Sashigane addiction

Travelling again, enjoying the German autobahn and it's punctual train and lovely buses. Irony of course, god I'm starting to hate this country.

We visited first Pauli in Essen, then moved to Münster to plane some oak and "Platane" with Till (Pauli's bf) and then I was picked up to help a friend with some restoration work in his family farm. He said it had to do with wood so I said yes, but he didn't mention powertools and dust, then I said no and spend saturday in the kitchen making pizza and sunday mostly in bed playing with my sashigane. Not in a masturbatory way that is.

I didn't bring my computer and my fingers were itching for writing some posts about this magical little tool and all the stuff you can make with it.

It seems to me that this sashigane thing will go for a long time, together with the mathematics/geometry of roof explained by a phisicist so I'm making a new tag for it: sashigane.

Let's go to it then.

First picture: How to draw a given slope. Say 5/10. You take the edge of timber, mark 5 on one side, 10 on the other, and you got it.

If you don't get it the why (as I didn't) maybe you can use some trigonometry.

The two triangles, the one with sides 5 and 10m and the other with sides 10cos(alpha) and 10sin(alpha) are equivalent, they both have the same slope. Homework, prove this.

Now, for Gabe's last joinery layout the half-roof pitch was not explained so here it goes.

Since the rafter is angled at 45 degrees the run becomes 10sqrt(2) You take a perpendicular to it that connects with the edge (same length since it's at 45 degrees) and the hypotenuse of that triangle is the new run for the side cut of the keta. Look here if you don't know what I'm talkin about:

http://granitemountainwoodcraft.com/2015/10/17/layout-for-simple-japanese-hip-roof/

and check this picture in particular:

In Gabe's case the slope is not touching the edge but it's parallel to the line I draw, thus has the same slope. Questions just ask, I don't know your level of geometry so I assume you got the basics as pytagoras, thales theorems, and sin, cos, tan notions.

So, why addiction? well look at this. You just move your sashigane around and the joints start to appear by themselves. Crazy shit.

 Here I was playing with slopes and realized that the splayed leg problems is trivial with a square and some approximations.

So this is the second lesson of today, and it's how to compute the square root of a number without a computer (great, written in a computer, I see the irony).

So, first of all why you need to compute square roots? Because we live in an eculidian space in the local universe. What do I mean? That the shortest line between to points on the world is equal to the square root of the sum of the displacement in each dimension squared. (That is, Pytagoras theorem is not a theorem but an axiom about what kind of world we live in. In you draw a triangle on the surface of a sphere instead of a plane it does not hold anymore. {I told you that learning geometry from a physicist was crazy, they always take detours to explain things.})

Anyway. Sqrt(100) = 10. That is, 10*10 = 100. Sqrt(156) = ????

Let's say we have a splayed leg with the following diagonal slope of 8/10.

First we need to know the length of the leg eh? So by pitagoras the hypotenuse is sqrt(10^2+8^2).

EDIT: thanks to jason I found my math was horrible. 8*8 is not 56 but 64, so now I edit the approximation accordingly. sorry for that.

Exactly, sqrt(164). 13 times 13 is 169. So let's write it like this: Sqrt(164) = Sqrt(169*(1 - 5/169)) = Sqrt(13^2)*Sqrt(1 - 5/169) = 13 * (1 - 0.015)

CHAN!

The last equal sign is not actually an equal sign but an approx. I use the following formula: sqrt(1+x) = 1 + x/2 when x is way smaller than 1. That's called Taylor expansion but doesn't matter a ball the name, just bear with me or google it.

So now that we know the length of the hypotenuse we can compute how much to decrease the square leg so once splayed appears with a square cross section on the plan.

Say we have a 10x10 cross section beam. The new diagonal length needs to be 10/13 so when it's splayed it's again 10.  Let's write 10 = 13 - 3 so 10/13 = 1 - 3/13  = 1 - 1.5/13 - 1.5/13

So you divide the side in 13 parts by using your sashigane in diagonal so it covers 130mm and mark one vertical line every 10mm then you take 1.5cm from each side and pass it to the diagonal with compass and not a vertical line as I did in the drawing.

Now go and try it for a slope 7/10 and tell me the answer. I go and try to cut a gothic roof in the meantime.

Thanks Jason for catching the error and hope this solve your doubts. I may draw the triangles again so they show the right numbers