Construct geometric line with nested root

In summary, constructing a line with length sqrt(sqrt(2)) can be achieved by drawing a triangle with sides of length 1 and using that as a base to extend the line to different lengths, such as sqrt(2 + sqrt(2)) or sqrt(2 + sqrt(2 + sqrt(2))). Other methods, such as drawing a perpendicular of length 1 on the end of the line, can also be used to create different iterations of the line.
  • #1
raphael3d
45
0
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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  • #2
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.
 
  • #3
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)
 
  • #4
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 :(
 
  • #5
nevermind :)
but you gave me some new ways to think about this...
maybe there are some other viewpoints?
 

Related to Construct geometric line with nested root

1. What is a geometric line?

A geometric line is a straight path that extends infinitely in both directions. It is one of the basic concepts in geometry and is used to represent the shortest distance between two points.

2. What does "nested root" mean in the context of constructing a geometric line?

In this context, nested root refers to the use of square roots or other roots within the formula for constructing a geometric line. It allows for more complex and precise constructions.

3. Can a geometric line be constructed without using nested roots?

Yes, a geometric line can be constructed using basic geometric tools such as a straightedge and compass without the use of nested roots. However, using nested roots allows for more intricate and accurate constructions.

4. What are some practical applications of constructing geometric lines with nested roots?

Constructing geometric lines with nested roots is commonly used in architecture, engineering, and design. It allows for precise and accurate measurements and constructions of structures and objects.

5. Are there any limitations when using nested roots to construct a geometric line?

The use of nested roots can become more complex and time-consuming as the number of roots increases. It also requires a good understanding of mathematical concepts and formulas. Additionally, there may be limitations in the accuracy of the construction due to human error or limitations of the tools used.

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