# radial rafters

I was reading Fan of the Fan (VI) and became curious about the mathematical basis for the spacing on the rafters.

I developed a spreadsheet based on his numbers and cobbled together some formulas to understand what is going on. I eventually developed the following formulas:

• L is the length of the perpendicular (shortest) rafter
• st = L×q   is the "tangent spacing" of the rafters. It is the length of the short side of the right triangles in his diagram
• θ0 = 0
• Ln = L / cos(θn)
• θn+1 = θn + atan(st / Ln)
It turns out that the exact value of L does not affect the value of any θ, and is only relevant when it is time to actually cut the rafters. Any computer scientist can type up a function f(q,m) which will iteratively compute θm.

If you want to figure out the proper q value to make θm = 45° for any particular value of m (the number of rafter spaces) I am somewhat confident there is no closed-form solution, so you'll have to experiment with various values of q until you get it right. Newton's Method is an obvious choice and would allow you to develop the carpentry tables Chris is referring to. Chris Hall's diagram of rafter tip spacing