Oops! By Shirleen Stibbe
The Background
This is a story about how I killed my friend Nigel. It's an excerpt from a lecture on Mathematical Analysis given at an Open University Summer School. But don't worry! This version is strictly for non-mathematicians, and you can safely ignore any pictures which look even vaguely mathematical if maths makes you feel queasy. If you want to see how the story illustrates some of the basic concepts of mathematical analysis, you can find a copy of the entire lecture on the Summer School page of my website, shirleenstibbe.co.uk PS: If you've come across fractals before, you may recognise one of the main players in this sorry tale. His name is Mr Sierpinski.
My friend Nigel
Nigel is a carpet seller, He's not just any old carpet seller, though. He happens to be the best carpet seller in the entire universe! There's nothing he doesn't know about carpets. And nothing he doesn't know about selling. The trouble is, there's quite a lot he doesn't know about mathematics. His big mistake was coming to me for assistance. Although we'd been close at one time, I hadn't seen him for quite a while. So I was surprised when he turned up one day seeking my help with a problem. He was negotiating the sale of his most expensive carpet to a new client, a Mr Sierpinski, and was having difficulty working out the implications of the contract.
Nigel's most expensive carpet
The carpet was stunningly beautiful, and Mr Sierpinski was blown away by it. The problem was he also loved his polished wooden floor with a deep and abiding passion. Mr Sierpinski, being a mathematician, had come up with a really smart solution to his dilemma. Suppose you divide the carpet (which is square) into nine equal squares, and remove the middle square, so the beautiful floor shows through. Then divide each of the remaining squares into nine equal squares, and remove the middle one – and so on. You should be able to do it at least a hundred times, since the room is so big. Nigel had agreed to give it a go, and that Mr Sierpinski would pay only for the actual amount of carpet finally laid. Seemed OK to Nigel – it was an eye-wateringly expensive carpet. He asked me to calculate how much carpet he would have to supply, and therefore how much money he would make on the deal.
Mr Sierpinski's big idea
OK- down to business. Size of room: 900 square feet easy start. The next task was to work out how much carpet would be left after 100 steps - a bit harder. So I hauled out my maths kit (reference books, stack of paper, five pencils, three erasers, hot towels, bucket of coffee) and plunged in. It wasn't long before I spotted a pattern which made it a doddle to find the proportion left after any number of steps. A bit of finger work on the calculator gave this result: Steps
Proportion left
Area left
20
0.095
85 sq feet
50
0.0028
2.5 sq feet
100
0.0000076
1 sq inch
Oh good grief! One square inch? I wasn't looking forward to breaking the bad news to Nigel ‌
Nigel's Carpet Problem 30 ft
Total area = 30 × 30 = 900 sq feet = 1 unit an = units remaining after the nth step
a0 = 1 a1 = 8/9 × a0 = 8/9
…
a2 = 8/9 × a1 = (8/9) 2 an = 8/9 × an–1 = (8/9) n Aha! The proportion left after n steps is 8/9 multiplied by itself n times.
It wasn't quite as bad as I expected. Of course, Nigel went ballistic when he saw the bottom line. But fortunately, when he's angry he lapses into an obscure Quenyan dialect, so my ears weren't offended by his comments. There was nothing I could do to comfort him, and he left – a deeply unhappy man. I thought I'd never see him again after that – even though it was hardly my fault! But blow me if he wasn't back again only three days later. Apparently, he'd gone to see Mr Sierpinski to turn down the contract. But before he had a chance to say a single word, Mr S said that he'd come up with another idea. It sounded all right, and Nigel thought that it might retrieve the situation. He just needed my help to do the maths – he didn't want to be caught out again.
Nigel's comments
✯❈ ❂◆❇❇❅❒✁ ✷❈❁▼ ❁ ❃❏❍❐●❅▼❅ ❂❁▲▼❁❒❄✁ ★❅ ❋■❅◗
Mr S had been worried that the cut edges of the carpet might look somewhat ragged and unfinished. So his idea was to sew a beautiful gold braid around all the edges. He invited Nigel to undertake the work. So it was back to the maths kit for me, to work out how much gold braid Nigel would need. OK. So the perimeter of the room is 40 yards – call that one unit. Now … It took three whole days and several buckets of coffee to come up with the formula. And then I crunched the numbers: Steps
Units
Length of braid
10
3,636
82 miles
20
66,130,931
1,502,976 miles
50
3 × 1020
9 × 1018 miles
To Alpha Centauri and back 190,000 times
Nigel's Carpet Opportunity 10 yards
Perimeter = 4 × 10 = 40 yards = 1 unit bn = units of braid needed at the nth step b1 = 1/3 b2 = b1 + 8 x b1 /3 = (1/3)[1 + 8/3] b3 = b2 + 64 x b1 /9 = (1/3)[1 + 8/3 + (8/3) 2]
… bn = (1/3)[1 + 8/3 + (8/3) 2 + … + (8/3) n-1] = (1/3) [(8/3) n - 1] / (8/3 – 1) = (1/5) [(8/3) n – 1]
I thought Nigel would shoot me when I told him the bad news. He was absolutely delighted! "But Nigel," I said, "don't you see? It's absolutely impossible. There just isn't that much gold braid in the entire universe" His reply alarmed me. "Ah – but Mr Sierpinski doesn't know that! "This is a job for life. I'll never have to sell another carpet and I'll still be rich. That gold braid doesn't come cheap, you know!" He was euphoric. He had visions of a golden future as a wealthy and well-connected socialite, being invited to the Queen's garden parties – the works! I didn't have the heart to spoil his dreams with mere reality, and he left - a truly happy man.
Nigel's vision
The next time I had news about Nigel, he had - in a sense - achieved his goal. He was in fact a guest of Her Majesty – but not quite in the way he'd expected. HMP Broadmoor is a pretty poor substitute for Buckingham Palace. So what happened to land Nigel in jail? Well, it appears that he started work on Mr Sierpinski's carpet, and at the end of the first month he asked for his month's pay. But Mr S is a cunning old mathematician. "Oh no," he said. "You'll only be paid when the job is complete." Nigel isn't one to hold back on his emotions. He really lost it this time! The pictures of Mr Sierpinski from the police files, before and after Nigel had finished with him, tell the whole story.
Crime Scene Report Case number: 347/26A!
Waclaw Sierpinski 1882 - 1969
About three weeks later, I had a letter from Nigel, asking (again!) for some mathematical help. He was planning a breakout – couldn't stand the food in Broadmoor – and needed a formula to program a flightpath for his carpet. He sent me a picture of a peak he'd have to fly over.
4.7 km
5 km
It was a nightmare trying to find a mathematical function which would generate a shape like that – but I'm not one to give up easily, and eventually, only a few weeks later, I had it! I sent the formula to Nigel straight away, along with a suitably filled cake – anything to help an old mate! AND I NEVER HEARD FROM HIM AGAIN!
The formula f(x) = 3e-3x (tan (loge(x + 1) ) + 10x6)
The cake
I did get a call from the police, however. They explained that Nigel had broken out of prison. They'd been tracking his progress from a helicopter, but he'd suddenly disappeared completely off the radar screen. They were absolutely baffled. Could I help? I told them I knew nothing about it, of course. Well, you have to protect your friends, haven't you? But it did start me wondering whether my formula might have had something to do with it. It looked like a nice, well-behaved, continuous path but ‌. I'd initially plotted 50 points. Since the spreadsheet merely joins the plotted points, could the path misbehave in between? Just to be sure, I plotted 1000 points. A blip appeared between 3.7 and 3.9. Hmmmm ‌ So I plotted 10,000 points just in that interval, and OMG! The function is catastrophically discontinuous! The path splits into two parts. The left branch suddenly turns upward, and goes up forever, the other goes downward, and down for ever!
The catastrophic discontinuity f(x) = 3e-3x (tan (loge(x + 1) ) + 10x6)
50 points
1,000 points
Discontinuity at
x ≈ 3.81 3.7
3.9
10,000 points
And that is how I precipitated the demise of my friend Nigel – all for the want of a bit of mathematical rigour! And you know the worst part? It's not knowing which direction he was travelling in. Upward branch? He's headed towards UDFy-38135539 (look it up). Downward? Don't even think about it. And just in case you were wondering what happened to Mr Sierpinski – well that's not a pretty story either. When the medics finally got the gold braid off his mouth, the guy started calling urgently for ... a carpenter! He'd had this idea for a coffee table, you see. You take a large cube of wood, divide it into 27 equal cubes, and remove the centre cube. Then divide each of the 26 remaining cubes into 27 equal cubes ....... The men in white coats were able to secure a place for him in a very pleasant facility. And now he spends his days happily building a gasket by dividing a triangle into 4 equal triangles and removing …….. THE END
Which way did he go?
Mr S's coffee table
Mr S's gasket