A Circle Limit III Calculation

Douglas Dunham
Department of Computer Science
University of Minnesota, Duluth
Duluth, MN 55812-3036, USA
E-mail: ddunham@d.umn.edu
Web Site: http://www.d.umn.edu/ddunham/

Abstract
M.C. Eschers Circle Limit III is usually thought to be the most appealing of his four Circle Limit patterns. Two
artistic/mathematical questions seem to arise: (1) what angle do the white backbone lines make with the bounding
circle, and (2) are other such patterns of fish possible? H.S.M. Coxeter answered the first question and I described
a 3-parameter family of possible fish patterns in my 2006 Bridges Conference paper. In this paper, I combine those
questions by calculating the intersection angle for any such fish pattern.

1. Introduction
Figure 1 below shows a computer rendition of the Dutch artist M.C. Eschers hyperbolic pattern Circle
Limit III. Figure 2 shows a fish pattern from the combinatorial family of Circle Limit III patterns, but with
an angular fish motif in the style of Eschers Circle Limit I. In my 2006 Bridges paper [4], I introduced

Figure 1: A rendition of Eschers Circle Limit III.

Figure 2: A pattern in the general family of Circle

Limit III, but in the style of Circle Limit I.

the concept of a 3-parameter family of Circle Limit III patterns indexed by the numbers p, q, and r of fish
meeting at right fin tips, left fin tips, and noses respectively. Such a pattern was denoted by the triple (p, q, r).

So Circle Limit III and the pattern of Figure 2 would be named (4, 3, 3) and (4, 4, 3) respectively. Of course
r should be odd so that the fish swim head-to-tail, and p, q, and r should all be greater than or equal to 3.
In keeping with the characteristics of Circle Limit III, we place some restrictions on the patterns in this
family. The first is that right fin tips should be at the center of the bounding circle (for patterns with fin tips
at the center). The second condition is that colors of the fish should obey the map-coloring principle: fish
that share an edge should be different colors. The fish should also be colored symmetrically and fish along
the same backbone line should be the same color. Figures 3 and 4 show (3, 4, 3) and (5, 3, 3) patterns.
Note the differences between these patterns and Circle Limit III. In particular, requiring that a right fin tip
be at the center allows us to distinguish between (p, q, r) and (q, p, r) when p 6= q.

Figure 3: A (3, 4, 3) fish pattern.

Figure 4: A (5, 3, 3) fish pattern.

As has been recounted before, Escher was inspired to create his Circle Limit patterns by a figure
showing a tessellation of the hyperbolic plane in one of Canadian mathematician H.S.M. Coxeters papers.
Coxeter in turn, being intrigued by Circle Limit III, wrote two papers on the geometry of the backbone lines
[2, 3]. In the issue of The Mathematical Intelligencer containing Coxeters second paper, an anonymous
editor wrote the following caption for the cover of that issue, which showed Circle Limit III:
Coxeters enthusiasm for the gift M.C. Escher gave him, a print of Circle Limit III, is understandable. So is his continuing curiosity. See the articles on pp. 3546. He has not, however
said of what general theory this pattern is a special case. Not as yet. [1]
Coxeter did not describe such a general theory, or at least did not publish it. In my 2006 Bridges paper [4],
I provided a formula for the angle that the backbone lines make with the bounding circle for a (p, 3, 3)
pattern. This result generalized the calculations in Coxeters papers (which only considered (4, 3, 3)).
The main goal of this paper is calculate the intersection angle between the bounding circle and a
backbone line of a general (p, q, r) pattern (all backbone lines of a pattern make the same angle with the
bounding circle). First we review some hyperbolic geometry that is used in the calculation. Then we proceed
through the calculation, list some results and show a couple of new patterns. Finally, we indicate directions
of further research.

2. Hyperbolic Geometry
Eschers Circle Limit patterns can be interpreted as repeating patterns of the hyperbolic plane, which is
often useful in analyzing their geometry. The hyperbolic plane is a surface of constant negative (Gaussian)
curvature and can be considered to be dual to the sphere, which has constant positive curvature. This duality
can sometimes be exploited to gain insight into facts about hyperbolic geometry. However, unlike the sphere,
the entire hyperbolic plane has no smooth, isometric (distance preserving) embedding in Euclidean 3-space
as was proved by David Hilbert in 1901 [6]. Thus, we must rely on Euclidean models of hyperbolic geometry
in which distance is measured differently and concepts such as hyperbolic lines have interpretations as
Euclidean constructs.
We will use two models of hyperbolic geometry: the Poincare disk model, and the Weierstrass model. In
the Poincare disk model the points are just the (Euclidean) points within a Euclidean bounding circle, which
we will take to be the unit circle in the xy-plane. Hyperbolic lines are represented by circular arcs orthogonal
to the bounding circle (including diameters). For example, the backbone lines lie along hyperbolic lines in
Figure 2. The disk model is conformal: the hyperbolic measure of an angle is the same as its Euclidean
measure. As a consequence, all fish in a Circle Limit III pattern have roughly the same Euclidean shape.
However equal hyperbolic distances correspond to ever smaller Euclidean distances toward the edge of the
disk. So all the fish in Circle Limit III patterns are the same (hyperbolic) size. The Poincare disk model
is appealing to artists (and appealed to Escher) since an infinitely repeating pattern could be shown in a
bounded area and shapes remained recognizable even for small copies of the motif, due to conformality.
A careful examination of the backbone arcs of the fish in Circle Limit III reveals that they are not hyperbolic lines they make an angle of about 80 with the bounding circle. They are actually equidistant curves
in hyperbolic geometry: curves at a constant hyperbolic distance from the hyperbolic line with the same
endpoints on the bounding circle. For every hyperbolic line and a given distance, there are two equidistant
curves, called branches, at that distance from the line, one each side of the line. In the Poincare disk model,
those two branches are represented by circular arcs making the same (non-right) angle with the bounding
circle on either side of the corresponding hyperbolic line. Equidistant curves are the hyperbolic analogs of
small circles in spherical geometry: a small circle of latitude in the northern hemisphere is equidistant from
the equator (a great circle or line in spherical geometry), and has a second corresponding small circle at
the same latitude in the southern hemisphere. Escher used only one branch for fish backbones from each
pair of equidistant curves in Circle Limit III.
The points in the Weierstrass model are the points on the upper sheet of the
of two sheets

hyperboloid
x2
x1

x2 + y 2 z 2 = 1. The hyperbolic distance between two points y1 and y2 is given by:

z2
z1
1
cosh (z1 z2 x1 x2 y1 y2 ). Each hyperbolic line in this geometry is the intersection of a Euclidean
plane through the origin with this upper sheet, and sois onebranch of a hyperbola. As in spherical geom`x

etry, a line can be represented by its pole, a 3-vector `y on the dual hyperboloid `2x + `2y `2z = +1,
`z
so that the line is the set of points satisfying x` x + y`y + z`z = 0. Again, in analogy to spherical geometry,
equidistant curves are represented by x` x + y`y + z`z = d, where d is the hyperbolic distance between
the equidistant curve and its line. There is a simple relationship between the Weierstrass model and the disk
model: stereographic

projection onto the xy-plane toward

the
vertex
of the lower
sheet of the hyperboloid
0
x
x/(1 + z)

of two sheets, 0 , which is given by the formula: y 7 y/(1 + z) .

1
z
0

3. The Calculation of the Intersection Angle

The calculation of the angle a backbone line of a (p, q, r) pattern makes with the bounding circle proceeds
through several steps. We first use a hyperbolic trigonometry formula to locate points on a fundamental
region for a fish motif. We note that a fundamental region for a fish can be taken to be a kite, a quadrilateral
2

with two opposite angles equal, the angles being 2

p , r , q , and r . Such tessellations by kites are shown
for the Circle Limit III pattern in Figure 5 and for the Circle Limit III pattern with a nose/tail point at the
origin in Figure 6. Then we use the computed points on the Weierstrass model to find one of the points on
the equidistant curve. Finally we project that point back down to the Poincare disk, and use symmetry and
Euclidean geometry to find cos .

Figure 5: The kite tessellation superimposed on the

Circle Limit III pattern.

Figure 6: A nose-centered version of Figure 5.

If p = q, the backbone lines are hyperbolic lines and = 90 . So we can assume p 6= q and, in fact,
p < q since the backbone lines of (p, q, r) and (q, p, r) make the same angle. To simplify the calculations,
we assume that the tail point of one of the kites is at the origin and the tail angle is bisected by the positive
x-axis, with the p-fold point P above the axis and the q-fold point Q below the axis. This configuration is
shown in Figure 7, which also shows the hyperbolic line ` determined by P and Q, and the equidistant curve
through the origin and the other r-fold point, R,of the kite. Figure 8 shows a detailed blowup of the area
around the kite.
We start by solving the hyperbolic triangle OP Q for the side lengths d p and dq of OQ and OP
respectively. Actually, we never need the values of d p or dq themselves, only the values of their hyperbolic cosines which, for notational convenience, we will call coshp and coshq p
(not to be confused
(coshp2 1) and
with cosh(p)
or
cosh(q),
neither
of
which
are
useful).
Similarly,
we
let
sinhp
=
p
sinhq = (coshq 2 1). Using the conventional notation for a triangle (a, b, and c denote the lengths of
the sides opposite angles A, B, and C), one of the standard hyperbolic trigonometry formulas is: cosh c =
cos A cos B+cos C
[5, page 406]. We apply that formula to the triangle OP Q, to obtain:
sin a sin b
coshp =

cos(/q) cos(/r) + cos /p

,
sin(/q) sin(/r)

coshq =

cos(/p) cos(/r) + cos /q

sin(/p) sin(/r)

(1)

`
P
P
O

/2r
/2r

Figure 7: The nose-centered (4, 3, 3) tessellation

showing the bisecting line ` of the OP RQ kite, the Figure 8: A blowup of the OP RQ kite area of Figure 7.
backbone line through O and R, and radius OB.
We can use these values to find the Weierstrass coordinates of P and Q:

cos(/2r)sinhq

P = sin(/2r)sinhq
coshq

cos(/2r)sinhp

Q = sin(/2r)sinhp
coshp

(2)

Then the coordinates of the pole of the line ` determined by P and Q are given by:

`x
P Q

` = `y =
|P Q|
`z

(3)

Where the hyperbolic cross-product P Q is given by:

Py Qz P zQy

P Q = Pz Qx P xQz
Px Qy + P yQx

(4)

(note the change of sign on the last component), and the norm of a line pole vector V is given by:
|V | =

(Vx2 + Vy2 Vz2 )

(5)

(again note the minus sign before the last term).

Before computing the matrix representing reflection across
`, we consider a simpler case. The pole

sinh d

vector representing the hyperbolic line through the point 0

and perpendicular to the x-axis is
cosh d

cosh d

0
, and the matrix Ref representing reflection of Weierstrass points across that line is given by:

sinh d

cosh 2d 0 sinh 2d

Ref =
0
1
0

sinh 2d 0 cosh 2d
where d is the the hyperbolic distance from the line (or point) to the origin. Thus reflection across a line
whose nearest point to the origin is rotated by angle from the x-axis is given by:
Rot()Ref Rot()

cos sin 0

where, as usual, Rot() = sin cos 0 . From ` we can identify sinh d as `z , and cosh d as
0
0
1
p 2
2
(`x + `x ), which we denote . Then cos = `x / and sin = `y /. Thus sinh 2d = 2 sinh d cosh d =
2`z and cosh 2d = cosh2 d + sinh2 d = 2 + `2z . We can now compute Ref` , the matrix for reflection across
` as:
`

`
`y
x
x
`y 0
0
(2 + `2z ) 0
2`z

`y `x

`x
0
1
0
Ref ` = `y

0
0

2`z
0 (2 + `2z )
0
0 1
0 0 1
Finally we use Ref` to reflect the origin to R since the kite OP RQ is symmetric across `:

0
2`x `z

R = Ref ` 0 = 2`y `z
1
2 + `2z

(6)

Now we project Weierstrass point R to the Poincare model:

v =
0

2`x `z
1+2 +`2z
2`y `z
1+2 +`2z

(7)

u
u

The three points v , v , and the origin determine the (equidistant curve) circle centered at w =
0
0
2
2
(u + v )/2v on the y-axis, as found by simple geometry. By easy algebra, we find the y-coordinate of the
intersection points of this circle, x 2 + (y w)2 = w2 , with the unit circle to be yint = 1/2w = v/(u2 + v 2 ).
As in Figure 7, let B denote the right-hand intersection point. Then the central angle, , made by the radius
OB with the x-axis is the complement of , the angle of intersection of the equidistant curve with the
bounding circle, as shown in Figure 7. This can be seen since the equidistant circle is symmetric across the
perpendicular bisector of OB. Thus y int = sin = cos , so that
cos = yint = v/(u2 + v 2 )
which is the desired result.

(8)

4. Results
The above calculations q
for cos have been programmed and have been compared with the values given by

) given in my 2006 Bridges paper for (p, 3, 3) patterns [4]. This

the formula cos = 21 1 3/4 cos 2 ( 2p
was done for p = 4, 5, . . . , 9, 10, 100, and 1000. In each case the values agreed to within 12 or 13 decimal
places (about the limit of double precision on our computers, given roundoff errors in the calculations).
Also, a program was designed to draw one of the equidistant curves on a kite tessellation based on a (p, q, r)
pattern. In each case we tested, the equidistant curve seemed to pass through the appropriate vertices, as is
shown for the particular cases (3, 4, 5) and (4, 5, 3) in Figures 9 and 10 respectively.

Figure 9: The kite tessellation and a backbone line Figure 10: The kite tessellation and a backbone line
equidistant curve for a (3, 4, 5) pattern.
equidistant curve for a (4, 5, 3) pattern.
Figure 11 shows the (5, 3, 3) pattern of Figure 4 translated so that it is nose-centered like the Circle
Limit III pattern of Figure 6. In each of patterns of Figures 4 and 12, the backbone lines form a Euclidean
equilateral triangle. All four of these patterns show that the backbone lines are not hyperbolic lines, since
hyperbolic lines are represented by straight Euclidean lines if and only if they pass through the center of the
bounding circle.

5. Conclusions and Future Work

For any (p, q, r) pattern, we have shown a calculation that computes the angle an equidistant backbone
curve makes with the bounding circle. Another unanticipated outcome was to figure out how to transform a
fin-centered Circle Limit III pattern to a nose-centered pattern, as was done in Figures 6 and 11.
However, there is still work to be done. I would certainly like to know if it is possible to simplify
the calculation above down to a single formula as was done for the special (p, 3, 3) that was treated in my
2006 Bridges paper [4]. It would also be useful to be able to transform one (p, q, r) pattern to another one
with different values of p, q, and r. A seemingly difficult problem is to automate the process of coloring a
(p, q, r) pattern so that it has the same color along any line of fish and adheres to the map-coloring principle
that adjacent fish have different colors. Currently I determine colorings by hand, and although it may be

Figure 11: A nose-centered version of the (5,3,3) Figure 12: A (3,5,3) pattern related to our patterns
pattern.
of Figures 4 and 11.
possible to program symmetric colorings of any repeating pattern, the requirement that fish along a backbone
line be the same color adds an extra degree of difficulty to coloring (p, q, r) patterns.

Acknowledgments
I would like to thank Lisa Fitzpatrick and the staff of the Visualization and Digital Imaging Lab (VDIL) at
the University of Minnesota Duluth.

References
[1] Anonymous, On the Cover, Mathematical Intelligencer, 18, No. 4 (1996), p. 1.
[2] H.S.M. Coxeter, The Non-Euclidean Symmetry of Eschers Picture Circle Limit III, Leonardo, 12
(1979), pp. 1925.
[3] H.S.M. Coxeter. The trigonometry of Eschers woodcut Circle Limit III, Mathematical Intelligencer, 18, No. 4 (1996), pp. 4246. This his been reprinted by the American Mathematical Society
at: http://www.ams.org/featurecolumn/archive/circle_limit_iii.html and also in
M.C. Eschers Legacy: A Centennial Celebration, D. Schattschneider and M. Emmer editors, Springer Verlag, New York, 2003, pp. 297304.
[4] D. Dunham, More Circle Limit III Patterns, in Bridges London: Mathematical Connections in Art,
Music, and Science, (eds. Reza Sarhangi and John Sharp), London, UK, 2006, pp. 451458, 2006.
[5] M. Greenberg, Euclidean & Non-Euclidean Geometry, Third Edition: Development and History, 3nd
Ed., W. H. Freeman, Inc., New York, 1993. ISBN 0716724464

[6] David Hilbert, Uber

Flachen von konstanter gausscher Krummung, Transactions of the American Mathematical Society, pp. 8799, 1901.