Elephants in Stilletos

I was intending to post this yesterday but my brief intoduction to Inktober ended up taking a lot longer than planned, so I postponed this one…

Yesterday’s prompt word for Inktober was “Pressure”. As usual, I thought about several possible avenues for interpreting this prompt, one of which was the scientific definition of pressure as force over area. This reminded me of a fun fact I learned while I was at school, namely that a woman in stiletto heels will do more damage to a wooden floor than an elephant, because although she weighs a lot less than the elephant (assuming average-sized women and elephants), all her weight is concentrated over a very small area compared to the elephant’s feet.

While thinking about this yesterday, it occurred to me that this was all based on the assumption (to be fair, probably a fairly safe one) that the elephant isn’t also wearing stiletto heels. This set me off on a rare foray into cartoon-style illustration:

Probably not one of my best sketches ever, but it was quite fun to make. I also took a bit of time to do my own calculations to verify the assertion that a woman in stilettos exerts more force than a barefoot elephant.

To do this, I looked up a few figures and estimated a few others.

According to an article appearing in the Independent in 2017, the average weight for a woman in the UK is 11 stone. That’s slightly higher than I expected, but I decided to go with that figure. Converting to metric units, that’s close enough to 70kg, so we can use that for our woman’s weight.

Except that it’s actually her mass, since weight is a force (gravity) acting on a massive object (i.e. an object that has mass, not necessarily a particularly large one) and is dependent on the strength of the gravitational field it’s in. We actually need the weight for our calculation (as pressure is force/area), but that’s easy enough to calculate from the mass. Newton’s 2nd law says that force is mass times acceleration (F=ma if you like equations, as I do) and in this case the acceleration is that due to gravity. That varies from place to place around the world but it’s roughly 9.81 metres per second squared. For my rough calculations, I decided that a nice round figure of 10m/s2 would do fine. So our average woman weighs about 700 Newtons.

I didn’t make a note of where I found the figures for an elephant but apparently a female African Bush Elephant weighs on average around 3 tons. I’m not sure if that’s supposed to be long tons or short tons, but either way I decided that just calling it 3 metric tonnes (3000kg) would be close enough. Again, that’s actually the elephant’s mass (everyday language tends to be shockingly imprecise when it comes to such things), and her weight would be 30,000N using the same figure of 10m/s2 for the acceleration due to gravity. Incidentally, I decided that since our woman is (by definition) female, I’d go with a female elephant too (they tend to be a bit smaller than the males) and since I tend to think of African savannas before African forests or any part of India when thinking of elephants, I opted for an African Bush Elephant (a species that’s generally somewhat bigger than the the other two varieties).

That bit was relatively easy. Working out the areas was slightly more problematic, especially for the woman in stilettos. You will probably be relieved to hear that I don’t have any stilettos in my own shoe collection, and I was too lazy to go out and find a woman with high heels so I could measure the surface area of her heels and toes together or figure out how much of her weight would be concentrated on each part of her foot. For the initial calculation, at least, I wanted to work on the assumption that both the woman and the elephant would be standing with their weight evenly distributed across all their legs (that sounds a bit weird for the woman – obviously “all” is just “both” in her case!). A bit of online research revealed that stiletto heels usually have a diameter of no more than one centimetre, but I couldn’t find anything out about the area of the front part of the foot that would be in contact with the ground and presumably bear its share of the weight. I settled for a rough estimate of about 1cm2 for the surface area of each heel and 50cm2 for the surface area of the toe/ball of each foot. For convenience I tweaked the latter down to 49cm2, giving a total surface area of 100cm2 for both feet (heels and toes combined).

The elephant’s foot size was actually a bit easier to determine. Apparently a typical African Elephant has feet between 40 and 50cm in diameter. I decided to give the woman a bit of a helping hand by assuming our elephant had relatively small feet (hence providing less area to spread the weight) and therefore a 40cm diameter, or 20cm radius which, if we assume that the feet are circular, gives a surface area of about 1250cm2 per foot or 5000cm2 for all four feet.

To ensure our final units are correctly expressed as Pascals, or Newtons per square metre, it’s handy at this stage to convert those areas into square metres rather than square centimetres. The woman, standing with both feet firmly on the floor is putting all her 700N of weight through 0.01m2 of the floor, while the elephant’s 30,000N is being spread across 0.5m2 with the net result that the woman is exerting 70,000N/m2 or 70kPa of pressure on the floor, while the elephant is exerting only 60,000N/m2 or 60kPa. So our average woman is indeed liable to do a bit more damage to our delicate wooden floor than our average elephant, though the figures are actually quite close.

The difference gets more pronounced if they both put all their weight on a smaller area. I’m not sure how practical it would be to rest all your weight on one heel while wearing stilettos (mind you, I’m not convinced it’s very practical to wear stilettos in the first place) but suppose she’s able to do so, our woman is now channeling 700N through an area of just 1cm2 or 0.0001m2 which makes for 7MPa of pressure (that’s 7 Megapascals, 7 million Pascals or 7×106Pa if you’re not afraid of scientific notation – it’s definitely much more convenient than long trails of zeroes at either end of your numbers). Assuming that it’s enough of a challenge for our elephant to stand on just one foot, without going up on her toes or heels, she would be putting 30kN through 0.125m2, which amounts to 2.4×105Pa, which is 240kPa or 0.24MPa – significantly less than the woman on one heel.

Since my cartoon was based on the idea that an elephant wearing stilettos would do more damage to the floor than a woman in stilettos, I couldn’t leave this set of calculations without considering the pressure exerted by our elephant if she were to don a set of stiletto heels. Presumably these would have to be custom made and I’ve no idea how big they would be, nor whether she’d wear them on all four feet or just two, so let’s assume that the heels themselves culminate in points the same size as the woman’s ones, i.e. 1cm2 each and the elephant has somehow managed to contrive to stand with all her 30kN of weight bearing down on just one of these heels. That would make for a pressure of 3×108Pa, or 300MPa. As we would expect, our elephant in stilettos would do considerably more damage to any floor than our woman. It’s probably just as well that elephants are not, as far as I’m aware, in the habit of wearing stiletto heels.

I should probably add that it’s been a good few years since I last did this sort of calculation, so I hope I haven’t made any major mistakes with my units or figures, or any assumptions that are too crazy (apart from the basic premise itself, perhaps). Still, I’m fairly confident, at least that the claim made by my cartoon is fundamentally correct:

An average-sized woman in stiletto heels exerts more pressure on the floor than an average-sized elephant…

… unless, of course, the elephant is also wearing stilettos!

(Magnus Forrester-Barker, 2021-10-09)

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).

Update:

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 🙂