making it up).

Apparently the first recorded use of the equals sign was in a book called The Whetstone of Witte, by Welsh mathematician Robert Recorde, published in 1557.  It is believed that Recorde invented this sign; before this, people used to just write “is equal to” (or words to that effect) when they wanted to indicate equality, so the sign was definitely a very convenient shorthand.

The same book is also credited with introducing the plus (+) and minus (-) signs to the English speaking world, though they (unlike =) were already known in other parts of the world so presumably Recorde became acquainted with them through perusing literature in other languages, or perhaps corresponding with other mathematicians, rather than re-inventing them independently.  In any case, the book definitely had a significant impact on the development of mathematical notation – and the importance of having good notation for being able to develop mathematical ideas should not be underestimated.

(*) Actually, my statement that “the equals sign was invented in Wales” is probably not quite accurate (the original article phrases it as “the equals sign was born in Wales”, which is little better).  Robert Recorde was indeed Welsh (born in Tenby, Pembrokeshire) but he seems to have spent most of his adult life in Oxford, Cambridge and London (where he was a physician as well as a mathematician) so it’s more likely that the equals sign was born/invented in one of those places.  Still, I think it’s fair to credit it as a Welsh invention.


Happy Birthday, Albert

Today is Pi Day.  I have blogged about it for the past couple of years, so this time I’ll content myself with wishing you a Happy Pi Day.

Today is also Albert Einstein‘s 135th birthday.

A few months ago, I came across an excellent quote by (or at least attributed to) Einstein.  Apparently he said:

Everybody is a genius.  But if you judge a fish by its ability to climb a tree, it will live its whole life believing that it is stupid.

I’ll leave you to think about that one.

Incidentally, Pi Day was the subject of  a Google Doodle back in 2010, which was shown in quite a few countries but not, apparently, in the UK.  Einstein’s birthday was the subject of a Doodle way back in 2003, which was shown globally (although I don’t remember seeing it).


I actually wrote this post shortly after discovering that quote, although I decided I’d save it for this year’s Pi Day / Einstein’s birthday.  Since then, I’ve found another cool Einstein quote which I thought I’d also share (with analysis left as an exercise for the reader):

If a cluttered desk is a sign of a cluttered mind, of what then, is an empty desk a sign?

NB I’m not sure about that punctuation (or the provenance of the quote) but that’s how it appears in the picture I saw it in on Facebook, so I’ll leave it as it stands.

Happy Pie Day

In case you’re wondering if I’ve got the date wrong and forgotten how to spell, today is not Pi Day (an international celebration of the mathematical constant π).  I discovered earlier today that it is National Pie Day in the United States of America.

This annual festival, which I’ve never previously heard of, is organised by the American Pie Council (I kid you not!), an organisation which, according to Wikipedia, is committed to “preserving America’s pie heritage and promotes America’s love affair with the food”.

I suppose it’s no stranger really to have a day celebrating a food than it is to have one celebrating a number.  As far as I can tell, though, the date of this celebration is entirely arbitrary whereas Pi Day is celebrated on a date of special significance to the thing being celebrated (14th March, or 3.14 according to one way of writing the date).

Although it is not officially an international celebration, I see no reason not to celebrate Pie Day outside the United States.  After all, pies have been in existence since long before the Pilgrim Fathers sailed to their brave new world (the first ones, apparently, were found in stone-age Egypt) and so, while I don’t deny that the US has a rich pie heritage, they certainly can’t claim that pie is a uniquely or originally American invention.  (OK, I suppose they could claim it, but they’d be wrong!)

If I’d had more time to plan things, I might have gone for a full banquet of pies of the world.  As it was, I had to settle for a few miniature pork pies and apple pies and a slightly larger Bakewell tart with which to mark this auspicious day.  Perhaps next year, I’ll push the boat out a bit further.

Incidentally, this is the second time this week I’ve come across an American national celebration which seems worth importing.  It was National Hug Day (aka National Hugging Day) on Tuesday.  This, like National Pie Day (as far as I can tell) is an entirely unofficial celebration and not a public holiday but it does have its own website and seems to be taking off in other countries.  Interestingly, at least a couple of languages there seem to describe it as International Hugging Day (Международен ден на прегръдката, which is apparently in Bulgarian –  I thought it was in Russian until I looked up the last word, which I didn’t recognise) or World Hug Day (Weltknuddeltag, which is in German).

I think pies and hugs are both things worth celebrating – though both should be enjoyed a lot more than once a year!

On The Fine Art of Compromise

This year I have celebrated (and blogged about) both Pi Day and Tau Day.

If you read slightly between the lines of my Tau Day post, you may have correctly got the impression that, in principle, I’m in favour of the idea of  τ, which is the  same as 2π (i.e. the ratio of the circumference of a circle to its radius), as the more fundamental constant (mainly because it gets rid of the factor 2 in quite a few formulas and therefore renders them a little bit more concise and beautiful) but, because I tend to be (or at least think of myself as) quite pragmatic (or maybe it’s because I’m a pessimist), I don’t see any great likelihood of τ replacing π in general usage anytime soon (and, looking on the bright side, at least π gives us the opportunity to make jokes about pumpkins).

With all that in mind, it’s perhaps not surprising that I particularly enjoyed today’s installment of the xkcd comic.

Of course, pau isn’t a Greek letter.  According to my favourite fount-of-much-knowledge, however, it is an alternative name for bao (aka baozi), a type of Chinese steamed bun which, co-incidentally cropped up in an episode of Firefly (just to link this into yet another recent post on my blog).  Therefore, if we were to adopt the compromise solution of pau instead of pi or tau, we could celebrate by eating bao (and perhaps watching Firefly, or at least the episode “Our Mrs Reynolds”).  It’s an unfortunate linguistic coincidence that the word bao sounds very much like the Welsh word baw, meaning mud and often used as a euphemism for certain other similarly coloured but somewhat less pleasant substances, as in the phrase baw ci (“ci” being Welsh for “dog”).

There is apparently also an Indian bread, from Goa, called pau, and a Hawaiian feather skirt called a pāʻū.   These could also make an appearance in a celebration of Pau Day.

‘Tis the season…

We’re about as far as we can get from Pi Day, and a fairly long way from the related Tau Day and Pi Approximation Day.  However, we are now fairly well into pumpkin season, which gives me a good excuse to share this mathematical / culinary riddle that I came across the other day: What do you get if you divide the circumference of a pumpkin by its diameter?   (I’ll put the answer at the bottom of the post in case you haven’t worked it out by then – although the first two sentences should give you a pretty hefty clue.)

Speaking of pumpkins, I decided this year (for the first time) to have a go at cooking with one and, having concocted this idea about a week ago, actually got round to buying and cooking a small pumpkin today.

There are many things you can do with a pumpkin and I decided to go for one of the standard ones – pumpkin soup.  Rather than be boring and follow a recipe, I opted for my usual experimental cookery approach.

I started by chopping up a couple of spring onions and a clove of garlic and lobbing them into my soup pot (which is to say, my biggest saucepan) and letting them simmer for a bit, with just a little oil.  In retrospect, that was perhaps a mistake as it took much longer to chop up the pumpkin (which was slightly underripe and quite tough to cut through, even with my nice big, reasonably sharp cleaver) and the oniony bits were ready long before the pumpkin was.  In future it would probably be good to prepare the pumpkin first, or at least make a start on it, before putting the onions onto the heat.

I removed the pumpkin seeds, which I put aside for further attention, and the skin, which I put in the compost bin, and then chopped the pumpkin flesh into fairly small chunks and put them into the pot, along with some boiling water.  For seasoning, I added a small amount of salt, a fairly generous amount of black pepper and paprika and a little bit of nutmeg, as well as a couple of bayleaves and a thickish slice of lemon, cut into quarters.  I brought that lot to the boil and then left it to simmer for about three quarters of an hour, before mushing it up with a potato masher (in the absence of a blender) and adding a bit of cornflour to thicken it.  After ten minutes more simmering it was ready to eat with some nice, fresh bread.

It seemed a shame to throw all the pumpkin seeds away, so I gave them a quick wash and then toasted them lightly in a dry frying pan.  There were too many to fit in the pan all in one go, so I did them in two batches.  The first I seasoned with a bit of salt.  For the second batch, I added some black pepper and paprika as well.  They seem to have turned out quite well, although I haven’t tested a very large sample just yet.

The answer to the riddle, in case you were still wondering, is pumpkin pi 🙂

Happy Pi Approximation Day

Back in March, we (or at least some of us) celebrated Pi Day and in June we (or probably rather fewer of us) celebrated Tau Day.

Today is the turn of Pi Approximation Day.

As I wrote on Pi Day, the mathematical constant π is an irrational number, which means that its exact value can’t be written down as a fraction or a finite decimal expansion.  A decimal approximation that’s good enough for many practical purposes is π≈3.14, which is why Pi Day is celebrated on 14th March (3-14 in ISO date format, without the year).  A slightly more accurate figure is 3.14159 (and beyond that, I’d have to look it up to be sure).

There are, of course, many fractional approximations to π (i.e. ones expressed as a ratio of two integers, aka whole numbers).  Probably the simplest one is π≈22/7.   That’s 3.14286 to 5dp, so it’s pretty close and certainly good enough for everyday use (essentially, any situation where 3.14 or 3.142 is a close enough approximation, which is most of the time in non-high-precision situations).  It’s also, conveniently, today’s date in standard British date format (omitting the year), which gives us another excuse to eat pies and do all the other stuff we do to celebrate our favourite mathematical constant.