Graphing Continued Fractions of Quadratic Irrationals

a

1

c

1

b

2

d

1

S

5

1

2

+

5

{2,{1,2,1,3}}

Let

x=

a

b

+

c

d

S

,

a,b,c,d,S∈

. The continued fraction of

x

is either finite (when

S

is a perfect square so that

x

is rational) or eventually periodic (when

S

is not a perfect square so that

x

is irrational).

If

x

is rational, the elements of its continued fraction are plotted.

If

x

is irrational, let its continued faction be

x=[

x

0

;

x

1

,

x

2

,…,

x

m

,

x

m+1

,

x

m+2

,…,

x

m+n

]

, where the repeating part under the bar starts as soon as possible. In that case, the plot is of the repeating part

w={

x

m+1

,

x

m+2

,…,

x

m+n

}={

w

1

,

w

2

,…,

w

n

}

, with the initial elements

{

x

0

;

x

1

,

x

2

,…,

x

m

}

ignored.

Sometimes

w

is a palindrome; that is,

w

is the same read from right to left as from left to right,

{

w

n

,…,

w

2

,

w

1

}={

w

1

,

w

2

,…,

w

n

}

, and its graph is symmetric. Often

w

is the concatenation of two palindromes, like

1

3

+

5

3

=[1;

1,4,1,2,6,2

]

. If

r>1

is rational and not a perfect square, then

r

=

x

0

;

x

1

,

x

2

,…,

x

2

,

x

1

,2

x

0



; that is,

w

is a palindrome concatenated with twice the integer part of

r

, which is a trivial palindrome. Finally, there are cases where

w

is not a palindrome. Colors distinguish the various cases.

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

}}

.