ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 20 (2021) 37–50
https://doi.org/10.26493/1855-3974.2046.cb6
(Also available at http://amc-journal.eu)
From Farey fractions to the Klein quartic
and beyond*
Ioannis Ivrissimtzis †
Department of Computer Science, Durham University, DH1 5LE, United Kingdom
David Singerman , James Strudwick
Mathematical Sciences, University of Southampton, SO17 1BJ, United Kingdom
Received 11 July 2019, accepted 21 September 2020, published online 14 July 2021
Abstract
In a paper published in 1878/79 Klein produced his famous 14-sided polygon repre-
senting the Klein quartic, his Riemann surface of genus 3 which has PSL(2, 7) as its au-
tomorphism group. The construction and method of side pairings are fairly complicated.
By considering the Farey map modulo 7 we show how to obtain a fundamental polygon
for Klein’s surface using arithmetic. Now the side pairings are immediate and essentially
the same as in Klein’s paper. We also extend his work from 7 to 11 as Klein also did in a
follow-up paper of 1879.
Keywords: Riemann surfaces, Klein quartic, regular maps, Farey tessellation, modular group, prin-
cipal congruence subgroups.
Math. Subj. Class. (2020): 30F10, 20H10, 51M20
1 Introduction
The Klein quartic was introduced in one of Felix Klein’s most famous papers, [5] of
1878/79. A slightly updated version appeared in Klein’s Collected Works [7], while for
a translation of this see the book The Eightfold Way, the Beauty of Klein’s Quartic Curve
edited by Silvio Levy [8]. This algebraic curve, whose equation is x3y + y3z + z3x = 0,
gives the compact Riemann surface of genus 3 with 168 automorphisms, the maximum
number by the Hurwitz bound.
*We thank the referees for their careful reading of this paper and their helpful suggestions.
†Coresponding author.
E-mail addresses: ioannis.ivrissimtzis@durham.ac.uk (Ioannis Ivrissimtzis), D.Singerman@soton.ac.uk
(David Singerman), J.Strudwick@soton.ac.uk (James Strudwick)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
38 Ars Math. Contemp. 20 (2021) 37–50
Let H denote the upper-half complex plane and let H∗ = H ∪ Q ∪ {∞}. Klein’s
surface is H∗/Γ(7), where Γ(7) is the principal congruence subgroup mod 7 of the classical
modular group Γ = PSL(2,Z). (For this concept see [4, p. 301].) Klein studies the
Riemann surface of the Klein quartic by constructing his famous 14-sided fundamental
region with its side identifications. See sections 11 and 12 of [5] for the construction and
between pages 448 and 449 of [5], page 126 of [7], or page 320 of [8] for the figure itself.
Our approach is to construct a fundamental region for Klein’s surface using the Farey
tessellation M3 of H∗, a triangular tessellation of H∗ which we define in §2, and which
was shown to be the universal triangular tessellation [10]. In §3 and §4, we study the level
n Farey map M3/Γ(n), through the correspondence of its directed edges with the elements
of Γ/Γ(n) and the correspondence of its vertices with the cosets of Γ1(n) in Γ. In §5 and
§6, we study the level 7 Farey map M3/Γ(7). As M3 ⊂ H∗, M3/Γ(7) ⊂ H∗/Γ(7), this
Farey map is embedded in the Klein surface. In a sense, we will show that this Farey map
is the Klein surface.
In §7 and §8, we review Klein’s original construction, computing Farey coordinates
on Klein’s 14-sided fundamental region and discussing the differences between the two
approaches. In volume 15 of Mathematische Annalen in 1879 [6], Klein extended his work
to study the surface H∗/Γ(11), which has PSL(2, 11) of order 660 as its automorphism
group and is somewhat more complicated. He did not draw a fundamental region for the
case n = 11 as he did for n = 7. However we are able to draw the corresponding Farey
map in §9.
2 The Farey map
The vertices of the Farey map M3 are the extended rationals, i.e. Q∪{∞} and two rationals
a
c and
b
d are joined by an edge if and only if ad − bc = ±1. These edges are drawn as
semicircles or vertical lines, perpendicular to the real axis, (i.e. hyperbolic lines). Here
∞ = 10 . This map has the following properties.
(a) There is a triangle with vertices 10 ,
1
1 ,
0
1 , called the principal triangle.
(b) The modular group Γ = PSL(2,Z) acts as a group of automorphisms of M3.
(c) The general triangle has vertices ac ,
a+b
c+d ,
b
d .
This forms a triangular tessellation of the upper half plane. Note that the triangle in (c) is
just the image of the principal triangle under the Möbius transformation corresponding to
the matrix
(
a b
c d
)
.
In [10] it was shown that M3 is the universal triangular map. This means that if M is
any triangular map on an orientable surface then M is the quotient of M3 by a subgroup
Λ of the modular group. A map is regular if its orientation preserving automorphism group
acts transitively on its darts, (i.e. directed edges) and M3/Λ is regular if and only if Λ is a
normal subgroup of Γ. The subgroup Λ here is called a map subgroup.
(In general if ∆(m,n) is the (2,m, n) triangle group, then every map of type (m,n)
has the form M̂/M where M̂ is the universal map of type (m,n) and M is a subgroup of
Γ. In our case we are thinking of the modular group Γ as being the (2, 3,∞) group. The
infinity here means that we are not concerned with the vertex valencies; we just require the
map to be triangular. For the general theory we refer to [3].)
We now consider the case when Λ = Γ(n), the principal congruence subgroup mod
n of the modular group Γ. The corresponding maps are denoted by M3(n). As Γ(n) is a
I. Ivrissimtzis et al.: From Farey fractions to the Klein quartic and beyond 39
Figure 1: The Farey map, (drawn by Jan Karabaš).
normal subgroup of Γ these maps are regular.
3 The map M3(n)
The map M3(n) is a regular map that lies on the Riemann surface H∗/Γ(n). The auto-
morphism group of M3(n) is Γ/Γ(n) ∼= PSL(2,Zn) whose order µ(n) for n > 2 is
µ(n) =
n3
2
∏
p|n
(
1− 1
p2
)
.
The product is taken over all prime factors of n, see [3, Chapter 6, Exercise 6L].
Also, µ(2) = 6.
Now µ(n) is the number of darts of M3(n) so the number of edges of this map is
µ(n)/2, and the number of faces is equal to µ(n)/3. Note that 10 is joined to
k
1 for k =
0, . . . , n− 1 so that 10 has valency n and by regularity every vertex has valency n. Thus the
number of vertices is equal to µ(n)/n. For example, µ(5) = 60, µ(7) = 168, µ(11) = 660,
so the numbers of vertices of M3(n), for n = 5, 7, 11, are 12, 24, 60, respectively. We can
now use the Euler-Poincaré formula to find the well-known formula for the genus g(n) of
M3(n);
g(n) = 1 +
n2
24
(n− 6)
∏
p|n
(
1− 1
p2
)
. (3.1)
3.1 Farey coordinates for M3(n)
If (a, c, n) = 1 then the projection of ac from M3 to M3(n) is denoted by [
a
c ], or simply
a
c when there is no room for ambiguity, To be precise, a Farey fraction
a
c is an equivalence
class of ordered pairs (a, c) ∈ Z2n with (a, c, n) = 1 under the equivalence relation (a, c) ≡
40 Ars Math. Contemp. 20 (2021) 37–50
(b, d) if b = ua, d = uc ∈ Zn and u = ±1 ∈ Zn. This is sometimes referred to as a Farey
coordinate of a vertex in M3(n).
See §4.1 for the case n = 5, where we give the Farey coordinates for the icosahedron.
4 The quasi-icosahedral structure of Farey maps
We now show that every Farey map has a quasi-icosahedral structure. Let us give some
definitions from [12].
1. The (graph-theoretic) distance δ(f1, f2) between two vertices f1 and f2 of a graph
is the least number of edges joining these two vertices.
2. A Farey circuit is a sequence of Farey fractions f1, f2, . . . , fk where fi is joined by
an edge to fi+1 with the indices taken mod k.
3. A pole of a Farey map is any vertex with coordinates a0 .
The following theorem was proved in [12].
Theorem 4.1. Let ac ,
b
d be distinct vertices of M3(p), where p is prime, and let ∆ =
ad− bc. Then:
δ
(
a
c
,
b
d
)
=

1 if and only if |∆| = 1,
2 if and only if |∆| ≠ 0, 1,
3 if and only if ∆ = 0.
Now let us call 10 the north pole N of M3(p). Then by the above theorem δ(N,
a
c ) = 1
if and only if c = ±1, δ(N, ac ) = 2 if and only if c ̸= 0,±1, and δ(N,
a
c ) = 3 if and only
if c = 0. That is, the vertices of M3(p) form four disjoint subsets: the north pole N at 10 , a
set of size n consisting of vertices whose graph-theoretic distance from N is 1, another set
of points at distance 2 from N , and other poles at distance 3 from N . In Theorem 4.2, we
will show that these two sets are in fact circuits. As the icosahedron has this property we
refer to these Farey maps as having a quasi-icosahedral structure.
(In [12] it was also shown that M3(n) has diameter 3 for all n ≥ 5.)
4.1 The icosahedron
M3(5) is an icosahedron [12] with vertex set{
1
0
,
2
0
,
0
1
,
1
1
,
2
1
,
3
1
,
4
1
,
0
2
,
1
2
,
2
2
,
3
2
,
4
2
}
;
see Figure 2. The north pole N at 10 , there is a Farey circuit of length 5 of points whose
denominator is equal to 1 and have distance 1 from N and a second circuit of length 5 of
points whose denominator is equal to 2 and have distance 2 from N . We also have the pole
2
0 at distance 3 from N .
For a quasi-icosahedral structure on M3(p) let N = 10 ∈ M3(p). The circuit of points
of distance 1 from N is
S1(p) =
0
1
,
1
1
, . . . ,
p− 1
1
.
The set of points at distance 2 from N is more complicated and we now construct it. To
make the calculation clearer we start with the example p = 7. From Theorem 4.1, we see
I. Ivrissimtzis et al.: From Farey fractions to the Klein quartic and beyond 41
Figure 2: Drawing of M3(5) with Farey coordinates.
that the points of distance 2 from 10 have the form
b
d where d = ±2 or ±3. Thus the points
1
3 ,
1
2 ,
2
3 ∈ S2(7) all have distance 2 from N . As the transformation t 7→ t+ 1 fixes N and
preserves distance, all points in S(7) + k have distance 2 from N , for k = 1, . . . , 6. Thus
we find the set
S2(7) =
1
3
,
1
2
,
2
3
,
4
3
,
3
2
,
5
3
,
0
3
,
5
2
,
1
3
,
3
3
,
0
2
,
4
3
,
6
3
,
2
2
,
0
3
,
2
3
,
4
2
,
3
3
,
5
3
,
6
2
,
6
3
consisting of points at distance 2 from N , see Figure 3. We now generalize this. Let p ≥ 5
be a prime and let
S(p) =
1
(p− 1)/2
,
1
(p− 3)/2
, . . . ,
1
3
,
1
2
,
2
3
, . . . ,
(p− 3)/2
(p− 1)/2
.
Then
Theorem 4.2. The concatenation of sequences
S2(p) = S(S + 1)(S + 2) . . . (S + p− 1),
where S = S(p), is the Farey circuit consisting of those points of distance 2 from N . The
length of S1(p) is p and the length of S2(p) is p(p− 4). There are (p− 1)/2 poles.
Proof. We first observe that the points in S2(p) do have distance 2 from N . Indeed, the
points 1k and
m−1
m for 2 ≤ k,m ≤
p−1
2 have distance 2 from N =
1
0 as
1
k ↔
0
1 and
m−1
m ↔
1
1 and none of these points have distance 1 from
1
0 . (The symbol ↔ means
adjacent to.)
The transformation t 7→ t+ 1 fixes 10 and preserves distance so that all points in S + k
have distance 2 from N = 10 . We now show that S2(p) is a Farey circuit. Clearly there are
edges between 1k and
1
k+1 for k ≥ 2 and between
k
k+1 and
k+1
k+2 for k ≥ 2. So, we only
42 Ars Math. Contemp. 20 (2021) 37–50
need to show that there is an edge between the last vertex in S + k and the first vertex in
S + k + 1. The last vertex of S + k is
k +
(p− 3)/2
(p− 1)/2
=
(p− 3 + kp− k)/2
(p− 1)/2
.
The first vertex of S + k + 1 is
k + 1 +
1
(p− 1)/2
=
(kp− k + p+ 1)/2
(p− 1)/2
.
As
[(p− 3 + kp− k)/2][(p− 1)/2]− [(kp− k + p+ 1)/2][(p− 1)/2] = −p+ 1,
we see that the last vertex of S + k is adjacent to the first vertex of S + k+1. Thus, S2(p)
is a Farey circuit consisting of points of distance 2 from 10 .
Now S1(p) clearly has p points, and the set S(p) has p−4 points, thus S2(p) has p(p−4)
points. The poles are 10 ,
2
0 , . . . with
k
0 =
−k
0 , and so the number of poles is
p−1
2 .
5 Drawing M3(7)
The map M3(7) has 24 vertices with Farey coordinates{
1
0
,
2
0
,
3
0
,
0
1
,
1
1
,
2
1
,
3
1
,
4
1
,
5
1
,
6
1
,
0
2
,
1
2
,
2
2
,
3
2
,
4
2
,
5
2
,
6
2
,
0
3
,
1
3
,
2
3
,
3
3
,
4
3
,
5
3
,
6
3
}
.
The first circuit is S1(7) = 01 ,
1
1 , . . . ,
6
1 , and we draw a polygonal curve C1(7), surrounding
1
0 , containing the points of S1(7). We draw a bigger simple closed curve C2(7), also
surrounding 10 , containing the points of S2(7). In Figure 3, C2(7) passes through the
points 13 ,
6
3 ,
6
2 ,
5
3 , . . . .
Finally, we can draw a simple closed curve C3(7) exterior to both C1(7) and C2(7)
which contains the poles 20 and
3
0 , see the dotted line in Figure 3. The pole
2
0 is a vertex
of seven triangles whose base is on the second circuit. One of these triangles is 63 ,
2
0 ,
1
3 and
the others are found by adding 1, 2, 3, 4, 5, 6 to these three points. For example, adding 1
to 63 ,
2
0 ,
1
3 gives
2
3 ( =
9
3 ),
2
0 ,
4
3 . (Adding the integer k has the geometric effect of rotating
M3(7) by 2πk .) The pole
3
0 is a vertex of seven quadrilaterals which are unions of two Farey
triangles, and also have one edge on C2(7). One of these is 13 ,
5
2 ,
3
0 ,
1
2 and we get the other
six by adding 1, 2, 3, 4, 5, 6. We end up with a 42-sided polygon pictured in Figure 3 (for
now ignore the dashed curves). It is interesting that exactly the same polygon was obtained
by E. Schulte and J. M. Wills in [9] by purely geometric methods.
6 The 14-sided polygon
We now show how to obtain a 14-sided polygon out of the Farey map M3(7) with the same
side-pairings as the Klein surface. As M3(7) has 42 edges and we need a 14-sided polygon
we define a new-edge to be a union of three consecutive edges which include vertices with
Farey coordinates 20 and
3
0 .
For example, our first new-edge goes from 20 to
5
3 to
3
2 to
3
0 and our second new edge
goes from 30 to
6
2 to
6
3 to
3
0 , see Figure 3. We now replace the new-edges by dashed lines.
I. Ivrissimtzis et al.: From Farey fractions to the Klein quartic and beyond 43
Figure 3: Drawing of M3(7) with Farey coordinates.
In Figure 3 the dashed line labelled 1 goes from a vertex labelled 20 to a vertex labelled
3
0 surrounding the vertices
5
3 and
3
2 of the first new edge, and similarly the dashed line
labelled 2 goes from 30 to
2
0 surrounding the vertices
6
2 and
6
3 . Notice that the dashed lines
are not part of the map M3(7), they are just a convenient way of representing our 14-
sided polygon. We can associate four Farey fractions to each dashed edge. For example,
associated to the dashed edge 1 we have the Farey fractions 20 ,
5
3 ,
3
2 ,
3
0 . We pair two new-
edges if their associated Farey fractions are the same. For example, consider the new-edge
labelled 6 in Figure 3. The associated Farey fractions are 30 ,
3
2 ,
5
3 ,
2
0 . These are the same
Farey fractions, but in reverse order as for the new-edge 1. This means we identify the
new edges 1 and 6 orientably. Similarly we get the other six identifications. Thus the
identifications are
1 ↔ 6, 3 ↔ 8, 5 ↔ 10, 7 ↔ 12, 9 ↔ 14, 11 ↔ 2, 13 ↔ 4.
This is exactly the same side-pairing as found by Klein from his 14-sided polygon which
shows that our 14-sided polygon does give the Klein quartic. Our way of finding the side
identifications is much more straightforward than the method used in Klein’s paper, which
we will summarize in §8.
7 Farey Coordinates for the Klein map
A regular map has type {m,n} if every face has size m and every vertex has valency n.
(We are following [1] here and not [3] where these numbers are reversed.) Now M3(n)
is a regular map of type {3, n} because 10 is adjacent to
0
1 , . . . ,
n−1
1 . Now M3(7) is the
44 Ars Math. Contemp. 20 (2021) 37–50
Klein map, or, in the standard notation in [1], the map {3, 7}8. The ‘8’ here is the length
of a Petrie polygon. (For Petrie polygons and how we find the lengths of Petrie polygons
using Farey fractions see [11].)
As noted in the introduction, the Klein map M3(7) is embedded in the Klein surface
H∗/Γ(7). The term “Klein map” comes from the drawing on page 320 of [7], or page 120
of [8], of Klein’s 14-sided polygon. After the given side identifications this does give a map
on a surface of genus 3. See Figure 4 (and just ignore the Farey coordinates in this diagram
for now). This is not the Klein map, for it is not regular, having vertices of different valency.
It consists of 336 triangles while the Klein map M3(7) has 56 triangles. Nevertheless, we
can easily obtain the Klein map from Figure 4. The vertices of the map are the vertices of
valency 14. Before we describe the Klein map structure on this surface we show how to
associate the 24 Farey fractions modulo 7 to the 24 vertices.
First, we assign the Farey coordinate 10 to the centre point. We note that there are
two circuits of seven vertices centred at 10 . We give the first circuit the Farey coordinates
0
1 ,
1
1 , . . . ,
6
1 . If we extend the perpendicular bisector from
1
0 to the hyperbolic line between
0
1 and
1
1 we get to another vertex of valency seven to which we assign the coordinate
0+1
1+1 =
1
2 . Similarly, we extend the perpendicular bisector from
1
0 to the hyperbolic line
between 11 and
1
2 to a vertex of valency seven which we give the Farey coordinate
3
2 . By
continuing, we find all vertices with Farey coordinates 12 ,
3
2 ,
5
2 ,
0
2 ,
2
2 ,
6
2 . Thus we have now
found all vertices with Farey coordinates xi for i = 1, 2 and we just have to find the vertices
with Farey ccordinates x0 or
x
3 which lie on the boundary of K. After Klein’s identifications
shown in Figure 3, we see that the 14 corners of K belong to two classes, which we can
label 20 ,
3
0 . Between any two of these vertices there is precisely one more vertex of M3(7).
(After side identifications these vertices also have valency 14.) We can assign to them
Farey coordinates of the form x3 just by reading them off from Figure 3. In fact, each
x
3
occurs exactly twice and we can now pair sides of K that have exactly the same value of x.
Again, this gives exactly the same side pairing as Klein found. We thus have two methods,
in sections 7 and 8, of using Farey coordinates to get Klein’s pairings just by observation.
Figure 4 gives a description of Klein’s work using Farey coordinates. We see that each
of the 14 sides of the boundary of K consists of a Farey edge and a non-Farey edge. The
segment from 20 to
x
3 is a Farey edge whilst the segment from
x
3 to
3
0 is not a Farey edge.
There is no automorphism of K mapping one segment to the other since all elements of Γ
map Farey edges to Farey edges. Note that by section 3, the Klein map has 24 vertices, 56
faces and 84 edges.
We now give the map structure. The vertices of the map are the points of valency 14 in
Figure 4, that is, those points that have been given Farey coordinates. An edge joins points
with Farey coordinates ac iand
b
d if and only if ad− bc ≡ 1 (mod 7). Three vertices with
Farey coordinates ac ,
b
d and
e
f form a triangular face if and only if e ≡ a + b (mod 7),
f ≡ c + d (mod 7). For example, there is a triangle with vertices 41 ,
4
3 and
6
3 for
6
3
represents the same point as 14 , for
1
4 =
−6
−3 .
8 What Klein did
Here we review Klein’s original construction of his fundamental domain of the congruence
subgroup Γ(7), and show how this construction can be interpreted in terms of the Farey
machinery we described above.
By the end of section 10 of [5] Klein had obtained the equation of his quartic curve and
I. Ivrissimtzis et al.: From Farey fractions to the Klein quartic and beyond 45
Figure 4: Farey coordinates on the Klein surface.
in section 11 he started to discuss the Riemann surface of this algebraic curve and also the
corresponding map. In fact, this was one of the first publications to use maps (or in today’s
language dessins d’enfants) in a profound way, pointing up the deep correspondence be-
tween maps and algebraic curves. While this correspondence was not properly understood
until Grothendieck’s Esquisse d’un programme some 105 years later [2], we note that in an
interesting anticipation of Grothendieck’s programme, Figure 2 of Klein’s follow-up paper
[6] shows the ten planar dessins of type (2, 3, 11) and degree 11.
Klein’s quite complicated construction of his fundamental domain comes from con-
sidering fundamental regions for subgroups of indices 7 and 8 in the modular group. In
section 12, he writes (in German) “In order not to make these considerations too abstract
I will resort to the ω-plane”; this is the upper-half plane on which the modular group acts.
46 Ars Math. Contemp. 20 (2021) 37–50
In Figure 6, he constructs a hyperbolic polygon corresponding to his 14-sided polygon
describing his surface. Then, in Figure 7, he draws semicircles (hyperbolic lines) in the
upper-half plane with rational vertices, which correspond to the edges of his 14-sided poly-
gon. Now consider this polygon as being inscribed in the unit disc so the vertices all lie on
the boundary circle. As the unit disc is conformally equivalent to the upper-half plane the
boundary circle corresponds to the real axis and so, every point of the circle has some real
coordinate. He starts with one edge (labelled 1) of his 14-sided polygon corresponding to
two consecutive edges of the polygon in the upper-half plane with vertices 27 ,
1
3 and
1
3 ,
3
7 .
(As we already noted above, 27 ,
1
3 is a Farey edge while
1
3 ,
3
7 is not, therefore we cannot
map one to the other by an element of Γ). A second edge (labelled 6) is given as the pair of
consecutive edges 187 ,
8
3 and
8
3 ,
19
7 . The Möbius transformation corresponding to the matrix(
113 −35
42 −13
)
in Γ(7) maps edge 1 (i.e. 27 ,
1
3 ,
3
7 ) to edge 6 (i.e.
18
7 ,
8
3 ,
19
7 ), and one more explicit example
of edge pairing is given. He states that in total seven such matrices can be found that give
all the side pairings. We feel that our technique of just using Farey coordinates is much
easier.
9 M3(11)
About a year after Klein wrote his paper [5] on the quartic curve, he wrote a further paper
[6] with the same title but with ‘siebenter’ replaced with ‘elfter’, i.e. ‘seventh’ replaced
with ‘eleventh’; basically, he was considering H∗/Γ(11). In that paper he did not draw a
diagram of the fundamental region equivalent to his drawing of K in [5]. Here we show
how to draw the Farey map M3(11) in a similar way to how we drew M3(7). This Farey
map will be embedded in the surface H∗/Γ(11).
The first circuit of vertices at distance 1 from 10 is
S1(11) =
0
1
,
1
1
, . . . ,
10
1
.
Now consider the sequence of vertices
S(11) =
1
5
,
1
4
,
1
3
,
1
2
,
2
3
,
3
4
,
4
5
;
and then the second circuit is
S2(11) = S(11) (S(11) + 1) . . . (S(11) + 10).
The orientation-preserving automorphism group of M3(11) is PSL(2, 11) of order 660
so the Farey map M3(11) has 660/2 = 330 edges, 660/3 = 220 triangles and 660/11 =
60 vertices. The Farey coordinates of the vertices are 10 ,
2
0 ,
3
0 ,
4
0 ,
5
0 and all Farey fractions
of the form rs for r = 0 to 10 and s = 1 to 5.
To draw the map we just need to find the 220 triangular faces. Because z 7→ z + 1 is
an automorphism of M3(11), which acts as a rotation about the centre 10 of the map, we
see that this map is divided into eleven congruent sectors each containing 220/11 = 20
triangles each. We construct one such sector W , shown in Figure 5, by starting from the
I. Ivrissimtzis et al.: From Farey fractions to the Klein quartic and beyond 47
Figure 5: The sector W .
central triangle 10 ,
0
1 ,
1
1 and adding 19 distinct triangles whose vertices lie in S(11). Exactly
8 of these 19 triangles have a vertex on the first circuit S1(11) ( 01 or
1
1 in particular) and
are uniquely determined. For the remaining 11 triangles, which either have three vertices
on S2(11), or two vertices on S2(11) and a pole vertex, there are several choices satisfying
the condition that they are distinct under rotation about 10 .
Figure 5 shows one such solution as the union of 20 triangles. The actual Farey coordi-
nates are
P0 =
1
0 P1 =
0
1 P2 =
1
5 P3 =
1
4 P4 =
1
3 P5 =
2
5
P6 =
1
2 P7 =
5
0 P8 =
6
2 P9 =
7
4 P10 =
3
5 P11 =
6
3
P12 =
4
0 P13 =
2
3 P14 =
6
4 P15 =
3
0 P16 =
3
4 P17 =
4
2
P18 =
4
5 P19 =
2
0 P20 =
6
5 P21 =
1
1
Each point Pi is labelled i in Figure 5 to reduce clutter.
Now let
W ∗ = W ∪ (W + 1) ∪ · · · ∪ (W + 10)
where W + k is defined as in Section 5, that is, geometrically, is the rotation of W by
2π
k . Then W
∗ is the union of 220 triangles as required and its boundary is a polygon with
11× 18 = 198 sides. A diagram of the map W ∗ is given in Figure 6.
Table 1 in the Appendix shows a list of the 198 boundary vertices of W ∗ arranged in
11 rows. The first row corresponds to W and the kth row is just the first row plus (k − 1).
We now notice that we have an orientable side pairing. For example, the first edge in row
1 going from 15 to
1
4 is paired with the edge in row 5 going from
1
4 to
1
5 , the next edge in
row 1 going from 14 to
1
3 is paired with the edge in row 8 going from
1
3 to
1
4 . Proceeding
in this way we find that all the 198 edges of the polygon are paired orientably which shows
that this polygon represents an orientable surface which must be M3(11). As the map W ∗
has 60 vertices, 220 edges, and 330 triangles, by the Euler-Poincaré formula the genus of
the surface is 26.
48 Ars Math. Contemp. 20 (2021) 37–50
Figure 6: The map W ∗.
ORCID iDs
Ioannis Ivrissimtzis https://orcid.org/0000-0002-3380-1889
David Singerman https://orcid.org/0000-0002-0528-5477
References
[1] H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, volume 14
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[2] A. Grothendieck, Esquisse d’un programme, in: L. Schneps and P. Lochak (eds.), Geomet-
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ries, pp. 5–48, 1997, doi:10.1017/cbo9780511758874.003, with an English translation on pp.
243–283.
[3] G. A. Jones and D. Singerman, Theory of maps on orientable surfaces, Proc. London Math.
Soc. 37 (1978), 273–307, doi:10.1112/plms/s3-37.2.273.
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[4] G. A. Jones and D. Singerman, Complex Functions: An Algebraic and Geometric Viewpoint,
Cambridge University Press, Cambridge, 1987, doi:10.1017/cbo9781139171915.
[5] F. Klein, Ueber die Transformation siebenter Ordnung der elliptischen Functionen, Math. Ann.
14 (1878), 428–471, doi:10.1007/bf01677143.
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(1879), 533–555, doi:10.1007/bf02086276.
[7] F. Klein, Gesammelte Mathematische Abhandlungen, Volume 3: Elliptische Funktionen, Ins-
besondere Modulfunktionen Hyperelliptische und Abelsche Funktionen Riemannsche Funktio-
nentheorie und Automorphe Funktionen, Springer, Berlin Heidelberg, 1923.
[8] S. Levy, The Eightfold Way: The Beauty of Klein’s Quartic Curve, volume 35 of Mathemat-
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http://library.msri.org/books/Book35/contents.html.
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amuc/article/view/913.
50 Ars Math. Contemp. 20 (2021) 37–50
A Appendix
Table 1: The boundary vertices of W ∗. The last vertex of a row is repeated as the first
vertex of the row below. Each row represents a sector; the first row represents sector W in
Figure 5. Vertices in bold belong to edges which are paired with edges in the first row W .
1
5
1
4
1
3
2
5
1
2
5
0
6
2
7
4
3
5
6
3
4
0
2
3
6
4
3
0
3
4
4
2
4
5
2
0
6
5
6
5
5
4
4
3
7
5
3
2
5
0
8
2
0
4
8
5
9
3
4
0
5
3
10
4
3
0
7
4
6
2
9
5
2
0
0
5
0
5
9
4
7
3
1
5
5
2
5
0
10
2
4
4
2
5
1
3
4
0
8
3
3
4
3
0
0
4
8
2
3
5
2
0
5
5
5
5
2
4
10
3
6
5
7
2
5
0
1
2
8
4
7
5
4
3
4
0
0
3
7
4
3
0
4
4
10
2
8
5
2
0
10
5
10
5
6
4
2
3
0
5
9
2
5
0
3
2
1
4
1
5
7
3
4
0
3
3
0
4
3
0
8
4
1
2
2
5
2
0
4
5
4
5
10
4
5
3
5
5
0
2
5
0
5
2
5
4
6
5
10
3
4
0
6
3
4
4
3
0
1
4
3
2
7
5
2
0
9
5
9
5
3
4
8
3
10
5
2
2
5
0
7
2
9
4
0
5
2
3
4
0
9
3
8
4
3
0
5
4
5
2
1
5
2
0
3
5
3
5
7
4
0
3
4
5
4
2
5
0
9
2
2
4
5
5
5
3
4
0
1
3
1
4
3
0
9
4
7
2
6
5
2
0
8
5
8
5
0
4
3
3
9
5
6
2
5
0
0
2
6
4
10
5
8
3
4
0
4
3
5
4
3
0
2
4
9
2
0
5
2
0
2
5
2
5
4
4
6
3
3
5
8
2
5
0
2
2
10
4
4
5
0
3
4
0
7
3
9
4
3
0
6
4
0
2
5
5
2
0
7
5
7
5
8
4
9
3
8
5
10
2
5
0
4
2
3
4
9
5
3
3
4
0
10
3
2
4
3
0
10
4
2
2
10
5
2
0
1
5