Building yurts using vectors

At granny's house, it always seems to go the same way after lunch: Bill and his cousin chase each other around the dining room, while the adults try to make head or tail of their toys.

This Sunday was no different: the toy in question comprised a large number of red straws and fewer, but still many, balls with holes in. The holes were spaced interestingly: around an equator, eight holes were evenly spaced; the two poles each had a hole; and halfway between the equator and each pole were four evenly-spaced holes.

Put another way, if you looked along the equator, there were eight holes. If you looked around one polar circumference, there were also eight holes; at right angles to both of those, there were eight holes, too.

Of course, the instructions didn't put it in such clear terms -- in fact, there weren't really any instructions. Just a picture of a yurt made using the materials before us.

Stand back, I thought, I'm a topologist!

And then I thought, oh crap. I'm a theorist. Actually building stuff? Ut-oh.

I managed to identify the triangles and squares from the diagram. The first layer went together just fine. The second, more complicated layer, worked after a fashion. But then... it seemed that the top layer required the struts to come out of the balls at angles where there weren't holes -- no matter how the thing was orientated.

I've not proved this yet: there were five struts meeting at these balls, and relative to some coordinate system, they needed to leave in the following directions:

$\bi$
$\bj$
$\bi + \bj + \bk$
$\bi-\bk$
$\bj-\bk$

(We could find several arrangements where the two lower diagonals and the horizontals fitted, but the upper holes weren't aligned properly.) If you can get the angles right, I'd love to know about it!

After bending a few struts and watching them immediately flip out of their holes, narrowly avoiding injury to bystanders, I came up with another arrangement that involved a square base rather than hexagonal. This was where the Core 4 came in: I wanted to know the angle between the diagonal struts (in, say, the $\bi + \bj$ and $\bi + \bk$ directions).

So, without paper -- remember, I'm at a family gathering, and while it's socially acceptable to be puzzling over a children's toy, whipping out a notebook to do sums is slightly frowned on, even around people who know me -- I realised I could do it with a dot product!

The dot product of those two vectors works out to be $1\times 1 + 1 \times 0 + 0 \times 1 = 1$ -- who needs paper! The moduluses of the vectors are both $\sqrt{2}$, so the cosine of the angle between them is $\frac{1}{\sqrt{2}\sqrt{2}} = \frac 12$. A good argument for knowing your cosines, this: it enables you to say 'oo! I can build equilateral triangles!' at granny's house without writing anything down.

Bill's favourite game at granny's was chasing Jessie the very messy dog around. Now it's pulling himself up to standing on a magnificent, triangle-enforced yurt designed by his dad. Hooray!

Colin

Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.

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I teach in my home in Abbotsbury Road, Weymouth.

It's a 15-minute walk from Weymouth station, and it's on bus routes 3, 8 and X53. On-road parking is available nearby.

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