
#A52 INTEGERS 24 (2024)

MÖBIUS MAPS AND CONTINUED FRACTIONS

Carminda M. Mennen
School of Mathematics, University of the Witwatersrand, Johannesburg, South

Africa
carminda.mennen@wits.ac.za

Received: 6/9/23, Accepted: 5/15/24, Published: 5/20/24

Abstract

The connection between the partial quotients of the regular continued fraction and
the number of left and right cuts of the cutting sequence of a geodesic across the
triangles of the Farey tessellation has been established by Series. To find a simi-
lar connection for the nearest integer continued fraction, we require a completely
new approach that combines aspects of well-known work on continued fractions.
Firstly, we discuss the impact of actually truncating the continued fraction and
explore alternate methods that use Möbius maps. We then animate the roles of
various vertices, geodesics, and Farey triangles, in the geometrical figures that are
commonly used, by developing a geometrical animation of the process that under-
lies the continued fraction of an irrational real number ξ. Finally, we reveal the
interplay between the orbits of three vertices, namely zero, one, and infinity, as
they move toward ξ, through rational values, under the action of successive partial
products of Möbius maps derived from the nearest integer continued fraction.

1. Introduction

Any irrational number ξ has an infinite continued fraction representation,

ξ = b0 +
1

b1 +
1

b2 +
.. .

= [b0, b1, b2, · · · ], (1)

where the partial quotients bk satisfy b0 ∈ Z, bk ∈ N for k ∈ N = {1, 2, 3 · · · }, if

it is derived from the Euclidean algorithm [8], but bk ∈ Z if the nearest integer

algorithm is used. The continued fraction of Equation (1) is call a regular continued

fraction (RCF ) in the former case and a nearest integer continued fraction (NICF )

in the latter. Traditionally, one finds rational approximants of ξ by truncating the

DOI: 10.5281/zenodo.11221703


INTEGERS: 24 (2024) 2

continued fraction at the nth partial quotient bn to get convergents Cn =
pn
qn

=

[b0, b1, · · · , bn], determined as a rational in reduced form. We sacrifice accuracy for

speed [7] when generating approximants by truncating the NICF instead of the

RCF . We use Beardon’s suggestion [3, 4, 6] to analyze the behavior of continued

fractions in hyperbolic space where Möbius maps act as isometries.

In this paper, we compare the process for the RCF and the NICF . We give

notation and basic definitions in Section 2. In Section 3, we use a new dynamic

approach to show how parabolic Möbius maps derived from a continued fraction

link the partial quotients of the continued fraction and the triangles of the Farey

tessellation when ξ ∈ R has NICF = RCF . In Section 4, we consider ξ with

NICF 6= RCF . The roles of geometric elements evolve and the dynamic is adapted

accordingly. We link the number of triangles of the Farey tessellation directly to the

coefficients of the Möbius maps derived from the NICF in Section 5. Continued

fractions of quadratic surds always have periodic partial quotients, a result discussed

in the work of Beardon in [6]. This periodicity will not affect the generality of the

discussion in this paper. We state our main theorem in Section 6, describing the

link between the partial quotients of the NICF and the triangles of the Farey

tessellation of hyperbolic space. In Section 7, we briefly discuss further areas of

research.

2. Basic Notation and Definitions

We introduce the geometry underpinning the NICF in terms of Möbius maps acting

on hyperbolic space.

2.1. The Farey Tesselation of Hyperbolic Space

The Farey tessellation F , used by Series [13] in 1985, is a tiling of the upper half

plane model of hyperbolic space H into hyperbolic triangles, where H = {z ∈ C :

Im(z) > 0} is endowed with the Poincare metric ds2 = dx2+dy2

y2 . The closure of

H includes the extended real line R∞ = R ∪ {∞} (the circle at infinity) as the

boundary ∂H. The following properties [8] hold in the Farey tessellation F of H.

• Each endpoint of a Farey geodesic represents a reduced rational with denom-

inator in N0 = N ∪ {0}.

• All geodesics are Farey geodesics λ( a
b ,

c
d ) joining reduced rational endpoints

a

b
and

c

d
(with a, b, c, d ∈ Z and b, d > 0). The endpoints are Farey neighbors,

denoted
a

b
∼ c

d
, as the Farey neighbor condition |ad− bc| = 1 is satisfied.


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• The three vertices of a triangle

(
a

b
,
e

f
,
c

d

)
in F , with

a

b
<
e

f
<
c

d
, satisfy the

Farey sum condition
e

f
=
a

b
⊕ c

d
=
a+ c

b+ d
. Since

a

b
∼ c

d
,
a

b
∼ e

f
, and

e

f
∼ c

d
,

all triangles in F are Farey triangles bounded by three Farey geodesics.

T0

0
1

1
1

1
2

1
3

2
3

1
4

2
5

1
5

4
5

3
5

3
4

Figure 1: The Farey tessellation of H and the fundamental triangle T0.

In the diagram of Figure 1, we introduce the fundamental triangle T0 which is

the shaded hyperbolic triangle with vertices 0, 1, and ∞ that are joined pairwise

by Farey geodesics. The vertical lines z = 0 and z = 1 are Farey geodesics joining

the vertex at∞ to the vertices at 0 and 1, respectively. The part of F shown in the

diagram of Figure 1 consists of all ideal geodesics joining Farey neighbors a
b and c

d ,

with 0 ≤ a
b ,

c
d ≤ 1 and 0 ≤ b, d ≤ 5.

2.2. The Action of Möbius Maps in Hyperbolic Space

We define Möbius maps as follows, and discuss how they are central to our analysis.

Definition 1. Möbius maps are functions of the form g(z) =
az + b

cz + d
, a, b, c, d ∈ C

with ad− bc 6= 0, which map the extended complex plane C∞ = C∪{∞} onto itself

with g(∞) =
a

c
and g

(
−d
c

)
=∞.


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In the work of Beardon [6], the Möbius map g(z) =
(b0b1 + 1) z + b0

b1z + 1
expresses

the continued fraction

f(z) = b0 +
1

b1 +
1

z

.

We convert the continued fraction [b0, b1, · · · ] into the product of maps

T = τb0φτb1φτb2 · · · , where τbk(z) = bk + z (with k ∈ N0 = N ∪ {0}) and φ(z) =
1

z
.

These are both Möbius maps, where τbk , with ad−bc = 1, leaves the hyperbolic plane

H = {z ∈ C : Im(z) > 0} invariant, but φ, with ad− bc = −1, interchanges the up-

per half-plane {z ∈ C : Im(z) > 0} with the lower half-plane {z ∈ C : Im(z) < 0},
while leaving R∞ invariant [9].

The modular group Γ =
{
z 7→ az+b

cz+d : ad− bc = 1, a, b, c, d ∈ Z
}

contains the

isometries of the hyperbolic plane [6, 8], so we wish to use only elements of Γ. In

order to achieve this, we parse the maps of T into the alternate factors τbk and

φτbkφ, where φτbkφ(z) =
z

bkz + 1
, so φτbkφ ∈ Γ.

The classification of Möbius maps requires that every g(z) = az+b
cz+d ∈ Γ be repre-

sented by a normalized matrix. We represent g by A =
(
a b
c d

)
since ad− bc = 1 for

g ∈ Γ. Using the work of Beardon [2] and Anderson [1], we classify Möbius maps

as follows.

Definition 2. Let g be a Möbius transformation in Γ with associated normalized

matrix A =
(
a b
c d

)
. Then:

i) g is parabolic if and only if tr2(A) = 4 if and only if g has a unique fixed point

in ∂H;

ii) g is elliptic if and only if 0 ≤ tr2(A) < 4 if and only if g has a unique fixed

point in H;

iii) g is loxodromic (hyperbolic) if and only if tr2(A) > 4 if and only if g has two

fixed points in ∂H.

Since all possible values for tr2(A) are exhausted in Definition 2, these three types

of Möbius maps are the only possible types of Möbius maps found in Γ.

In the following definition, we rewrite T as a product of parabolic Möbius maps

whose factors alternate between the maps τbn and φτbnφ, that fix ∞ and 0, re-

spectively. We also define Tn, the partial product of parabolic Möbius maps, as

follows.

Definition 3. Let [b0, b1, b2, · · · ] be the RCF (or NICF ) for some irrational num-

ber ξ. We let T = t0t1t2 · · · be the product of parabolic Möbius maps associated

with the RCF (or NICF ) of ξ, with tk = τbk for k even and tk = φτbkφ for k odd,


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where bk ∈ N (or bk ∈ Z) for each k ∈ N0. We call Tn = t0t1 · · · tn the partial

product of parabolic Möbius maps of the RCF (or NICF ) of ξ and it has n + 1

factors.

Since
m

n
∼ p

q
implies g

(m
n

)
∼ g

(
p

q

)
for all g ∈ Γ, all Möbius maps in Γ

preserve Farey neighbors, and every Farey triangle in F is the image of T0 under

some Möbius map in Γ. Using the given metric, every factor tk of T preserves H
and is an isometry of H, with d(u, v) = d(g(u), g(v)) for all u, v ∈ H. As parabolic

Möbius maps, each tk is a conformal map that maps circles to circles [5] and Farey

neighbors to Farey neighbors, properties which we use in our analysis.

2.3. The Convergence of Continued Fractions

In the work of Beardon [4], it is stated that convergence of the continued fraction

of Equation (1) is equivalent to convergence of the sequence Sn(0), where Sn =

s0s1 · · · sn with s0(z) = b0 + z and sn(z) = 1
bn+z for n ∈ N. When using Tn of

Definition 3, convergence of the continued fraction of Equation (1) is equivalent to

convergence of the sequence

T0(0), T1(∞), T2(0), T3(∞), T4(0), T5(∞) · · · .

From the work of Beardon [4], we state Pringsheim’s Theorem without proof.

Theorem 1. Suppose that for all k, |bk| ≥ 1 + |ak|. Then the continued fraction
a1

b1 +
a2

b2 +
. . .

converges to some value v with |v| ≤ 1.

The continued fraction of Equation (1) satisfies Theorem 1 if it is an NICF ,

as each ak = 1 and each partial quotient bk is derived as the integer closest to
1

bbk−1e − bk−1
, where bbk−1e is the integer closest to bk−1. Since |bbk−1e − bk−1| <

1
2 for each partial quotient of an NICF , all |bk| ≥ 2 and the NICF of ξ − b0 given

by
1

b1 +
1

b2 +
.. .

= [0, b1, b2, · · · ] converges to a value between −1 and 1.

3. Generating New Dynamics for the Continued Fraction

We use the images of the fundamental triangle T0 (with vertices 0, 1, and∞) under

the application of the consecutive partial products Tn, for n ∈ N0, to establish the

geometry that underlies the process of using RCF s and NICF s to approximate

real numbers.


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Example 1. The RCF and theNICF for
√

5−2 ≈ 0.23606 . . . is [0, 4, 4, 4, 4, . . . ] =

[0, 4], from which we derive the product of parabolic maps

T = τ0 (φτ4φ) τ4 (φτ4φ) τ4 (φτ4φ) · · · = (φτ4φ) τ4.

The convergents of
√

5− 2 are

C0 = 0, C1 =
1

4
, C2 =

4

17
, C3 =

17

72
, · · · .

The diagram in Figure 2 shows all Farey triangles1 crossed by the geodesic λ(P,
√
5−2)

(from point P on the imaginary axis to
√

5− 2 on the real axis). We show the five

Farey triangles that share a common vertex at each convergent. Each of these

triangles is linked directly to the coefficients in the product of Möbius maps derived

from the continued fraction as follows.

P

∞

0
√

5− 2

1
5

2
9

3
13

4
17

21
89

38
161

55
233

17
72

1
4

5
21

9
38

13
55

1
3

1
2

1
• • • •

λ(P,
√
5−2) T0

A

B

C

Figure 2: All Farey triangles cut by the geodesic λ(P,
√
5−2).

• For C0 = 0: From point P , the geodesic λ(P,
√
5−2) first passes through the

triangle T0 (T0) = T0 = (0, 1,∞). The geodesic λ(P,
√
5−2) then passes through

the three shaded triangles
(
0, 12 , 1

)
,
(
0, 13 ,

1
2

)
, and

(
0, 14 ,

1
3

)
, which are the

1The pair of ideal end points of each Farey geodesics represent a pair of Farey neighbors in
R. All Farey fractions that appear in the diagram of Figure 2 are drawn equidistant from each
other. This adjustment in the diagram allows us to work with rational values that are very close
to
√

5− 2 more easily.


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images of T0 under (φτ1φ), (φτ2φ) and (φτ3φ), respectively, to get to T1 (T0) =

(φτ4φ) (T0) =
(
0, 15 ,

1
4

)
, which is labeled A in the diagram of Figure 2. Each

of these Farey triangles has 0 as a vertex, since 0 is the image of the 0 vertex

of T0 under each of the given maps. The vertex 1
4 is the image of ∞ under

T1 = τ0 (φτ4φ).

• For C1 = 1
4 : Continuing from T1 (T0) =

(
0, 15 ,

1
4

)
, the geodesic λ(P,

√
5−2)

passes through the three shaded triangles
(
1
5 ,

2
9 ,

1
4

)
,
(
2
9 ,

3
13 ,

1
4

)
, and

(
3
13 ,

4
17 ,

1
4

)
,

which are the images of T0 under (φτ4φ) τ1, (φτ4φ) τ2 and (φτ4φ) τ3, respec-

tively, to get to T2 (T0) = (φτ4φ) τ4 (T0) =
(

4
17 ,

5
21 ,

1
4

)
, which is labeled B in

the diagram of Figure 2. Each of these Farey triangles has 1
4 as a vertex,

since 1
4 is the image of the ∞ vertex of T0 under each of the given maps. The

vertex 4
17 is the image of 0 under T2 = τ0 (φτ4φ) τ4.

This process continues indefinitely, matching each partial quotient ‘4’ to the four

Farey triangles generated under the associated partial product of Möbius maps Tn
for all n ∈ N.

3.1. Fixed Points

As we build up successive images of T0 under the Tn transformations, the images

of 0 and ∞ are alternatively fixed points of the last map in the composition as the

map τbn fixes ∞ and φτbnφ fixes 0. Each iteration generates a fixed point with

respect to the previous iteration. We define the fixed points under Tn as follows.

Definition 4. Let Tn be a partial product of parabolic Möbius maps. If Tn−1(p) =

Tn(p), n ∈ N, then Tn(p) is the fixed point in the progression of triangles from

Tn−1 (T0) to Tn (T0).

For each partial product of Möbius maps Tn, the value Tn(0) is the fixed point in

the progression of triangles from Tn−1 (T0) to Tn (T0) when n is odd, and the value

Tn(∞) is the fixed point in the progression of triangles from Tn−1 (T0) to Tn (T0)

when n is even.

3.2. Intermediate Farey Triangles

We define the intermediate images of T0 for each Tn as follows.

Definition 5. In the partial product of Möbius maps Tn = t0t1 · · · tn−1tn, with

tn = τbn for n even and tn = φτbnφ for n odd, we replace tn with tk to get

Tn,k = t0t1 · · · tn−1tk (for all 1 ≤ k ≤ bn − 1, k ∈ N), where tk = τk for n even

and tk = φτkφ for n odd. The Farey triangles Tn,k (T0) = t0t1 · · · tn−1tk (T0),

k = 1, . . . , bn− 1, are called the intermediate Farey triangles between Tn−1 (T0) and

Tn (T0).


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Example 2. In the diagram of Figure 2, the shaded triangles

T1,1 (T0) =

(
0,

1

2
, 1

)
, T1,2 (T0) =

(
0,

1

3
,

1

2

)
and T1,3 (I0) =

(
0,

1

4
,

1

3

)
share the vertex T1 (0) = 0, as it is the fixed point in the progression of triangles

from T0 (T0) to T1 (T0). These three triangles are the three intermediate Farey

triangles between T0 (T0) and T1 (T0).

3.3. The Link between Partial Quotients and Intermediate Farey Trian-
gles

The following proposition states the link between partial quotients and the number

of intermediate Farey triangles for each Tn.

Proposition 1. For each n ∈ N, there are bn − 1 intermediate Farey triangles

between Tn−1 (T0) and Tn (T0), and each pair of consecutive intermediate triangles

Tn,k−1 (T0) and Tn,k (T0), share a common edge, for all k ∈ N with 1 < k ≤ bn− 1.

Proof. By simple calculation, we have that

φτ1φ (0, 1,∞) =

(
0,

1

2
, 1

)
and τ1 (0, 1,∞) = (1, 2,∞) ,

so

Tn,1 (0, 1,∞) = Tn−1 (φτ1φ) (0, 1,∞) = Tn−1

(
0,

1

2
, 1

)
, for n odd,

and

Tn,1 (0, 1,∞) = Tn−1τ1 (0, 1,∞) = Tn−1 (1, 2,∞) , for n even.

Thus, with v = 0 for n odd and v = ∞ for n even, we have that Tn,1 (v) =

Tn−1 (v) and Tn,1
(
1
v

)
= Tn−1 (1). The Farey triangles Tn−1 (T0) and Tn,1 (T0)

share the edge Tn−1

(
λv, 1v

)
= Tn,1 (λv,1). Similarly, Tn,k (v) = Tn,k−1 (v) and

Tn,k
(
1
v

)
= Tn,k−1 (1). Thus Tn,k−1 (T0) and Tn,k (T0) share the edge Tn,k

(
λv, 1v

)
=

Tn,k−1 (λv,1) as required.

Since 1 < k ≤ bn − 1, k ∈ N, it follows that there are bn − 1 intermediate Farey

triangles between Tn−1 (T0) and Tn (T0).

4. Nearest Integer Continued Fractions with Negative Partial Quotients

For n ∈ N, the continued fraction in Equation (1) has partial quotients bn that are

natural numbers when it is an RCF , but integers when it is an NICF . We show

the difference in the geometry when an NICF has negative partial quotients.


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Example 3. For
√

3 − 1 ≈ 0.73205 . . . the RCF is [0, 1, 2] and the function Tn
becomes:

Tn(z) =

{
τ0 (φτ1φ) τ2 (φτ1φ) · · · (φτ1φ) τ2 (φτ1φ) (z), for n odd
τ0 (φτ1φ) τ2 (φτ1φ) · · · τ2 (φτ1φ) τ2(z), for n even

(2)

where each product has n + 1 factors. The values for Tn(0), Tn(1), and Tn(∞),

calculated using Equation (2), are shown in Table 1. The corresponding Tn (T0) are

labeled in the diagram of Figure 3.

In the diagram of Figure 3, there is one shaded intermediate triangle between

Tn (T0) and Tn+1 (T0) for n odd, and none between Tn (T0) and Tn+1 (T0) for n even,

which satifies the link between partial quotients and the number of intermediate

Farey triangles established in Proposition 1 in Section 3.3.

z T0(z) T1(z) T2(z) T3(z) T4(z) T5(z) T6(z)
τ0 τ0 (φτ1φ) τ0 (φτ1φ) τ2 T2 (φτ1φ) T3τ2 T4 (φτ1φ) T5τ2

0 0 0 2/3 2/3 8/11 8/11 30/41
1 1 1/2 3/4 5/7 11/15 19/26 41/56
∞ ∞ 1 1 3/4 3/4 11/15 11/15

Table 1: The images of 0, 1, and ∞ under Tn for n = 0, · · · , 6.

P

∞

0
√

3− 1

11
2

2
3

5
7

8
11

19
26

30
41

71
97

112
153

3
4

11
15

41
56

153
209

λ(P,
√
3−1) T0 (T0)

T1 T2T3 T4T5 T6T7

• ••••• • • •

Figure 3: The images of T0 under successive Tn derived from the RCF .

Using the nearest integer division algorithm for
√

3− 1, we get the NICF

√
3− 1 = [1,−4, 4,−4, 4, ...] = [1,−4, 4],


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and Definition 3 gives us Tn for the NICF as follows:

Tn(z) =

{
τ1 (φτ−4φ) τ4 (φτ−4φ) · · · (φτ−4φ) τ4 (φτ−4φ) (z), for n odd
τ1 (φτ−4φ) τ4 (φτ−4φ) · · · τ4 (φτ−4φ) τ4(z), for n even

(3)

where each product has n + 1 factors. The values for Tn(0), Tn(1), and Tn(∞),

calculated using Equation (3), are shown in Table 2. The corresponding Tn (T0) are

shaded in the diagram of Figure 4.

In the diagram of Figure 4, since there are three Farey triangles between Tn−1 (T0)

and Tn (T0) for each n ∈ N, we see that the link between the bn (the partial quotients

of the NICF ) and the number of intermediate Farey triangles (between Tn−1 (T0)

and Tn (T0)) still satisfies Proposition 1 in Section 3.3.

z T0(z) T1(z) T2(z) T3(z) T4(z) T5(z)
τ1 τ1 (φτ−4φ) τ1 (φτ−4φ) τ4 T2 (φτ−4φ) T3τ4 T4 (φτ−4φ)

0 1 1 11/15 11/15 153/209 153/209
1 2 2/3 14/19 30/41 194/245 · · ·
∞ ∞ 3/4 3/4 41/56 41/56 · · ·

Table 2: The images of 0, 1, and ∞ under Tn for n = 0, · · · , 5.

P

∞

0 √
3− 1

11
2

2
3

5
7

8
11

19
26

30
41

71
97

112
153

3
4

14
19

11
15

41
56

153
209

λ(P,
√
3−1) T0 T0 (T0)

T1
T2*T3T4

•

••••

Figure 4: The images of T0 under successive Tn derived from the NICF .

This result, however, is not as satisfactory in the case of the NICF as it is in the

case of the RCF for the following two reasons.

i) The vertices of T2 (T0) =
(
3
4 ,

14
19 ,

11
15

)
are all larger than

√
3− 1.

ii) The Farey triangle marked with the *, in the diagram of Figure 4, is the third

intermediate triangle (T2,3 (T0)) between T1 (T0) and T2 (T0), but it is also


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the first intermediate triangle (T3,1 (T0)) between T2 (T0) and T3 (T0), which

means that this Farey triangle is a repeated image among the intermediate

Farey triangles of the product of parabolic Möbius maps in the Tn derived

from the NICF .

The two conditions, where all vertices of the Farey triangle are larger than the target

and Farey triangles are repeated, occur for all even values of n in this example.

However, it is clear from Tables 1 and 2 that the NICF produces approximants

with greater speed.

We avoid the problems of ‘all vertices larger than target’ and ‘repeated images’,

as seen in the NICF of Example 3, by recognizing and isolating the elliptic Möbius

maps from the parabolic Möbius maps in Tn.

5. Dynamics Generated by the NICF

For the product of parabolic Möbius maps derived from the NICF we introduce

the following notation. The parabolic Möbius maps that fix ∞ and 0 are

p∞ = τ1 and p0 = (φτ1φ) ,

respectively. The elliptic Möbius maps are

s = τ1 (φτ−1φ) , and its inverse s−1 = (φτ1φ) τ−1.

In the following theorem, we show that s and s−1 act as elliptic Möbius maps on

H and summarize the procedure for finding and isolating the elliptic maps in the

product of parabolic Möbius maps of Definition 3.

Theorem 2. The maps s and s−1 are maps of order 3 that permute the vertices 0,

1, and ∞. The maps s and s−1 are the only elliptic maps found among the factors

of the product of parabolic Möbius transformations generated from an NICF .

Proof. The maps s and s−1 both appear among the factors of the product of

parabolic Möbius maps as a product of two parabolic maps, p∞p
−1
0 = τ1 (φτ−1φ)

and p0p
−1
∞ = (φτ1φ) τ−1, which are s and s−1, respectively.

A simple calculation shows both s and s−1 fix the single point
1 +
√

3i

2
in H.

The calculations

s(0) = 1, s(1) =∞, s(∞) = 0 and s−1(0) =∞, s−1(1) = 0, s−1(∞) = 1

show that s and s−1 are maps of order 3 that permute the vertices 0, 1, and ∞.

We express

Tn(z) =

{
τb0 (φτb1φ) τb2 (φτb3φ) · · ·

(
φτbn−2

φ
)
τbn−1

(φτbnφ) (z), for n odd
τb0 (φτb1φ) τb2 (φτb3φ) · · · τbn−2

(
φτbn−1

φ
)
τbn(z), for n even,

(4)


INTEGERS: 24 (2024) 12

derived from the NICF , in terms of the Möbius maps p0, p∞, s and s−1 as follows.

Clearly the map φτbnφ can be writen as φτ±1τbnτ∓1φ = (φτ±1+bnφ) (φτ∓1φ) =

(φτ±1φ) (φτbn∓1φ) since φ2 and τ±1∓1 = τ0 are both the identity map. Anywhere

in the product Tn(z) where bk > 0 and bk+1 < 0, we rearrange the factors involving

bk and bk+1, and make the elliptic Möbius maps explicit. There are two cases.

• When k is even, the factors · · · τbk
(
φτbk+1

φ
)
· · · are rearranged to become

· · · τbk−1 [τ1 (φτ−1φ)]
(
φτbk+1+1φ

)
· · · which is equal to · · · pbk−1∞ sp

bk+1+1
0 .

• When k is odd, the factors · · · (φτbkφ) τbk+1
· · · are rearranged to become

· · · (φτbk−1φ) [(φτ1φ)] τ−1]τbk+1+1 · · · which is equal to · · · pbk−10 s−1p
bk+1+1
∞ .

After that, all remaining (φτbφ) are replaced with pb0 and all remaining τb are

replaced with pb∞ to produce the product of Möbius maps called TM . The product

Tn derived from the NICF yields only s and s−1 as elliptic factors.

In Example 4, we show that the use of Möbius maps allows us to identify new

geometrical aspects embedded in the process of finding rational approximants using

continued fractions.

Example 4. For
√

3− 1, we replace T = τ1 (φτ−4φ) τ4 (φτ−4φ) · · · for
√

3− 1 with

the product of Möbius maps

T = [τ1 (φτ−1φ)] (φτ−3φ) τ3[τ1 (φτ−1φ)] (φτ−3φ) τ3[τ1 (φτ−1φ)] · · · , (5)

which becomes

TM = sp−30 p3∞sp
−3
0 p3∞sp

−3
0 p3∞ · · · (6)

or more simply

TM = sp−30 p3∞. (7)

The partial products derived from Equation (7) are

TM(n) = sp−30 p3∞sp
−3
0 p3∞sp

−3
0 p3∞ · · · tM(n),

where tM(n) represents the nth factor in the product of Möbius maps TM . We now

track the images of the vertices 0, 1, and∞ of T0 under the application of successive

TM(n).

Each line under the diagram of Figure 5 shows the images of T0 under successive

TM(n) where TM(1) = s, TM(2) = sp−30 , TM(3) = sp−30 p3∞, TM(4) = sp−30 p3∞s and so

on. We show the movement of images of T0 under the TM(n) generated from the

NICF using maps p0 and p∞ next to the arrow that represents the movement that

they cause in the diagram of Figure 5. An s is placed in the triangle whose vertices

have been permuted by the s map. The effects of TM(2) and TM(3) are elaborated

over three lines below the image.

For each progression shown in the diagram of Figure 5 under successive TM(n),

we describe the associated movement as follows.


INTEGERS: 24 (2024) 13

0
√

3− 1
s TM(1)∞ 0 1

1 ∞ 0

1 ∞ 0

1 ∞ 0

sp−10

sp−20

TM(2)sp−30

1 ∞0

1 ∞0

1 ∞0

sp−30 p1∞

sp−30 p2∞

TM(3)sp−30 p3∞

0 1∞sp−30 p3∞s TM(4)

1 ∞1
2

2
3

5
7

8
11

19
26

30
41

3
4

11
15

•••••

s

ss

••p−30

••p3∞
•••p−30

•••p3∞

Figure 5: Tracing the iterates TM(n)(0), TM(n)(1) and TM(n)(∞) for the NICF of√
3− 1 to count movements of triangles about fixed points for

√
3− 1.

• Start with the vertices 0, 1, and ∞ of T0.

• For TM(1): Of the three vertices, 0, 1, and ∞, 1 is the closest to ξ, so we

permute the vertices using s. In the line with s on the left hand side in the

diagram of Figure 5, we note that images of 0, 1, and∞ are the three vertices

permuted by TM(1) = s, so s(∞) = 0 and s(0) = 1. Here s(1) = ∞ as

indicated.

• For TM(2): The image of 0 under TM(1) is now closest to the target ξ, so it

should be fixed under t2. Since t2 = φτ−3φ, the image of 0 is fixed under

TM(2) as illustrated by the next three lines in the diagram of Figure 5, while


INTEGERS: 24 (2024) 14

the images of 1 and ∞ approach TM(1)(0) = TM(2)(0) = 1 from the left.

• For TM(3): The image of ∞ under TM(2) is closest to ξ, and this point will be

fixed by t3 = τ3. The images of 0 and 1 approach TM(2)(∞) = TM(3)(∞) = 3
4

from the left under TM(2) as shown in the next three lines in the diagram of

Figure 5.

• For TM(4): After the last iteration, TM(3)(1) is closest to ξ. The map s is

used again to permute the images of 0, 1, and ∞. The vertex TM(4)(0) is now

closest to ξ and the process is continued in the same way.

Example 4 serves to highlight the new roles adopted by vertices, geodesics and

Farey triangles in the process of generating approximations. Changing the focus

from the RCF to the NICF changes the following parts of the process.

• The Euclidean division algorithm is replaced by the nearest integer division

algorithm.

• Reading off values of partial quotients of the RCF is replaced by an analysis of

Möbius maps derived from the NICF , requiring that we separate the elliptic

maps s and s−1 from the parabolic maps.

• Counting left and right cuts along a geodesic is replaced by counting move-

ments of hyperbolic triangles about successive fixed points.

6. Partial Quotients and the Link between Adjusted Coefficients and
Counting Triangles

In order to simplify the link between coefficients in TM and triangles in the Farey

tessellation, we define the adjusted product as follows.

Definition 6. Let b∗n be the adjusted coefficient of the partial quotient bn after

all the elliptic factors s and s−1 have been accommodated in the factors of T to

form TM . The adjusted product of T for the NICF is TA = t∗0t
∗
1t
∗
2 · · · , where each

t∗n, n ∈ N0 is a product of an elliptic map (e (the identity map), s, or s−1) and a

parabolic map τb∗n or φτb∗nφ. The partial adjusted product of T for the NICF is

TAn = t∗0t
∗
1t
∗
2 · · · t∗n.

In the following example, we show the adjusted product of
√

3− 1 of Example 4,

in comparison with the original product of parabolic Möbius maps T .

Example 5. For
√

3− 1:
T = τ1 (φτ−4φ) τ4 (φτ−4φ) · · · ,
TA = τ0 s (φτ−3φ) τ3 s (φτ−3φ) · · ·

= p0∞ sp−30 p3∞ sp−30 · · · .
Each coefficient is adjusted by the factor s wherever it is accommodated.


INTEGERS: 24 (2024) 15

When using continued fractions to find rational approximations of irrational reals,

the RCF is useful to get the configuration of the Farey triangles for the irrational

number, but the NICF gives us a quicker method to find approximations. By

reinterpreting the Möbius maps of the NICF , we become aware of the roles of the

images of 0, 1, and ∞. Using the partial product TM(n) = t1t2 · · · tn it is easy to

establish which of the three images is closest to the target by looking at tn+1.

• If tn+1 = pb∞ then TM(n)(∞) is closest to the target and becomes the fixed

point of TM(n+1).

• If tn+1 = pb0 then TM(n)(0) is closest to the target and becomes the fixed point

of TM(n+1).

• If tn+1 = s or tn+1 = s−1 then TM(n)(1) is closest to the target. Since neither

of the parabolic maps uses 1 as a fixed point the images of 0, 1, and ∞ must

be permuted.

In the final example, we see the ‘semi-periodic’ pattern of the partial quotients

associated with the number e. We give the conversion from the product of maps T

to the adjusted product of Möbius maps TA for e− 2.

Example 6. For e− 2 we have:

T = τ1 (φτ−4φ) τ2 (φτ5φ) τ−2 (φτ−7φ) τ2 (φτ9φ) τ−2 · · · ,

TA = τ0 s (φτ−3φ) τ2 (φτ4φ) s−1τ−1 (φτ−7φ) τ2 (φτ8φ) s−1τ−1 · · ·

= p0∞ sp−3
0 p2∞ p40 s−1p−1

∞ p−7
0 p2∞ p80 s−1p−1

∞ · · · .

All fractions on the number line in the diagram of Figure 6 are approximations
derived from the process of Farey subdivision [8], while the fractions 0

1 ,
1
1 ,

2
3 ,

3
4 ,

5
7 , · · ·

(connected with solid geodesics) are convergents of the RCF . Approximations that
are marked by bullets in the diagram of Figure 6 are convergents of the NICF .
The exponents of the p∞ and p0 give the number of triangles that we move through
about each fixed point, under successive TM(n) derived from the NICF , and the
sign of the exponent indicates the direction of the movement for the triangles about
the fixed point.

Once the Möbius maps for the NICF of ξ are established, the adjustment to

the exponents of p∞ and p0, when incorporating the maps s and s−1 to form the

product TM , ensure that the following theorem is satisfied.

Theorem 3. For the product of Möbius maps TM , derived from the NICF , the

partial products of Möbius maps TM(n) generate images of the fundamental triangle

T0 where

a) the exponents of p∞ and p0 count triangles that each share the fixed point of

a parabolic map as a common vertex;


INTEGERS: 24 (2024) 16

0 1
2

2
3

5
7

28
39

79
110

130
181

181
252

232
323

283
394

334
465

719
1001

e

TM(n) = sp−30 p2∞p
4
0s
−1p−1∞ p−70 p2∞p

8
0s
−1p−10 p−11∞ p2∞p

12
0 s
−1 · · ·

385
536

51
71

23
32

18
25

13
18

8
11

3
4

1

∞

••• • • •

s

s−1

p−70

•p2∞•
p−30

•p40
••p−1∞

Figure 6: Tracking images of T0 for e− 2.

b) the target ξ lies between the smallest and the largest of the 3 vertices TM(n)(0),

TM(n)(1), and TM(n)(∞) of triangle TM(n)(T0), if tM(n) is equal to either p∞
or p0, raised to a non-zero integer;

c) the 3 vertices TM(n)(0), TM(n)(1), and TM(n)(∞) of triangle TM(n)(T0) are

permuted if the map tM(n+1) is equal to s or s−1.

This result establishes the link between the partial quotients of the NICF and

the number of triangles encountered in the Farey tessellation near ξ in the same

way that Series [12, 13] and Schmidt [11, page 18] did for the regular continued

fraction.

7. Conclusion

The understanding of how a process works is often linked to the images we use to

visualize the process. In the case of continued fractions, the dominant images are

those that involve the regular continued fraction. This paper offers insight into the

procedure for using the nearest integer continued fraction. The interpretation uses

Möbius maps to maintain the level of accuracy for approximations of real numbers


INTEGERS: 24 (2024) 17

using the NICF .

The examples so far have all had periodic or semi-periodic partial quotients in

their NICF s. The RCF of π (given by [3, 7, 15, 1, 292, 1, 1, . . . ]) and the NICF

of π, have no pattern for their partial quotients. Is there a way to extend the

geometrical method of this paper to any of the well known, beautifully patterned

continued fractions for π? For instance:

π = 3 +
12

6 +
32

6 +
52

6 + · · ·

and, from [10],
π

2
= 1 +

1

1 +
1

1
2 +

1
1
3 + · · · .

The geometry for continued fractions with numerators not all equal to 1, and those

with numerators equal to 1, but denominators rational, need to be explored further.

As we extend work on continued fractions to higher dimensions, the NICF pro-

vides a more viable option, so it is essential that we have suitable images that

illustrate the underlying process of the NICF . In higher dimensions, simply trun-

cating the NICF has a negative impact on accuracy of the approximations. In the

case of approximating reals, we already need to track the orbits of three vertices

(namely∞, 0, and 1) to visualize the process. As we move into approximating com-

plex numbers using NICF s the number of vertices whose orbits we need to track

increases. This will be the subject of our next paper. Thereafter we will explore

the extension of these ideas to the approximation of quaternion numbers.

References

[1] J. W. Anderson, Hyperbolic Geometry, Springer-Verlag, London, 2005.

[2] A. F. Beardon, The Geometry of Discrete Groups, Springer, Berlin, 1983.

[3] A. F. Beardon, Continued fractions, discrete groups and complex dynamics, Comput. Methods
Funct. Theory 1 (2001), 535-594.

[4] A. F. Beardon, The geometry of Pringheim’s continued fractions, Geom. Dedicata 84 (2001),
125-134.

[5] A. F. Beardon, Algebra and Geometry, Cambridge, Cambridge University Press, 2005.

[6] A. F. Beardon, Möbius maps and periodic continued fractions, Math. Mag. 88 (2015), 272-277.

[7] D. Hensley, Continued Fractions, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ,
2006.

[8] M. Hockman, Continued fractions and the geometric decomposition of modular transforma-
tions, Quaest. Math. 29 (2006), 427-446.

[9] M. Hockman and R. Van Rensburg, Simple continued fractions and cutting sequences, Quaest.
Math. 36 (2013), 437-448.


INTEGERS: 24 (2024) 18

[10] E. Scheinerman, T. J. Pickett and A. Coleman, Another continued fraction for π, Amer.
Math. Monthly 115 (2008), 930-933.

[11] A. L. Schmidt, Diophantine approximation of complex numbers, Acta Math. 134 (1975),
1-85.

[12] C. Series, The modular surface and continued fractions, J. Lond. Math. Soc. (2) 2 (1985),
69-80.

[13] C. Series, The geometry of Markoff numbers, Math. Intelligencer 7 (1985), 20-28.


	Introduction
	Basic Notation and Definitions
	The Farey Tesselation of Hyperbolic Space
	The Action of Möbius Maps in Hyperbolic Space
	The Convergence of Continued Fractions

	Generating New Dynamics for the Continued Fraction
	Fixed Points
	Intermediate Farey Triangles
	The Link between Partial Quotients and Intermediate Farey Triangles

	Nearest Integer Continued Fractions with Negative Partial Quotients
	Dynamics Generated by the NICF
	Partial Quotients and the Link between Adjusted Coefficients and Counting Triangles
	Conclusion