Construct geometric line with nested root

[raphael3d]

well it is easy to construct sqrt(2) with a triangle with two sides of length 1.
but what about sqrt(2 + sqrt(2)) or the iteration sqrt(2 + sqrt(2 + sqrt(2))).

the question is how to construct a line with length sqrt(sqrt(2)) i guess(beginning with lines of length 1), but i am not sure.

 

[sjb-2812]

well it is easy to construct sqrt(2) with a triangle with two sides of length 1.
but what about sqrt(2 + sqrt(2)) or the iteration sqrt(2 + sqrt(2 + sqrt(2))).

the question is how to construct a line with length sqrt(sqrt(2)) i guess(beginning with lines of length 1), but i am not sure.

Just as you can construct a line of length root2 by drawing a triangle with sides 1 and 1, you can then extend the line of root2 to e.g. 5 + root2 by adding a segment of length 5 to the end, or convert it into root(root2 + 1) by drawing a perpendicular of length 1 on the end of the root2 line and creating a new hypotentuse.

 

[raphael3d]

a prependicular of length 1 on the end of the root2 line means root((root2)^2 + 1^2) = root3 which is not equal to root(root2 +1)

 

[sjb-2812]

a prependicular of length 1 on the end of the root2 line means root((root2)^2 + 1^2) = root3 which is not equal to root(root2 +1)

Ah yes, my bad. I thought your way was right, but somehow managed to convince myself it was wrong :(

 

[raphael3d]

nevermind :)
but you gave me some new ways to think about this...
maybe there are some other viewpoints?