Loading...
1. What Are Sangaku? Sangaku (算額, literally "calculation tablet") are colorful wooden tablets depicting geometric problems, often solved and dedicated to Shinto shrines or Buddhist temples in Japan. They were created by people from all walks of life—samurai, farmers, merchants, and professional mathematicians (called wasanka )—from the early 17th to the late 19th century (the Edo period).
Distance between centers of (R) and (r) = (R + r) (external tangency): [ \sqrt{(d-R)^2 + (r-R)^2} = R + r ] Simplify: [ (d-R)^2 + (r-R)^2 = (R+r)^2 ] [ (d-R)^2 + R^2 - 2Rr + r^2 = R^2 + 2Rr + r^2 ] [ (d-R)^2 - 2Rr = 2Rr ] [ (d-R)^2 = 4Rr ] [ d - R = 2\sqrt{Rr} \quad (\text{positive since } d > R) ] [ d = R + 2\sqrt{Rr} ] sangaku math
From first equation: [ (h - R)^2 + (x - R)^2 = (R + x)^2 ] [ (h - R)^2 + x^2 - 2Rx + R^2 = R^2 + 2Rx + x^2 ] [ (h - R)^2 - 2Rx = 2Rx ] [ (h - R)^2 = 4Rx ] [ h - R = 2\sqrt{Rx} \quad \Rightarrow \quad h = R + 2\sqrt{Rx} ] They were created by people from all walks
Center = ((h, x)), tangent to line at ((h,0)). Tangency to circle (R): distance between centers = (R + x): [ \sqrt{(h - R)^2 + (x - R)^2} = R + x ] Tangency to circle (r): distance between centers = (r + x): [ \sqrt{(h - (R+2\sqrt{Rr}))^2 + (x - r)^2} = r + x ] tangent to line at ((h
