Friday, 16 July 2010

A physicists view of teetering high heels

Recently, I stumbled across this website: and was reminded of an idea that I had a while ago: relating the ideas of physics and fashion. OK, it sounds lame, but if Dr. Debbie Berebichez (PhD in physics, don’t you know?) can make a go calling herself “the science babe” then maybe there is hope for making it less lame than other people might.

I then found this article. My first reaction: "are you serious!? They actually press released this!" Now, alright, on the surface it looks like calculating the highest heel a woman can wear is a kind of funny thing to do, and clearly just done for fun. But then you read on and you realise that factors such as “the probability that wearing the shoes will help you 'pull' (in a range from 0 to 1, where 1 is pwhooar and 0 is stick to carpet slippers). If the shoes are a turn-off, there's no point wearing them.”

Well, lets be a bit more academically rigorous about this. Whether the shoes are in fashion or will help you ‘pull’ only changes the likelihood that you will wear them, it’s not physically impossible to!

So instead let’s model the female foot as a right-angled triangle, where the point is the toes and the ankle is the largest part. Now let’s assume that most women (not being ballet dancers) can rotate their foot so that the top of it is 5 degrees from vertical.

Now, theoretically you can fit some kind of shoe under that foot – and no, I don’t care for now if it’s in fashion or if I want to pull when I’m wearing it, as it doesn’t affect the height! So, lets calculate (given your shoe size) how tall the shoe could be.
The result of the theoretically tallest heel a woman can wear depending on her foot size is shown in the graph below. I’ve included my calculation for transparency, feel free to have a go at me about it. Just remember – at least I didn’t include the ‘pull’ factor. Tissh.

At a massive UK size 8, I can wear a heel of almost ten inches! OK, so I wouldn’t want to actually try wearing heels that high, but I’d like to believe it were theoretically possible for me to.

Well, according to Wikipedia

Adult shoe size =  3 * length in inches – 25

So rearranging this, length = (Shoe size +25 )/3

That is the base length of the triangle. Assuming that the distance from top of foot to base of heel is ~ 4 inches we can use pythagoras’ theorem to calculate:

Angle of foot triangle = arctan(4/length)
                                   = arctan(12/(shoe size+25))

We want to add to that the 5 degrees from vertical that we decided on before, so the angle that the base of the foot makes with vertical is:
total angle = 5 degrees + arctan(12/(shoe size+25))

Finally, to calculate the heel height in inches we take:
heel height=cos(total angle)*length

So for me (size 8 UK, biggest possible heel size!)

Heel height = cos(5 deg + arctan(12/33))*(33/3)
            = cos(5+19.98)*11
            = 0.9065*11
            = 9.97 inches!

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