Graphing Continued Fractions of Quadratic Irrationals
Graphing Continued Fractions of Quadratic Irrationals
Let , . The continued fraction of is either finite (when is a perfect square so that is rational) or eventually periodic (when is not a perfect square so that is irrational).
x=+
a
b
c
d
S
a,b,c,d,S∈
x
S
x
S
x
If is rational, the elements of its continued fraction are plotted.
x
If is irrational, let its continued faction be , where the repeating part under the bar starts as soon as possible. In that case, the plot is of the repeating part , with the initial elements ignored.
x
x=[;,,…,,,,…,]
x
0
x
1
x
2
x
m
x
m+1
x
m+2
x
m+n
w={,,…,}={,,…,}
x
m+1
x
m+2
x
m+n
w
1
w
2
w
n
{;,,…,}
x
0
x
1
x
2
x
m
Sometimes is a palindrome; that is, is the same read from right to left as from left to right, , and its graph is symmetric. Often is the concatenation of two palindromes, like +=[1;]. If is rational and not a perfect square, then ;,,…,,,2]; that is, is a palindrome concatenated with twice the integer part of , which is a trivial palindrome. Finally, there are cases where is not a palindrome. Colors distinguish the various cases.
w
w
{,…,,}={,,…,}
w
n
w
2
w
1
w
1
w
2
w
n
w
1
3
5
3
1,4,1,2,6,2
r>1
r
=[x
0
x
1
x
2
x
2
x
1
x
0
w
r
w
The continued fraction is shown under the plot in the Mathematica notation .
{,,…,,{,,…,}}
x
0
x
1
x
m
x
m+1
x
m+2
x
m+n
References
References
[1] E. R. Burger, "A Tail of Two Palindromes," The American Mathematical Monthly, 112, 2005 pp. 311–321. mathdl.maa.org/mathDL/22/?pa=content&sa=viewDocument&nodeId=3154.
External Links
External Links
Permanent Citation
Permanent Citation
George Beck
"Graphing Continued Fractions of Quadratic Irrationals"
http://demonstrations.wolfram.com/GraphingContinuedFractionsOfQuadraticIrrationals/
Wolfram Demonstrations Project
Published: July 17, 2012