Fundamentals
 

How to create an extreme overhang with toy bricks

  • 24 December 2007
  • NewScientist.com news service
  • Ben Longstaff
 
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Parabolic stack
Parabolic stack
Defying gravity
Defying gravity

Your festive dinner is nearly at an end and just when you think you're about to burst, out come the after-dinner mints. Instead of dicing with indigestion this year, why not try a spot of practical mathematics? Your challenge is to build a stack of mints on the edge of a table and see how far you can make the pile jut out without toppling. If you have already guzzled the mints, try it with dominoes or building blocks instead.

OK, it's hardly an extreme sport but you'll be amazed at what is possible. Just ask computer scientist Mike Paterson at the University of Warwick in the UK. Earlier this year, he and his colleagues demolished a long-standing result in mathematics that relates how far the overhang can extend with a given number of blocks - and demonstrated some extraordinary structures on the way. It's great news for anyone wanting to impress friends and family, or trying to arrange an unfeasibly large number of presents under the Christmas tree.

The overhang puzzle has been doing the rounds in engineering and mathematical circles since the mid-19th century. Most people have tended to focus their attention on the simplest stacks, in which the blocks are simply piled one on top of the other, with each block sticking out slightly further than the one below. To achieve the maximum overhang, each block is placed as far out as is possible without it toppling off or being unbalanced by those above it. It turns out that to do this, the top block has to poke out from the one below by a half its length, the second by a quarter, the third by a sixth, and so on, with the nth from the top overhanging the one below by 1/2n. The number of block lengths by which the nth block overhangs the table edge can thus be written as 1/2 (1 + 1/2 + 1/3+... 1/n). This so-called harmonic sum is nearly equal to the natural logarithm of n (written ln n), so the overhang is approximately 1/2 ln n.

Amazingly, you don't need glue or cement to build a free-standing tower with an overhang as big as you like, which is probably why the result has become established in mathematical lore. You'd better have plenty of bricks, though: for the overhang to extend out the length of five bricks, you'd need more than 22,000 blocks.

The overhang puzzle is established mathematical lore. The new finding torpedoes that result

So what happens if you get a bit more adventurous, and try counterbalancing your teetering tower by using more than one block per layer? The few mathematicians who dared to explore these possibilities faced a brick wall. "People looked at the problem with four and five blocks and thought that was complex enough," says Paterson. "Beyond that most people would give up and just say it's chaotic."

Enter Uri Zwick, a computer scientist at Tel Aviv University in Israel. "I'd been familiar with the problem since I was a schoolboy," says Paterson. "Then five years ago Uri brought it to my attention again. We spent many happy hours messing around with blocks. It's a bit embarrassing because we are supposed to be computer scientists, and this is recreational maths."

Working out how far a given number of bricks will get you if you are allowed to use some as a counterbalance was no picnic, however. Even with four blocks it requires some pretty fiddly maths to work out from first principles what the maximum overhang could be. The answer, Paterson and Zwick found, is to forget rigorous theoretical methods. Their experience with computer algorithms and programs inspired them to study the problem in terms of how the forces "flow" through the stack, rather than thinking of it as a static structure. For a given number of blocks, they looked at all the different possible arrangements and then used their programs to find the ones that produced the best overhangs.

Their numerical approach paid dividends. After three years working on the problem in their spare time, Paterson and Zwick smashed through the logarithm limit in a paper that showcased different kinds of stacks that race out far faster than the basic harmonic stack can manage (www.arxiv.org/pdf/0710.2357).

Teetering towers

First off, they looked at "spinal stacks" - essentially harmonic stacks with a counterbalancing backbone of blocks riding piggyback (see Diagram). Starting with a program for optimising the overhangs for spinal stacks, they worked out what distribution of forces would give the best overhang, and used this to find out which arrangements of blocks supplied the necessary forces. With these in place, the stack gains the poise needed to enable a few adventurous blocks to teeter dramatically over the abyss. They showed that the maximum overhang is twice as large as in the harmonic case: in other words, with n blocks to play with, the overhang is ln n and those 22,000 blocks would jut out 10 blocks from the edge.

This was just the start, though. Paterson and Zwick then turned their attention to the "parabolic stack", in which each layer of blocks is offset from the one below it by half a block, like bricks in the wall of a building. Unlike most brick walls, however, the bottom layer contains just one block: from there, the stack gradually widens, going up an increasing number of layers each time it ventures another block outwards (see Diagram).

The big surprise here is the rate of widening, which Paterson and Zwick proved follows the cube root of n. This torpedoes the logarithm result and means you can achieve a large overhang using fewer bricks. For example, to build a tower with an overhang of 15 block lengths, it turns out you would need a mere 17,200 blocks for a parabolic stack, compared to 1.2 million for a spinal stack and over 5000 billion for a harmonic stack.

When they announced their findings last year at the Symposium on Discrete Algorithms conference in Miami, Florida, the news was a surprise hit. "It turned out to be the most popular session of the conference," says Paterson. "The lecture room was overflowing."

The pair have also been building incredible structures called loaded stacks. These contain what are known as "point masses" at key locations - essentially weights of zero size. It might sound like a practical impossibility, but if you imagine the blocks are Christmas parcels, just think of a point mass as adding the odd lead fishing weight at strategic points inside this or that parcel.

Assuming Santa took this unusual approach, what sort of overhangs could he get on his sleigh? Adding only a modest number of point masses to their stacks, Paterson and Zwick demonstrated two astonishing possibilities, one shaped like a vase and the other like an oil lamp with a long tapered end (see Diagram). To cap it all, with the help of three collaborators they have now shown that the fastest rate at which the overhang can increase is as the cube root of the number of blocks (www.arxiv.org/pdf/0707.0093).

The researchers have one last tip for competitive stackers looking to go that extra millimetre with their after-dinner mints: skintling. All you do is angle all your blocks so that their diagonals are perpendicular to the edge of the table (see Photograph: left, top). Since the diagonal is longer than the edge, you'll gain a small advantage, and create an attractive pattern into the bargain.

So what are you waiting for? Sit back and skintle.

From issue 2635 of New Scientist magazine, 24 December 2007, page 42-45
 
There are 2 comments on 1 page   

No Pictures?

By Tim

Mon Dec 24 15:30:54 GMT 2007

I followed the link to see a cool picture of some amazing construction. Instead nearly 2 pages of dry text! How disappointing.

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Parabolic Stack

By E Ainscoe

Tue Dec 25 10:47:06 GMT 2007

Would the illustration stay stable with a 3 block overhang if there was just the first layer of 6 blocks, stabilised by a layer of 5 immediately above it? If not, why not? (I ran out of jenga blocks trying it practically, but it worked for the bottom photo on p44 of the paper magazine - a 1 1/2 block overhang. No jenga shops were open by that time on Christmas eve for additional supplies of bricks!)

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