My general answer would be: geometry - and specifically trigonometry and circles/arcs. I don't know what the syllabus is like these days, but in the mid-nineties I learned everything I've ever used in woodwork at GCSE; if the syllabus is the same these days I don't think there's any real reason to learn anything more advanced. (I did Maths and Further Maths A-Levels, then went on to a Computer Science degree that was full of maths, and the only geometry I encountered outside of GCSEs was a long way from practically applicable!)
And if you want to learn the mathematics of things in a
particularly useful way, then there's one simple rule: whenever someone shows you a neat trick that helps you do some marking-out or checking particularly easily without having to get a calculator out, then make sure you understand
why it works. It's quite probably the case that there's a simple mathematical principle behind it, and once you understand that you may well find some other things it's applicable to.
phil.p":2vdwgfrf said:
Sometimes 5 - 12 - 13 is more useful than 3 - 4 - 5.
Indeed! In case you're unaware: triangles with sides of 3, 4 and 5 or 5, 12 and 13 inches/metres/whatever will produce a right angle between the smaller two numbers - this is a practical application of Pythagoras' theorem.
In general, when you're using ratios between numbers to find a particular angle, it's always best to use the largest numbers that you can and the numbers with a ratio closest to the ratio of the thing you're measuring helps you do that. So if you're looking to get your 60mm by 450mm frame square, don't mark 60mm down one side and 80mm down the other and check that the distance between marks is 100mm (3x20, 4x20, 5x20) - mark 50mm down one side and 120mm down the other and check that the distance between is 130mm (5x10, 12x10, 13x10).
There are many such sets of whole numbers that work like this. They're called "Pythagorean Triples" and you can find a list (along with a load of Wikipedia's usual jargon-filled explanation) here:
http://en.wikipedia.org/wiki/Pythagorean_triple
Looking at that list, for the above example you may even be better off with the triple (9, 40, 41), which fits the 60/450 shape even better. So you'd measure 54mm down one side and 240mm down the other, and check that the diagonal is 246mm (9x5, 40x5, 41x5).
The reason for this is that you'll often be a millimetre or so out when you make your marks or measure from them, and the longer the distance you're measuring, the less significant that millimetre error is going to be. Two millimetre's difference over 100mm is a 2% error, while 2m difference over 246mm is only a 0.8% error.
(On the same topic of squares, I don't think I noticed anyone mention the easiest application of Pythagoras: if you're making a thing which is supposed to be rectangular, then assuming that all the sides are straight (hold them against each other one way around and the other to check for no gaps) and opposite sides are actually the same length (check with a tape measure), the distance from a corner to the opposite corner will be the same for both diagonals if it's all in square.)
Another useful thing to understand is the number of things you can do with a pair of compasses - which tend to revolve around circles and triangles. There's a lot of useful examples on this chap's YouTube channel, in amongst the various projects and a number of other neat tips:
https://www.youtube.com/channel/UCoCEoP ... 58O-l3ttDQ
If you ever need a smooth transition from one flat side to another, there's a useful trick that relies on the geometry of circles. Since the special property of circles is that the distance from the centre to the outside is the same all the way around the circle, there's also an easy way to find where to position a circle if you want to round a corner off to any particular radius. Simply set your combination square to the radius of your desired circle and use it to draw a line
one radius in from the edge of the two sides you want to round. Where those two lines meet is the point that you start your compasses to draw the circle - it will touch but not go past both the sides that you marked from, assuming they're both straight:
You can do this with inside corners as well - you just have to measure your lines and draw your circle into the offcut bit before you make the cut!
(The reason this works is that when you draw a line one radius away from an edge, any circle centred at a point along that line will just touch but not cross the edge that you referenced the line from. If you draw a circle that is centred on two such lines, therefore, it will simultaneously touch but not cross both edges.)
bugbear":2vdwgfrf said:
Geometry and mathematics are related (vide that nice Mr Descartes, back in the 17th C), and maths can do all that geometry can,
but not vice versa.
As Bugbear I think hints at here: the problem with learning maths to do woodwork is that the academic approach to maths assumes that all planes are perfectly flat, all angles are easily measured and marked, and so on - so it's worth knowing some geometry and trig, but it's also worth knowing some tools to transfer that academic knowledge into practical use. Generally when you're doing woodwork you don't want to be getting your protractor out and measuring angles unless you absolutely have to, because there will be some error in the process that could be eliminated or sidestepped if you take a different approach. Checking the ratio as above is a much easier-to-replicate and easier-to-measure process than sticking a ruler against a square and drawing a line while hoping that nothing moved, so it's a better answer in the practical situation of cutting a bit of wood. Similarly, once you've learned some trigonometry and can work out exactly what length the opposite side of a triangle with a 35-degree adjacent angle is, you'll also want to work out the other two side lengths so you can mark and check them using ratios and get the angle more accurately than you'd be able to do by marking dots around the outside of a plastic Helix protractor. And it's easy, because every triangle is made up of right angled triangles one way or another:
;-)
That was not meant to be a dig - honest!
