Construct geometric line with nested root

AI Thread Summary
Constructing geometric lines with lengths involving nested square roots, such as sqrt(2 + sqrt(2)) and sqrt(sqrt(2)), begins with the established method of creating sqrt(2) using a triangle with sides of length 1. The discussion highlights the challenge of extending this method to more complex expressions. It suggests that by adding segments or drawing perpendiculars, one can derive new lengths, though some calculations may lead to confusion, such as distinguishing between sqrt(3) and sqrt(sqrt(2) + 1). Participants express a willingness to explore different approaches and clarify misunderstandings. Overall, the thread emphasizes the geometric construction of nested roots and the potential for further exploration of these concepts.
raphael3d
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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.
 
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raphael3d said:
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.
 
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)
 
raphael3d said:
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 :(
 
nevermind :)
but you gave me some new ways to think about this...
maybe there are some other viewpoints?
 
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