h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

Entry: The Mathematician And The Blancmange - A17452136
Author: Icy North - U225620

I've tried to make this simple to understand. If it's not, please let me know.

Over to you...

It makes sense to me

Good one Icy. Very clearly explained.

Thanks guys,

Just added some h2g2 links.

An example of the difficulty of trying to explain something without diagrams. While easy to visualise the sawtooth early iterations in terms of doubling of pyramids halving in height "adding" their heights was mentally challenging .
These iterations became much easier to follow from the link to the Wolfram Mathworld site, though the iterations don't all show the function to the same scale.

Since there is no scale on these graphs I couldn't tell whether the blancmange was supposed to increase in height. It does - the middle crack shows the height of the original triangle (and plateau of second iteration), which due the doubling nature of iteration never gets any higher.

Yes, it is challenging when you do it manually - it's an interesting exercise to go through, though.

I confirmed the maths by writing an Excel spreadsheet to do the iterations. The resulting figure did indeed resemble a blancmange.

I'm trying to write one which does the parabola, but getting the formulae right is slowly turning my brain into something wobbly.

Icy

Flippin' 'eck, Icy! I understood this! Well done

Seconded, ma´gs - although I tried to draw it on a piece of paper - I'm just as bad at 'art' as I'm at mathematics, though.

Thanks both!

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I was enjoying it up until then, Icy. Then you set me up to get bored - which I then did, what with all those formulas and long words.

Maybe you might consider re-wording that section in case other people feel the same.

Apart from that, a good article. Very well written.

Thanks DDD,

I'd like to make it more understandable in that case. Can you tell me exactly where you lost the thread of it, so I can help to reword those bits.

I've tried to explain continuity as 'a line without any gaps in it', and differentiability as 'the ability to calculate what is the slope of a curve'. These are actually very complex topics - 2nd level degree course stuff - and I've taken some serious liberties in presenting them like this. I've invited the h2g2 maths group to comment, and I'm expecting them to rip it to shreds for being too simplistic.

The formulae like y=x^2 and y=1/x are GCSE-level curves, and I would guess that most readers of say 14 upwards would recognise them. I'd be happy to remove them (they're a pain to subedit for one thing), but I'd welcome an alternative way to express the ideas. The first is a quadratic curve, but that's yet another long word; the second is a reciprocal function, which I do mention. If I were to generalise the first, I'd be talking about polynomials, which again is not a common word. I'm sure it's possible - Hawking's 'Brief History of Time' contains no formulae except e=mc^2. What do you think?

Icy

I don't think you should go into technicalities about continuity or differentiability. After all, that's the way it was understood at the beginning until people came with epsilons and limits. I think you'll lose too many people if you say that continuity means the preimage of any open set is an open set

After all you want to show non-specialists why this curve is of any interest; and with a title like this you're bound to attract non-mathematicians!

I like mathematical entries without too many technicalities or formulas. Not always easy to achieve though.

I only got an idea of what was meant by "add the heights" when I looked at the shape given in the link; and even then a bit of shrewd inspection was needed. This needs to be spelled out for the general reader.

When you draw the two mountains, you then need to draw a fourth mountain whose height at each point is the sum of the heights of the first and second mountain at that point (or distance across the page), then the sum of the height of the first and third mountain. Mountain no 4 will accordingly rise twice as steeply, remain flat across a plateau, then fall twice as steeply as the first mountain.

It may be possible to put this in fewer words, but I think some such explanation is needed. "Add the heights" just doesn't say enough. This is another case where an illustration is absolutely required.

Thanks guys - good comments.

I'll try to simplify the continuity/differentiability sections as far as I can, and I'll give some more thought to the graphical construction...

It's a good entry, Icy. I can't think of any way to improve it.

Thanks Gnomon, but I've made a few changes in response to the other comments.

I've made the wording a little simpler in the "How To Draw A Blancmange" section. You are right that adding the heights is a challenge, but I don't think I can add a lot more other than say that it's a little tricky. It's a knack, really - once you've got the idea with sawtooth no 2 you'll not have any trouble completing further iterations. If you get the plateau then you've cracked it.

I've removed the lighthearted "bores me rigid" comment.

I've removed all the formulae for the curves, and simplified the "continuous functions" section a lot.

I've also simplified the "differentiable functions" section, and have extended the theme of imagining the slope of a curve as the gradient on a hill.

I can't really do a lot more to simplify the "Pathological functions" section, as I want to be able to use the quote.

That's it - back to you...

Icy

That addition to the "add the heights" should do the trick

It's not that it's difficult, it's just that you need warning that something extra is coming up, if you're not a

Ah. I see a potential problem.

"Now draw another sawtooth, but this time with four mountains. Again, each will be half the size"

- the word "again" suggests that these mountains are again half the size of the original single mountain, when in fact they are half the size of the mountains in the previous step.

Thanks Gnomon

I've been moving that "again" word from the beginning to the end of that sentence, and back, and it's still unclear. I've now rewritten it explicitly:

"Now draw another sawtooth, but this time with four mountains. Each will be one quarter the size of the original sawtooth..."

Icy

I've had to change the parabola section slightly. You draw your first sawtooth as normal, but at each subsequent stage you multiply the number of sawteeth by two, but divide their height by four.

I've tested it in Excel and it looks OK.

Icy