The Beauty of Blaschke Products

Ulrich Daepp, Pamela Gorkin, Gunter Semmler, and Elias Wegert

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Complex Arithmetic and Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Seeing Complex Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Hyperbolic Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Blaschke Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16Blaschke Products and Ellipses in the Euclidean Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Blaschke Products and Ellipses in the Poincaré Disk Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 23Compositions of Blaschke Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Abstract

This chapter is dedicated to showing how visual tools, using geometry and color,can be used to enhance the understanding of statements about complex functions.In particular, the focus will be on the class of finite Blaschke products, and therelevant geometries are Euclidean as well as hyperbolic. Some of the geometrictools that will be used include symmetry, tilings, and curves generated by linesconstructed using particular properties of Blaschke products. The focus thenturns to the possibility of visualizing when a Blaschke product is the compositionof two (nontrivial) Blaschke products. Color appears in the phase portraits thatare constructed, and the main results are then validated with these visual tools.

U. Daepp · P. Gorkin (�)Bucknell University, Lewisburg, PA, USAe-mail: udaepp@bucknell.edu; pgorkin@bucknell.edu

G. Semmler · E. WegertTechnische Universität Bergakademie Freiberg, Freiberg, Germanye-mail: semmler@math.tu-freiberg.de; wegert@math.tu-freiberg.de

© Springer Nature Switzerland AG 2020B. Sriraman (ed.), Handbook of the Mathematics of the Arts and Sciences,https://doi.org/10.1007/978-3-319-70658-0_88-1

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This chapter begins with an introduction to finite Blaschke products, the Poincarédisk model, and phase portraits of complex functions.

Keywords

Blaschke product · Phase portrait · Ellipse · Poincaré disk model · Hyperbolicgeometry · Symmetry · Composition of functions · Complex function

Introduction

Geometry is one of the first mathematical disciplines in history, and this is notjust happenstance. Geometric ideas are abstractions of objects commonly found inthe human visual experience, and its theorems were first discovered by looking atsimple sketches. In this chapter, the focus is on the geometry of so-called hyperbolicpolynomials, also known as (finite) Blaschke products. Though there are manygeometries, functions will be considered in one of two: the Euclidean plane andthe Poincaré disk model.

Euclidean geometry may be thought of as a deductive structure of thoughts; thiswas first realized in the 13 books of the Elements by Euclid. Although Euclid’s textshave shaped the way modern mathematics is developed and taught, he remains anobscure figure in Greek mathematics. The Elements start from definitions, axioms,and postulates, from which all propositions are derived through logical reasoningand (though not always perfectly) without recourse to visual perception. Here,axioms and postulates are basic statements that are accepted without proof.

The postulates are stated below as they appear in Euclid’s elements; Heath’stranslation, without the parenthetical statements, is from Euclid (1956). The pos-tulates are stated in terms of construction:

(1) To draw a straight line from any point to any point.(2) To produce (extend) a finite straight line continuously in a straight line.(3) To describe a circle with any center and distance (radius).(4) That all right angles are equal to one another.(5) (The parallel postulate): That, if a straight line falling on two straight lines

makes the interior angles on the same side less than two right angles, the twostraight lines, if produced indefinitely, meet on that side on which the angles areless than two right angles.

The fifth postulate is different than the others – for one thing, it is difficult tounderstand. Playfair’s postulate, which is equivalent to the parallel postulate, iseasier to understand and states that

(P) In a plane, given a line and a point not on it, at most one line parallel to thegiven line can be drawn through the point.

The Beauty of Blaschke Products 3

In fact, many mathematicians felt the fifth postulate could be derived from the otherfour. Failure to show that this postulate could be deduced from the others led toother geometries. One of these, called hyperbolic geometry is based on the following“fifth postulate”:

(5′) Through any point not on a given line more than one straight line can be drawnparallel to the given line.

These geometries are quite different; for example, in Euclidean geometry, the sumof the angles in a triangle is the sum of two right angles, and in hyperbolic geometry,the sum is less than two right angles. (There is also a third geometry, calledspherical geometry, in which the sum is more than two right angles.) This chapterwill consider familiar functions and symmetries in the Euclidean and hyperbolicgeometries.

Informally speaking, symmetry means that an object does not change when youmove it in some way. For example, in Euclidean geometry, symmetries come fromtranslations, reflections, and rotations; see Fig. 1. More details on symmetries, aswell as similarities, will be given later.

A geometric model is specified by its points and lines. While there are otherhyperbolic models, the focus here is on the Poincaré disk model in which the pointsare the points of the open unit disk (inside the bounding circle) and the lines arecircles (or diameters) that are orthogonal to the bounding circle. Thus, points closeto the bounding circle may be “far” from each other in this model, but close toeach other in the Euclidean geometry. Hyperbolic geometry was attractive to thegraphic artist M. C. Escher, for example, because it is restricted to a disk and thuslies inside a bounded region. It also preserves angles; more precisely, the anglesbetween intersecting lines are equal to the Euclidean angles between the tangents tothe hyperbolic lines at the points of intersection. This model is said to be conformal,and therefore objects retain their shape in a repeating pattern, roughly speaking.

Fig. 1 Translation, reflection, and rotation

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Functions that behave well in hyperbolic space are the Möbius maps (or linearfractional transformations). While these will be discussed further in the sectionentitled “Complex Arithmetic and Geometry”, at this point it is only important tomention that the Möbius transformations that map the open unit disk D to itself andthe unit circle ∂D to itself have a special form:

M(z) = cz − z0

1 − z0z,

where z, z0 ∈ D, and c is a point on the unit circle. These maps and their propertiesare relatively easy to investigate. A next natural step would be to investigateproperties of products of such maps. One goal of this chapter is to consider theseproducts, known as Blaschke products, not only from the point of view of formalmathematics but also from a visual point of view. This will be done in the sectionentitled Blaschke Products.

While real-valued functions of real variables are easy to visualize using a two-dimensional graph, Blaschke products are complex-valued functions of complexvariables that map the unit disk to the unit disk and the unit circle to itself. Therefore,visualization requires four dimensions. While this might sound impossible, there isa method for doing precisely this. This method relies on a particular coloring of thecomplex plane and consideration of lines of constant modulus and argument. Thisprocedure will be explained in detail in the section Seeing Complex Functions. Forthe moment, we only illustrate this with a particular example in Fig. 2. The numberof products of Möbius transformations used is called the degree of the Blaschkeproduct, and it is known that the Blaschke product winds the circle around itselfthe same number of times as its degree. This can be seen in Fig. 2 by looking atthe unit circle and noting that each color shows up the same number of times as thedegree: Each color appears 5 times and the degree of the Blaschke product is 5. What

Fig. 2 Representation of a degree-5 Blaschke product in the plane and on the Riemann sphere

The Beauty of Blaschke Products 5

happens when points of the same color on the unit circle are connected with linesin the Euclidean plane? What happens in the Poincaré disk model? The answer willappear in sections entitled Blaschke Products and Ellipses in the Euclidean Planeand Blaschke Products and Ellipses in the Poincaré Disk Model, respectively.

While these results are certainly interesting from a geometric point of view, theyalso have a curious function-theoretic application. For this, a little more backgroundis required: Given two polynomials, it is easy to compose them. But given apolynomial p, it is usually very difficult to decide whether or not there are twopolynomials (of degree greater than one) such that when they are composed, theresulting function is p. It would seem that the same is true for Blaschke products, yetit turns out that it is possible to “see” when a Blaschke product is a composition oftwo other Blaschke products of degree greater than one. The technique for decidingwhether or not a Blaschke product is a composition will be discussed in the sectionCompositions of Blaschke Products.

The geometry of Blaschke products provides not only a deep and rich mathe-matical theory but also a theory that can be understood through visualization. Theauthors of this chapter hope that this serves to convince the reader of the beauty ofBlaschke products.

Complex Arithmetic and Geometry

Complex numbers z = x + iy can be represented as points z in the complex planeC, often associated with the names of Wessel, Argand, or Gauss. In this context, theelementary arithmetic operations admit geometric interpretations.

In the following, it is convenient to consider the point z not as being fixed, butas a representative of any point in the complex plane. Adding a (fixed) complexnumber b to z moves z to z + b, which induces a parallel shift of the complex planethat moves the origin 0 to b. Similarly, multiplication of z by a positive number r

amounts to dilating (or contracting) the plane by a stretching (contracting) factorof r , keeping the origin fixed. Multiplication by complex numbers of the formeiϕ , where ϕ is a real number (these are called unimodular and are representedby points on the unit circle ∂D), is another special transformation; the mappingz �→ eiϕz rotates the plane by an angle of ϕ about the origin. So complexaddition and multiplication represent three similarity transformations: translation,dilation/contraction, and rotation.

These operations can be combined freely with each other. For instance, multipli-cation with an arbitrary complex number a = reiϕ (with r > 0) is the combinationof a rotation and a dilation (in either order) that both keep the origin fixed. This iscalled a spiral similarity. Another example is the mapping z �→ eiϕ(z−c)+c, whichdescribes a rotation by an angle ϕ about the point c.

Expressed in mathematical jargon, the complex operations z �→ z + b (withb ∈ C) and z �→ a·z (with a ∈ C\{0}) generate a group with respect to composition.An arbitrary element of this group is the mapping z �→ az + b, which is alsothe most general form of a bijective holomorphic self-map of the complex plane.

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This group is an arithmetic model of the group of orientation preserving similaritytransformations of the Euclidean plane.

The fourth of the elementary similarity transformations, reflection, can also beexpressed as an operation on complex numbers: the mapping z �→ z sends z = x+iyto its complex conjugate z = x − iy, which represents reflection in the real lineR. It is a similarity transformation, but in contrast to the other three it reversesorientation, which is the reason why it will be excluded from the game.

It is also possible to reflect an object in a curved surface. In the plane, thesimplest such transformation is reflection in the unit circle, also called inversion.This transformation sends a point z to a point z∗ on the same ray emanating fromthe origin in such a way that the product of the absolute values |z| and |z∗| is 1.This may be viewed using complex arithmetic via the mapping z �→ 1/z. As anorientation reversing transformation, inversion plays against the rules. However,when combined with reflection in the real line, the mapping z �→ 1/z is obtained,and this can be admitted. The action of this transformation is illustrated in Fig. 3.

In fact, the new player z �→ 1/z paves the way for consideration of two non-Euclidean geometries, called hyperbolic and spherical, respectively. To enter thespherical world, the complex plane must be complemented by the point at infinity,z = ∞. A model for this extended complex plane, C ∪ {∞}, is the Riemann sphereS. With appropriate definitions of the arithmetic operations, like 1/0 = ∞ and1/∞ = 0, the mappings z �→ z + a, z �→ a · z, z �→ 1/z can be extended to theRiemann sphere. These mappings can then be combined arbitrarily (composed), andany such composition is a Möbius transformation,

f (z) = az + b

cz + d, a, b, c, d ∈ C, ad − bc = 0.

The Möbius transformations form a group that is an arithmetic model of the groupof all orientation preserving conformal (angle preserving) automorphisms (bijectiveself-mappings) of the Riemann sphere.

Fig. 3 The transformation z �→ 1/z acting on a flower bed: original, transformed, zoomed out

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Seeing Complex Functions

Visualization is an important tool for understanding mathematical objects. In twodimensions, it is possible to visualize a real-valued function of a real variable viaits graph: If x is a real number in the domain of f , the value f (x) is associatedwith x and the point (x, f (x)) is plotted. If the domain and the range of f aresubsets of the complex numbers, then a point z in the domain can be written asz = x + iy. Therefore, the domain already requires two dimensions, as does therange. Visualization in four dimensions is tricky, and it is a central focus of thischapter.

Traditionally, complex functions are visualized by so-called analytic landscapesthat depict the graph of the modulus, |f |, of f . An example appears on the left ofFig. 4, which is an illustration of Euler’s gamma function from the famous bookof Jahnke and Emde (1909). A careful look at this picture shows that the surfacecarries additional lines. There are two families of such lines: along lines of the firsttype f has constant modulus, while on the other lines the argument (phase) of f isconstant.

Now that computer graphics are available, it is more convenient to use colors tocode the phase information; on the right, this method is applied to the same function.

Another common method of visualization considers complex functions f : D →G as transformations that map the domain D onto the range G. When D is endowedwith some additional structure, S, that can be transplanted to the range plane via f ,it may be possible to recover mapping properties of the function f from the shapeof the image f (S). A typical application is visualization of (univalent) conformalmappings, where S is a mesh formed by two families of orthogonal grid lines (seeArnold and Rogness 2008, e.g.). Conformality of the mapping, or preservation ofangles, is reflected in the orthogonality of the image mesh.

However, when f is not univalent, this technique of pushing S forward is limited,because the image f (S) is self-overlapping (see Fig. 5). Fortunately, there is aneasy way to overcome this dilemma: Instead of pushing structures forward from the

Fig. 4 A historical and a contemporary analytic landscape of the gamma function

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Fig. 5 Transplantation of a square mesh via complex functions

Fig. 6 Pullback of a flower bed (right) via f (z) = z5 (left) and f (z) = sin z (middle)

z-plane to the w-plane via w = f (z), it is possible to do this the other way around;that is, they can be pulled back.

An unconventional realization of this idea is depicted in Fig. 6. Here the periodic“flower bed” on the right is pulled back by the functions w = z5 (left) and w = sin z

(middle), revealing symmetries of these mappings.For serious investigations, it is better to use standardized structures. This is

illustrated in Fig. 7 for the rational function f (z) = (z−1)/(z2 + z+1). The imagein the middle of the figure is a standard polar mesh in the w-plane. Its pullback via f

is shown on the left, and a colored analytic landscape of f is depicted on the right.The colored analytic landscape uses only saturated colors to represent the phase.

So why not incorporate the modulus of functions by brightness and color-code thevalues f (z) completely? This is the basic idea of domain coloring. Early examplesappeared in the 1980s, but the method was popularized by Frank Farris in hispaper Farris (1998). The examples in Fig. 8 show domain coloring representationsof f (z) = (z − 1)/(z2 + z + 1), f (z) = sin z, and f (z) = e1/z.

Since domain coloring encodes the values f (z) uniquely, these images makeit possible to recover the depicted function – at least theoretically. In practice,however, it is often difficult to see the details clearly. In some cases it is even betterto discard the modulus of a function completely. This leads to the concept of phaseportraits (or phase plots), which are color-coded representations of the phase f/|f |

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Fig. 7 A grid mapping induced by f (z) = (z− 1)/(z2 + z+ 1) and its colored analytic landscape

Fig. 8 Domain coloring for f (z) = (z − 1)/(z2 + z + 1), f (z) = sin z, and f (z) = e1/z

on the domain of f . (For more detailed information, see Wegert (2012); Wegert andSemmler (2011)). There is also an annual calendar, Complex Beauties, that providesa phase portrait for each month and is available from its homepage http://www.mathcalendar.net/.

It can be verified easily that holomorphic (and, more generally, meromorphic)functions are uniquely determined by their phase up to a positive constant factor, sothat (in principle) all essential information can be reconstructed from phase plots.In order to reveal hidden structures, additional information can be incorporated.Figure 9 illustrates the construction of such enhanced phase plots. In the first step, astandard color scheme highlighting special features is defined in the w-plane (lowerrow). In the second step, every point z in the domain of f is assigned the same coloras the value f (z) in the w-plane (upper row). The leftmost column corresponds to aplain phase plot; the second has shaded contour lines of |f |; in the third, some linesof constant phase are emphasized; and the rightmost phase plot is equipped witha conformal tiling. The last color scheme combines the advantages of traditionaldomain coloring with mesh representations.

The location of zeros and poles can easily be seen in a phase plot: these are thepoints where all colors meet and the number of times each color appears indicatestheir multiplicity. Zeros and poles can be distinguished by the orientation of colorsin their neighborhood. Figure 10 shows some examples.

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Fig. 9 Different versions of phase plots (upper row) as pullback from the images below

Fig. 10 Appearance of zeros (upper row) and poles (lower row) in a phase plot

Aside from zeros and poles, there is a third class of points that is at least ofthe same importance for the structural analysis of functions: the critical points, thatis, the zeros of its derivative f ′. Critical points that are not zeros of f are saidto be saddle points, a name that is motivated by their appearance in the analyticlandscape. In a plain phase plot, it is difficult to locate saddle points. Since the valueof the function is nearly constant in a neighborhood of such points, the phase plotis almost monochromatic there and does not show much detail (see Fig. 18). Oneway to depict them more clearly is to enhance lines with constant phase. In fact,if z0 is a saddle point of order k (i.e., f ′ has a zero of multiplicity k at z0), thenz0 is the crossing point of exactly k + 1 such isochromatic lines. The functions

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Fig. 11 Saddle points of orders 1, 2, 3, and 8 as crossings of isochromatic lines

Fig. 12 Saddle points of orders 1, 2, 3, and 8 located in an exceptional tile

depicted in Fig. 11 from left to right have saddle points of orders 1, 2, 3, and 8,respectively. In these images some highlighted isochromatic lines run through z0(which is the exception rather than the rule). The four images in Fig. 12 show thetypical appearance of saddle points of orders 1, 2, 3 and 8, respectively, in enhancedphase plots with conformal tiling. Though it is nearly impossible to count how manyisochromatic lines cross in these pictures, the orders of the saddle points can bedetermined from the number of vertices of the large exceptional tile: A tile with4k + 4 vertices contains a saddle point of order k (or several saddle points withorders summing up to k).

Let T be a tile in an enhanced phase plot of a meromorphic function f thatis constructed from a (standard) polar conformal tiling of the w-plane (see Fig. 9,right). Then T is said to be regular if its closure does not contain a zero, a pole, or asaddle point of f . Each regular tile is the (univalent) conformal image of a square,and its vertices correspond to the corners of the square.

While all regular tiles are conformally equivalent (to a unit square), this is notso for the exceptional tiles. Moreover, these tiles have some intrinsic symmetry thatdepends on the location of the saddle point z0. This is visualized in Fig. 13. Thesquares in the lower row are embedded in a (Cartesian) chessboard tiling. The upperrow shows some prototypes of exceptional tiles T that are generated by holomorphicfunctions with a single critical point z0 inside T or on its boundary. The red dotsindicate the locations of the critical points z0 (upper row) and the critical value f (z0)

(lower row). Note that several (two or four) squares are involved in the constructionin the third and fourth column.

More subtle symmetries are revealed when the squares of the range tiling aresubdivided into four triangles, which are colored red, yellow, green, and blue,

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Fig. 13 Some prototypes of exceptional tiles (shown in black)

Fig. 14 Symmetries near critical points arising from subdivision of tiles

respectively, in Fig. 14. As in Fig. 13, the images in the upper row are constructedby pulling back the images in the lower row via w = f (z).

Hyperbolic Geometry

One strategy used to prove the parallel postulate was to replace it by its negationand then to search for a contradiction. Carl Friedrich Gauss first realized that thissearch would be in vain and that, in fact, there is a geometry that satisfies allEuclidean axioms but with the fifth postulate negated. Although he had developeda fairly advanced non-Euclidean geometry in the year 1816, Gauss did not publish

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it and confined himself to occasional remarks in private letters (see Stäckel 1933for details). The first publications about what is now called hyperbolic geometry areby Nikolai Ivanovich Lobachevsky (1829) and János Bolyai (1832). Yet, even later,whether or not this new and counterintuitive geometry might contain a contradictionremained an open question. These doubts could be dispelled by providing aEuclidean model for hyperbolic geometry; that is, as discussed in the introduction,the terms line and point, among others, are associated with certain objects fromEuclidean geometry that are then shown to satisfy the axioms. It follows that non-Euclidean geometry is consistent if Euclidean geometry is.

The first model described by Eugenio Beltrami in 1868 was the pseudospheremodel that represents only a portion of the hyperbolic plane and will thereforenot be discussed further. That same year Beltrami first described the Poincaré andKlein models, even in n dimensions (see Milnor 1982). In two dimensions thesemodels can be introduced most conveniently using projections from a sphere, as isvisualized in Fig. 15.

(1) The points of the hemisphere model are the points lying strictly in the upperhalf of a unit sphere in R

3; the lines are half-circles orthogonal to the equator. InFig. 15, one such line is depicted in yellow. It can be verified easily that each pairof distinct points determines exactly one line that meets them both. Moreover, to agiven line L and a point P not on L, there exist an infinite number of lines throughP that do not intersect L, so that this geometry does not satisfy Playfair’s axiom.

(2) The Poincaré disk model is obtained from the hemisphere model by stere-ographic projection from the south pole to the equator plane. Here points areidentified with points in the open unit disk D, and lines are either open arcsorthogonal to the unit circle or open diameters. In Fig. 15, points and lines in thePoincaré model are depicted in red.

Figure 16 on the left shows a few hyperbolic lines in the Poincaré model.Again, through two distinct points in D, there is exactly one (hyperbolic) line.The endpoints on the unit circle, which are not part of the line, are referred toas ideal points. Hence, two different lines terminating at the same ideal point are

Fig. 15 Isomorphismsbetween the hemispheremodel, the Klein model, andthe Poincaré models

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Fig. 16 Lines in the Poincaré disk model and the half-plane model

parallel because they have no common point in D. This distinguishes them fromultraparallel lines, which do not even tend to common ideal points. Figure 16 alsoillustrates the fact that Playfair’s axiom is violated.

(3) Projecting the hemisphere orthogonally onto the equator plane yields theKlein model, Cayley-Klein model, or Beltrami-Klein model. Its points are the pointsin the open unit disk, and its lines are open line segments connecting ideal pointson the boundary of the unit disk. A (hyperbolic) line is depicted in blue in Fig. 15.Composing the two mappings just introduced yields an isomorphism between thePoincaré and the Klein model. Its explicit form in complex coordinates is

F(z) = 2z

1 + |z|2 . (1)

For more details about the Klein model, the reader is referred to Baldus (1944) andGreenberg (2007).

(4) If, as in Fig. 15, the projection is performed from a point on the equator of theRiemann sphere onto a plane tangent to the sphere at an opposite point, the modelthat lives in a half plane, usually identified with the upper half plane H := {x + iy :y > 0} of the complex plane, is obtained. This model is the Poincaré half-planemodel in which lines are either open semicircles or open half-lines orthogonal tothe real line; see Fig. 16, right. Elementary considerations also show a mapping thatdirectly relates both Poincaré models. Such a mapping turns out to be the Cayleytransform, a special Möbius transformation that maps D onto H:

F(z) = i1 − z

1 + z. (2)

See, for example, the exposition in Anderson (2005), which is based on the Poincaréhalf-plane model.

Hilbert’s axioms for plane geometry introduce only five undefined or primitiveterms by the properties they are supposed to possess. These are “point,” “line,”

The Beauty of Blaschke Products 15

“incidence,” “betweenness,” and “congruence.” So, in the description of the models,all these terms must be addressed, not only points and lines. Incidence andbetweenness have their usual Euclidean meaning. It is therefore possible to define aline segment AB as the collection of all points on the line through A and B that arelocated between A and B or coincide with one of the points. Such a line segment iscolored in green in the two Poincaré models shown in Fig. 16.

It is also possible to define (hyperbolic) convexity: A set is hyperbolically convexif it contains all line segments between any two of its points. As in Euclideangeometry, the intersection of all convex sets containing a set M is again convexand will be called the hyperbolic convex hull of M .

The notion of congruence and length of line segments in the Poincaré model ismore subtle, since it must be compatible with Hilbert’s axioms. To this end, wedefine the hyperbolic distance between z and w in D by

d(z,w) = log1 + ρ(z,w)

1 − ρ(z,w), where ρ(z,w) =

∣∣∣∣

z − w

1 − wz

∣∣∣∣. (3)

The function ρ is called the pseudo-hyperbolic distance.This formula has interesting consequences: If z is fixed and w tends to the

boundary of the unit disk, their distance will go to infinity. In other words, thePoincaré disk is unbounded and ideal points have infinite distance from points insidethe disk. If both z and w tend to ∂D while their Euclidean distance remains fixed,their hyperbolic distance will go to infinity. The other way around also works: Anobject of fixed size in the hyperbolic world that moves toward the boundary of theunit disk, (which is not a boundary of the hyperbolic disk) becomes tiny for anobserver in the Euclidean plane.

The conformality of the Poincaré disk model and the Möbius map (2) imply thatthe Poincaré half-plane model is also conformal. On the other hand, the transitionmap (1) is not conformal. Therefore, the hyperbolic size of an angle between twolines in the Klein model is not equal to its Euclidean size.

The final relevant notion is that of isometries, that is, self-mappings of thehyperbolic plane that leave the length of line segments and, therefore, anglesinvariant. Isometries form a group under composition that acts on the hyperbolicplane. In the Poincaré disk model, there are two types of isometries: conformal self-maps of the unit disk, which are the special Möbius transformations that can bewritten in the form

b(z) = cz − z0

1 − z0z(4)

with |c| = 1 and z0 ∈ D, and anti-conformal self-maps of D, which can be writtenas the composition b(z). Recall that z �→ z is the reflection in the real axis and itreverses orientation. A function of the form (4) is called a Blaschke factor.

When considered in the complex plane, b has a pole at 1/z0, as can be seen onthe left of Fig. 17. The image in the middle of the figure shows a Blaschke factor.

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Fig. 17 A conformal self-map of the disk (domain on the left and in the middle, image at right)

This self-map of D is a hyperbolic translation; that is, there is a line that is invariantunder b (which is the black line with the white dots in the middle and right imagesof the figure).

Blaschke Products

A complex polynomial p of degree n has the form p(z) = anzn + · · · + a1z + a0

with complex coefficients a0, . . . , an, where an = 0 is assumed. It follows from thefundamental theorem of algebra that, for any w ∈ C, the equation p(z) = w hasexactly n solutions, if they are counted according to multiplicities. Since p is alsoholomorphic, we can say that a polynomial is a self-map of C that maps C onto ann-fold covering of itself. In view of the fact that lim|z|→∞ |p(z)| = ∞, polynomialshave a continuous extension onto the Riemann sphere by defining p(∞) := ∞.Recalling that z �→ 1/z interchanges 0 and ∞, it is even possible to speak of themultiplicity of ∞, which is then defined to be the multiplicity of the zero of theanalytic function z �→ 1/p(1/z), and this multiplicity is easily seen to be equal ton. Hence, polynomials are also self-maps of the Riemann sphere of valency n. (Itshould be noted that they are not the only ones.)

What are the self-maps of the unit disk of valency n? The answer is that these areexactly the functions

B(z) = c

n∏

k=1

z − zk

1 − zkz(5)

with |c| = 1 and z1, . . . , zn ∈ D. These functions are called Blaschke products orhyperbolic polynomials of degree n. Blaschke products share many properties withpolynomials. Again by the fundamental theorem of algebra, a polynomial of degreen can be factored as

p(z) = c (z − z1) · · · (z − zn)

The Beauty of Blaschke Products 17

where z1, . . . , zn ∈ C are its zeros, each repeated according to its multiplicity.Recalling that conformal self-maps of C are of the form z �→ az + b, it followsthat polynomials of degree n are exactly those functions that can be written as aproduct of n conformal self-maps of C. Comparing the conformal self-maps of Dappearing in (4) with the general form of a Blaschke product of degree n as in (5),it follows that Blaschke products are exactly those functions that can be written asa product of n conformal self-maps of D.

The derivative of a polynomial p of degree n is a polynomial of degree n −1 and therefore has n − 1 zeros, called critical points of p. The derivative of aBlaschke product B of degree n is not a Blaschke product, but it has n − 1 criticalpoints in D (counting, of course, possible multiplicities). If the points ζ1, . . . , ζn−1 ∈C are prescribed, and a polynomial p with these points as critical points is to beconstructed, it is apparent that p′(z) = a(z−ζ1) · · · (z−ζn−1) with a = 0 arbitrary,and thus

p(z) =∫

p′(z) dz = a

∫

(z − ζ1) · · · (z − ζn−1) dz + b;

that is, p exists and is uniquely determined up to post-composition with a conformalself-map z �→ az + b of the plane C. The analogous property for hyperbolicpolynomials is a nontrivial result that was first proved by Heins: Given n − 1points in D, there is a Blaschke product B of degree n with these points as criticalpoints, Heins (1962, 1986). The function B is unique up to post-composition withconformal self-maps of D. Semmler and Wegert (2019) gave a simple proof usingequilibriums of charge configurations. For further references on this topic, see Krausand Roth (2008, 2013) and Garcia et al. (2018, Chapter 6).

Blaschke products live most naturally in the Poincaré disk model because oftheir conformal character. This is demonstrated by the following theorem due toWalsh, which is the hyperbolic counterpart of the well-known Gauss-Lucas theorem,Walsh (1950, 1952). The Gauss-Lucas theorem states that the critical points of anycomplex polynomial lie in the convex hull of its zeros. Walsh’s theorem says thatthe critical points (in D) of a hyperbolic polynomial (i.e., a Blaschke product) lie inthe hyperbolic convex hull of its zeros (for an alternative proof, see Wegert 2011).

In the picture on the left of Fig. 18, the 10 zeros of a Blaschke product are given(black points), along with their hyperbolic convex hull and the critical points in D

of the Blaschke product (gray points).In fact, from Formula (5) it is possible to define the Blaschke product B not

only in the unit disk but also on the Riemann sphere. In this case, B is a rationalfunction that maps the Riemann sphere onto itself. More precisely, each of theupper hemisphere, the lower hemisphere, and the equator is mapped onto an n-foldcovering of itself, respectively. In particular, a Blaschke product of degree n mapsD onto itself n times, and it wraps the unit circle around itself n times with strictlymonotone argument.

The Blaschke product B also shows some nice symmetry: The value B(1/z) at apoint 1/z, which is the reflection of z in the unit circle, is the reflection of B(z) in

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Fig. 18 Walsh’s theorem on the location of critical points of a Blaschke product

Fig. 19 Blaschke products of degrees 3, 20, and 70 in the square |Re z| ≤ 2, |Im z| ≤ 2

the unit circle,

B(1/z) = 1/B(z), z ∈ C. (6)

This has the consequence that the zeros zk of B inside the unit circle correspondto poles of B at the reflected points 1/zk outside the unit disk. This is illustrated inFig. 19. (Not all poles can be seen because z is restricted to the square |Re z| ≤ 2,|Im z| ≤ 2.)

The perfect symmetry induced by relation (6) can be seen more readily byviewing the Blaschke product on the Riemann sphere, as shown in Fig. 20. Sincereflection reverses orientation, the orientation of the color spectrum in the upper andlower hemisphere is reversed, and the zeros in the upper hemisphere appear as polesin the lower hemisphere.

This brief discussion demonstrates the surprising analogy between Blaschkeproducts and polynomials, justifying the name hyperbolic polynomials. Moresubstantial results in this direction can be found in Ng and Tsang (2013).

The Beauty of Blaschke Products 19

Fig. 20 Blaschke products of degrees 3, 20, and 70 on the Riemann sphere

Fig. 21 A Blaschke product of degree 5 with some inscribed polygons

Blaschke Products and Ellipses in the Euclidean Plane

Recall from the previous section that a Blaschke product of degree n wraps theunit circle, ∂D, around itself n times. Therefore, given any color, there will beexactly n points on the unit circle in the phase plot that have this color; in otherwords, these points are mapped to the same λ ∈ ∂D. Now draw the convex n-gonwith these points as vertices. In Fig. 21, this process is applied once in the figureon the left and then repeated on the right, moving λ around the unit circle. Asthis happens, the envelope of the n-gons obtained produces a closed curve in theunit disk. The phase portrait of the Blaschke product appears to have no obvioussymmetries, but what about the curve? Can the curve be described in a nice way?Does it possess symmetries? And, if so, do these symmetries say anything about theBlaschke product?

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Fig. 22 Line segments with points on the unit circle identified by a Blaschke product of degree 2

To form a Blaschke curve, points of the same color on the unit circle are joinedto their nearest neighbors using line segments. The envelope of these line segmentsyields the curve. The curve appearing in Fig. 21 is obtained from a Blaschke productof degree 5, and, perhaps as expected, it does not have any obvious symmetry. Hereit is assumed that B(0) = 0. If this is not the case, it is possible to choose a Blaschkefactor M mapping B(0) to 0 and then to consider the Blaschke product defined byB1 := M ◦ B. Now B1 and B are of the same degree, B1(0) = 0, and the curveobtained from B1 is the same as the one obtained from B. One thing is surprisingthough: The four nonzero zeros of the Blaschke product (the white dots not in thecenter) lie inside the curve. It is not difficult to show that if B(0) = B(a) = 0 anda = 0, then for each λ ∈ ∂D the point a lies in the convex hull of the points z forwhich B(z) = λ, Gorkin et al. (1994, Proposition 4.8).

Consider the Blaschke product B(z) = i z (z − 1/2) / (1 − z/2). In the pictureof Fig. 22, there are two red points on the unit circle and these are the two points z1and z2 for which B(z1) = B(z2) = 1. The result of connecting these two points isthe picture on the left.

Continue connecting the two points on the circle of the same color and drawingthe line segments; something unexpected happens as a result of this process.Looking at the picture on the right in Fig. 22, all line segments seem to intersectin one point, namely, the point 1/2, the nonzero zero of B.

This is always true: Let B be a Blaschke product of degree 2 with one zero at0 and one zero at a point a in D, and consider the line segment joining the twopoints z1,λ and z2,λ for which B(z1,λ) = B(z2,λ) = λ. Then for all λ on theunit circle, these line segments will pass through the point a. In this simplest case,the enveloping curve will degenerate to a point. While this might be considered asymmetric object, it is not a very interesting one. The obvious question now is: What

The Beauty of Blaschke Products 21

Fig. 23 The circles associated to B1(z) = z3 and B2(z) = z6

happens if B has zeros 0, a, and b and the points B identifies on the unit circle areconnected?

Start with the simplest degree-3 Blaschke product, namely, B(z) = z3. As inFig. 23, the points B identifies are equally spaced on the unit circle, and it followsfrom this that the curve obtained from this process is a circle, a perfectly symmetriccurve. In fact, any Blaschke product of the form czn for |c| = 1 and a positiveinteger n ≥ 3 will be associated with a circle centered at the origin. The bigger then is, the larger the radius of the circle. This is illustrated by the picture on the rightof Fig. 23.

Now consider the case in which the degree-3 Blaschke product has a nonzerozero. For example, suppose the two zeros are a = 0.8 − 0.1i and b = 0.5 + 0.5i.

In Fig. 24, the three points B maps to the point 1 are connected to obtain thesingle triangle appearing in the picture on the left. For the picture on the right, 20such triangles are formed. As before, a and b lie inside all the triangles. But moreseems to be true here.

In fact, it seems that every triangle circumscribes an ellipse and that the foci of theellipse are located at these two zeros of B. A classical theorem explains why sucha result might be expected: Siebeck’s theorem says that given three noncollinearpoints z1, z2, and z3, the zeros of the function

F(z) = m1

z − z1+ m2

z − z2+ m3

z − z3, where m1,m2,m3 > 0

are the foci of the ellipse inscribed in the triangle formed by z1, z2, z3. ApplyingSiebeck’s theorem to the function

Fλ(z) = B(z)/z

B(z) − λ, for λ ∈ ∂D,

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Fig. 24 A Blaschke product of degree 3

one sees that for each λ ∈ ∂D, the triangle with vertices at the three points B

maps to λ circumscribes an ellipse with foci at the zeros of B(z)/z. But theseellipses may change as λ changes. So the question is: Why do these triangles alwayscircumscribe the same ellipse? For the answer, see Daepp et al. (2002, 2018). ForBlaschke products of degree 3, then, the result is a well-known curve with two axesof symmetry, and the two foci of the curve are the zeros of B(z)/z.

The next case to consider is, of course, degree 4. Here, when the convexquadrilaterals with vertices at the points the Blaschke product identifies are formed,the result is a curve – but not a familiar one, in general. However, there is a veryspecial case in which some symmetry appears: By a theorem of Fujimura, theassociated curve of a degree-4 Blaschke product is an ellipse if and only if theBlaschke product is the composition of two degree-2 Blaschke products, Fujimura(2013) (see also Gorkin and Wagner 2017). For example, the picture on the left ofFig. 25 is associated with a Blaschke product that is a composition, B = D ◦ C,where

C(z) = zz − (0.2 − 0.5i)

1 − (0.2 + 0.5i)zand D(z) = z

z − (0.5 + 0.3i)

1 − (0.5 − 0.3i)z.

On the right, an asymmetrical picture associated with a degree-4 Blaschke productoutlining a clearly non-elliptical form appears, and Fujimura’s result says that theBlaschke product cannot be written as a composition of two nontrivial Blaschkeproducts.

The Beauty of Blaschke Products 23

Fig. 25 Curves associated to degree-4 Blaschke products

Blaschke Products and Ellipses in the Poincaré Disk Model

So far, the focus has been on the envelope of the line segments joining the pointsthat the Blaschke product B identifies. But Blaschke products are hyperbolicpolynomials, so it is natural to view this in terms of hyperbolic geometry. Inparticular, what happens when the Euclidean lines of the previous section arereplaced with hyperbolic lines in the Poincaré disk model?

We again start with the simplest case, a Blaschke product B of degree 2. The idealpoints on the unit circle that are identified by B are now joined by the hyperbolicline of the Poincaré disk model. These lines all intersect in one point, but this timethey intersect at the critical point of B; see Fig. 26. In the Euclidean model, whenB(0) = 0, a Blaschke factor could be used to move points so that zero mapped tozero. In the hyperbolic case, there is no need to require that B(0) = 0, because thecritical points are invariant under post-composition by a Blaschke factor.

Formula (3) for distance in the Poincaré disk model was introduced in the sectionentitled Hyperbolic Geometry. In view of this formula, it is reasonable to ask what ahyperbolic circle with center a ∈ D is. The answer is that it is the set of points z forwhich d(z, a) = r , for a constant r with r > 0 – the same as in the Euclidean setting,but using a different distance function. Since d(z, a) = r if and only if ρ(z, a) = s

for some s with 0 < s < 1, a hyperbolic circle with center a and ρ-radius s is

Dρ(a, s) := {z : ρ(z, a) = s}.

If a = 0, then (as the reader should check) the hyperbolic circle centered at zero andof radius r is also a Euclidean circle with center zero and Euclidean radius s, where

24 U. Daepp et al.

Fig. 26 Hyperbolic lines with ideal points identified by a degree-2 Blaschke product

s = (er − 1)/(er + 1). Something like this is true in general: Consider Dρ(a, s) andthe Blaschke factor defined by

Ma(z) = z − a

1 − az

that maps a to 0. Since Ma preserves hyperbolic circles and distances, Ma maps thecircle Cρ(a, s) to the circle Cρ(0, s). But Ma also preserves Euclidean circles and sodoes M−1

a . Therefore Cρ(a, s) = M−1a (Cρ(0, s)) is also a Euclidean circle. Thus,

a hyperbolic circle is also a Euclidean circle (and conversely), though the centerand radius are not usually the same. Formulas for the center and the radius can befound in Garnett (1981). The Poincaré disk model bounded by the black unit circlewith five blue circles of hyperbolic radius 1 appears in Fig. 27. The blue hyperboliccenters are located at five distinct points, namely

(

1 − 1

2n

)

e(2n−1)π i/4 for n = 0, . . . , 4.

The black dots are the Euclidean centers of the circles, and it should be noted that theEuclidean and hyperbolic centers of the circle centered at the origin coincide. Thecurve in the figure on the right is also a hyperbolic circle produced by the Blaschkeproduct

B(z) =(

z − (1 + i)/5

1 − z(1 − i)/5

)3

.

In the previous section, the focus was on ellipses (Euclidean ellipses). It is time toconsider ellipses again, but this time in the hyperbolic setting. A hyperbolic ellipse

The Beauty of Blaschke Products 25

Fig. 27 Circles in the Poincaré disk model

is the set of points such that the sum of the distances to two fixed points a1 and a2is a constant. While this may sound familiar, the point is that the distance is now thehyperbolic distance.

Recall that lines in this model (the Poincaré disk model) are represented bycircular arcs (geodesics) inside the unit circle that are orthogonal to the circle at theirpoints of intersection. In the Euclidean model, starting with a Blaschke product ofdegree 3 with a zero at zero and connecting the points it identified, the resultingtriangle circumscribed an ellipse. Consider now the hyperbolic version of this:Connect the three points a Blaschke product of degree 3 (not necessarily with a zeroat 0) identifies with geodesics, forming three parallel lines. Is the result interesting?Indeed it is. On the left of Fig. 28, the ideal points that are identified by a Blaschkeproduct of degree 3 are connected by three parallel lines. On the right, these linesenvelope a curve when the procedure is repeated with other points on the unit circle.

In fact, Singer showed that if γ denotes the curve in D that is the envelope of thenon-Euclidean geodesics joining points a Blaschke product B of degree n identifies,then γ is part of an algebraic curve for which the real foci in D are the criticalpoints of B in D (together with their inverses with respect to D), Singer (2006). Ifthe Blaschke product is of degree 3, then the curve γ is a non-Euclidean ellipse, andthe geometric foci are the two algebraic foci in D, which are the two critical pointsof B in D.

The curve on the right in Fig. 28 is thus a hyperbolic ellipse with foci marked inthe figure. Our eye, trained by Euclidean geometry, is no longer able to recognizethe symmetry of this curve because the distance function seems “distorted”; thus,to understand this, it is necessary to expand the traditional vision (i.e., Euclideanvision) of an ellipse.

We now also have a better understanding of the circle on the right of Fig. 27.The curve associated with this Blaschke product is a hyperbolic ellipse, but thetwo critical points coincide, and thus the ellipse becomes a hyperbolic circle. This,

26 U. Daepp et al.

Fig. 28 A Blaschke product of degree 3 in the Poincaré disk model

Fig. 29 Curves associated to Blaschke products of degree 4 and 7 in the Poincaré model

however, is also a Euclidean circle, as discussed above. It may be difficult to see inthe picture, but the zero of the Blaschke product, which is also the critical point, isnot in the Euclidean center of this circle. It is the circle’s center with respect to thePoincaré disk model.

Figure 29 shows curves that correspond to Blaschke products of degrees 4 and7, respectively. These are no longer ellipses, but the critical points of the Blaschkeproducts that lie in the Poincaré disk are real foci of these curves. There is anotherpiece of the curve and other critical points outside the disk; the relation between thepoints is that those outside are the symmetric points, with respect to the unit circle,of those inside.

The Beauty of Blaschke Products 27

Compositions of Blaschke Products

This section demonstrates how visualization can provide new insight into thealgebraic structure of mathematical objects. Such structures arise, for instance, fromthe factorization of natural numbers, which is the source of the rich field of numbertheory. In contrast to this, factorization of Blaschke products in the usual sense istrivial. However, there is another algebraic operation that gives rise to a number ofchallenging questions – this is the composition g ◦ f of two functions f and g,defined by (g ◦ f )(z) := g(f (z)). It can easily be verified that the compositionB = g ◦ f of Blaschke products f and g of degree m and n, respectively, isa Blaschke product of degree mn, so composition is an operation on the class of(finite) Blaschke products.

In analogy to multiplication of integers, it is easy to build a composition – butto decide whether or not a given function is composite, and to decompose it into its“factors,” is a nontrivial task. There has been a great deal of interest in this problem:For polynomials a breakthrough appears in Ritt (1922). For a brief account of thehistory of the composition problem for Blaschke products, the reader, is referredto Cowen (2012). One approach to the problem will be described in detail below.For a different, but related, approach, see Chalendar et al. (2018).

Figure 30 shows enhanced phase plots of two Blaschke products. Knowing thatone is a composition and the other one is not, it is not too difficult to guess which iswhich: A composition should have more structure than an arbitrary function – andlooking at the phase plots with some patience, one finds that the candidate on theleft is the better choice.

In fact, the relevant structure is encoded in the exceptional tiles that have morethan 4 corners (see the section Seeing Complex Functions). The reason for this is

Fig. 30 Which of these phase plots represents a composition of Blaschke products?

28 U. Daepp et al.

explained in Fig. 31, which illustrates the composition of two Blaschke products.Starting with a standard polar pattern on the right-hand side, the phase plot of g

in the middle is its pullback via the mapping g. The n − 1 critical points of g arelocated in the exceptional tiles. The phase plot of g ◦ f on the left is the pullback ofthe image in the middle via f . Its exceptional tiles are of different origin: Tiles ofthe first type contain the critical points of f – they are pre-images of regular tiles ofthe plot in the middle – while tiles of the second type are generated as pullback ofexceptional tiles in the phase plot of g. Since the image of f is an m-fold coveringof the disk, each of the latter tiles is replicated m times.

As a consequence, the exceptional tiles in the phase plot of g ◦ f form n − 1groups, each consisting of m conformally equivalent tiles, and the remaining m − 1tiles contain the critical points of f (see Fig. 32). The arithmetic background of thisobservation is the chain rule

(g ◦ f )′ = (g′ ◦ f ) · f ′.

f→

g→

Fig. 31 Construction of the phase plot of a composition g ◦ f , to be read from right to left

Fig. 32 Generation of critical tiles in the phase plots of g ◦ f and g from tiles in the z plane

The Beauty of Blaschke Products 29

The explanations above are somewhat simplified and have to be modified if f

or g have multiple critical points or if their critical points and critical values are inspecial positions. Working out the details yields the following decomposition result.

A finite Blaschke product B is decomposable as B = g ◦ f with Blaschkeproducts f and g of degree m ≥ 2 and n ≥ 2, respectively, if and only if thecritical points of B can be partitioned into multisets1 A0, A1, . . . , An−1 such that:

(i) The set A0 contains m − 1 elements, and each set A1, . . . , An−1 contains m

elements.(ii) Two critical points of B have the same multiplicity whenever they belong to

the same set Ak for some k = 1, . . . , n − 1,(iii) Let f0 be (one and then any) Blaschke product of degree m with A0 as set of

critical points. Then f0 is constant on each Ak for k = 1, . . . , n − 1.

If these conditions are satisfied, then B can be decomposed as B = g0 ◦ f0, and thegeneral form of such decompositions is

B = (g0 ◦ h−1) ◦ (h ◦ f0)

with a conformal disk automorphism h, Daepp et al. (2015).In many cases the determination of candidates for the sets A1, . . . , An−1 is rather

easy: If B = g0 ◦ f0 with f0 satisfying condition (iii), then B must also be constanton each set, say B(z) = ck for all z ∈ Ak . Typically, these n− 1 critical values ck ofB are distinct, and they are also different from the critical values of B attained at thecritical points of f0. Thus, the splitting of critical points into the sets Ak correspondsto the values of B at these points. However, one should be aware that the existenceof such a partitioning is a necessary condition that does not by itself guarantee thatB is decomposable. Moreover, there are (exceptional) cases in which B also attainsthe critical value ck at critical points not belonging to Ak , for instance, at pointsin Aj = Ak or at critical points of f0. This happens, for example, if g is itself acomposition.

Verification of the crucial third condition relies on Heins’ theorem on theexistence and uniqueness of a Blaschke product f0 of degree m with prescribed setA0 of m − 1 (not necessarily distinct) critical points in D (see the section Blaschkeproducts). The algorithmic aspects of this result are quite challenging, but in the caseat hand, this can be circumvented in the following way: Since the set of Blaschkeproducts (of any fixed degree) is invariant with respect to pre- and post-compositionwith conformal automorphisms h of the disk, one can assume that B(0) = 0 andthen replace a (possible) composition B = g◦f by B = (g◦h−1)◦(h◦f ) = g0◦f0,such that also f0(0) = 0. This implies g0(0) = 0. Hence, if f0(z0) = 0 at a pointz0 ∈ D, then B(z0) = 0. Consequently f0 must be a sub-product of B, and it is onlynecessary to test the third criterion (iii) against all such f0.

1In contrast to sets, multisets may contain elements repeatedly.

30 U. Daepp et al.

Fig. 33 Some hidden symmetries in the phase plot of a composition

Fig. 34 Discrete conformal analysis of Blaschke products revealing symmetries

The governing principle behind these observations is a kind of hidden symmetrythat is inherent in the compositions of Blaschke products and can be read offfrom their enhanced phase plots. Such symmetries are visualized in Fig. 33, whichillustrates the fact that the critical points of B = g ◦ f belonging to the same set Ak

(see the decomposition result) are, in a sense, symmetric with respect to the criticalpoints of f . The reason for the appearance of these symmetries becomes plausiblein Fig. 34, which shows “polar chessboard plots” of g ◦ f (left) and g (right). Theyellow and the violet exceptional tiles contain the m − 1 = 2 critical points of f .Each of these belongs to a colored chain that connects opposite edges of that tile withthe boundary of the disk. These two chains split the disk into m = 3 subdomainsS1, S2, S3, and f maps each subdomain onto (a copy of) D (shown on the right).These mappings are injective, except for the (colored) tiles of the bounding chains,in which two symmetric tiles have the same image. The exceptional (yellow andviolet) tiles are mapped onto (double coverings) of the corresponding terminal tilesof the (yellow and violet) chains in the image on the right.

The Beauty of Blaschke Products 31

Fig. 35 Symmetric paths connecting tiles associated with the sets A1 (left) and A2 (right)

Each of the subdomains Sj contains n − 1 = 2 exceptional tiles (red, green,blue) that are the pre-images of the (correspondingly colored) critical tiles inthe chessboard plot of g with respect to the mapping f . Finally, each groupA1, . . . , An−1 of critical points in the decomposition result is represented by exactlyone member in each subdomain Sj . The color of the corresponding exceptional tileindicates the group to which its critical point belongs.

Polar chessboard plots are also convenient for finding the relevant symmetries,as illustrated in Fig. 35. This figure shows several chains of tiles that are symmetricwith respect to the critical points of f (contained in the yellow and violet exceptionaltile) and connect exceptional tiles (red, green, and blue) that are associated withthe same set A1, A2, and A3, respectively. If such symmetries are absent at somelevel of discretization, the Blaschke product is indecomposable. Theoretically, itwould also work the other way around, provided that one could verify thesesymmetries exactly. This requires two things: (1) finding appropriate symmetricpaths connecting the exceptional tiles and (2) verifying that the connected criticaltiles are conformally equivalent. The first step is basically a counting exercise,but the second is impracticable. On the other hand, if a Blaschke product is not acomposition, this can always be detected by choosing a sufficiently fine discretizedchessboard plot.

Conclusion

The goal of this chapter has been to demonstrate how visualization can helpto understand concepts and results of a mathematical theory without recourseto formulas. In fact, many mathematicians consider their research to be oneof discovery, and seeing the objects of their investigations can be a source of

32 U. Daepp et al.

inspiration. Beauty and symmetry are often guiding principles in these explorations.While beauty remains a matter of individual taste, experience, and convention, theconcept of symmetry has a precise mathematical meaning, formally expressed inthe language of group theory and reaching far beyond the common interpretation ofthis notion.

For centuries in the visual arts, beauty was a dominating criterion for thereception of artwork. In particular, during the Renaissance, strong compositionrules were often based on symmetry and mathematical principles, and ingeniousartists like Leonardo da Vinci or Albrecht Dürer made important contributions tomathematics.

During the twentieth century, the fundamental paradigm that art must be beautifulwas abandoned, and art was assigned a new role as part of the human project ofexploring nature, reality, and society. In the preface to the catalogue of CarstenNicolai’s exhibition “unidisplay,” Eva Huttenlauch writes: “Art has become acategory of thinking rather than seeing . . . ”, Huttenlauch (2013, p. 13). As the aimsof artists expanded, the border between mathematics and arts began to blur.

Artists like Victor Vasareli, Sol LeWitt, and Carsten Nicolai and scientistslike Heinrich Heesch, Anatoly Fomenko, and Isaac Amidror are building bridgesbetween arts and mathematics. Approaching a topic from opposite sides, emphasiz-ing different aspects, and using tools specific to each discipline, mathematiciansand artists sometimes seem to explore one and the same general phenomenon,indicating the presence of universal principles. For example, it is not an accidentthat the “stellar grating MI-RG-090” from the work “moiré index” (Nicolai, 2010b)resembles a black-and-white phase plot of the function f (z) = e1/z (Fig. 36).In the same vein, it would be interesting to do a comparative study of Niclolai’ssystematic listing of grids and tilings, Nicolai (2010a), (especially the semi-regular and irregular ones), with Frank Farris’ wallpaper patterns, Farris (2015),

Fig. 36 Carsten Nicolai’s stellar grating MI-RG-090 and a plot of f (z) = exp(1/z)

The Beauty of Blaschke Products 33

and Isaac Amidror’s theoretical foundation of moiré patterns, Amidror (2007,2009).

Visualization is important for both the arts and mathematics. For the former, thishas a long tradition. For the latter, tradition has focused on the subfield of geometry.However, advances of computer graphics opened up new possibilities for otherbranches of mathematics as well. In this chapter, it is the field of complex analysisthat benefits from visual representations of its objects. In many mathematicalsubjects, the visual tradition has only just begun.

Acknowledgments Since August 2018, Pamela Gorkin has been serving as a Program Directorin the Division of Mathematical Sciences at the National Science Foundation (NSF), USA, and asa component of this position, she received support from NSF for research, which included work onthis paper. Any opinions, findings, and conclusions or recommendations expressed in this materialare those of the authors and do not necessarily reflect the views of the National Science Foundation.

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