The Interplay Between HyperbolicSymmetry and History

Douglas DunhamDepartment of Computer ScienceUniversity of Minnesota, DuluthDuluth, MN 55812-3036, USA

E-mail: ddunham@d.umn.eduWeb Site: http://www.d.umn.edu/˜ddunham/

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Outline

• Introduction

• Hyperbolic Geometry and its History

• Symmetry, Discovery, Aesthetics, and New Knowl-edge

• Reinterpretation of Geometric Knowledge

• Integration of Knowledge to Create Repeating Hy-perbolic Patterns

• Creating Hyperbolic Art Using Computers

• Integration of the Cerebral Hemispheres in Creat-ing Patterns

• Future Work

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Introduction

There are 3 “Classical Geometries”:

• the Euclidean plane

• the sphere

• the hyperbolic plane

The first two have been known since antiquity

Hyperbolic geometry has been known for slightly lessthan 200 years.

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Hyperbolic Geometry DiscoveredIndependently in Early 1800’s by:

• Bolyai Janos

• Carl Friedrich Gauss (did not publish results)

• Nikolas Ivanovich Lobachevsky

4

Why the Late Discovery of HyperbolicGeometry?

• Euclidean geometry seemed to represent the phys-ical world.

• Unlike the sphere was no smooth embedding of thehyperbolic plane in familiar Euclidean 3-space.

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Euclid and the Parallel Axiom

• Euclid realized that the parallel axiom, the FifthPostulate, was special.

• Euclid proved the first 28 Propositions in his Ele-ments without using the parallel axiom.

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Attempts to Prove the Fifth Postulate

• Adrien-Marie Legendre (1752–1833) made manyattempts to prove the Fifth Postulate from the firstfour.

• Girolamo Saccheri (1667–1733) assumed the nega-tion of the Fifth Postulate and deduced many re-sults (valid in hyperbolic geometry) that he thoughtwere absurd, and thus “proved” the Fifth Postu-late was correct.

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New Geometries

• A negation of the Fifth Postulate, allowing morethan one parallel to a given line through a point,leads to another geometry, hyperbolic, as discov-ered by Bolyai, Gauss, and Lobachevsky.

• Another negation of the Fifth Postulate, allowingno parallels, and removing the Second Postulate,allowing lines to have finite length, leads to ellip-tic (or spherical) geometry, which was also investi-gated by Bolyai.

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A Second Reason for Late Discovery

• As mentioned before, there is no smooth embed-ding of the hyperbolic plane in familiar Euclidean3-space (as there is for the sphere). This was provedby David Hilbert in 1901.

• Thus we must rely on “models” of hyperbolic ge-ometry - Euclidean constructs that have hyperbolicinterpretations.

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The Beltrami-Klein Model of HyperbolicGeometry

• Hyperbolic Points: (Euclidean) interior points of abounding circle.

• Hyperbolic Lines: chords of the bounding circle(including diameters as special cases).

• Described by Eugenio Beltrami in 1868 and FelixKlein in 1871.

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Consistency of Hyperbolic Geometry

• Hyperbolic Geometry is at least as consistent asEuclidean geometry, for if there were an error inhyperbolic geometry, it would show up as a Eu-clidean error in the Beltrami-Klein model.

• So then hyperbolic geometry could be the “right”geometry and Euclidean geometry could have er-rors!?!?

• No, there is a model of Euclidean geometry within3-dimensional hyperbolic geometry — they are equallyconsistent.

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Henri Poincare’s Models of HyperbolicGeometry

• The circle model (described next).

• The upper half-plane model:

– Hyperbolic points: points in the xy-plane withpositivey-coordinate.

– Hyperbolic lines: upper semi-circles whose cen-ter is on thex-axis (including vertical half-linesas special cases).

– Used by M.C. Escher in his “line-limit” patterns.

• Both models areconformal: the hyperbolic mea-sure of angles is the same as their Euclidean mea-sure.

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Poincare Circle Model of HyperbolicGeometry

• Points: points within the (unit) bounding circle

• Lines: circular arcs perpendicular to the boundingcircle (including diameters as a special case)

• Attractive to Escher and other artists because itis represented in a finite region of the Euclideanplane, so viewers could see the entire pattern, andis conformal, so that copies of a motif in a repeat-ing pattern retained their same approximate shape.

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Relation Between Poincare andBeltrami-Klein Models

• A chord in the Beltrami-Klein model representsthe same hyperbolic line as the orthogonal circulararc with the same endpoints in the Poincare circlemodel.

• The models measure distance differently, but equalhyperbolic distances are represented by ever smallerEuclidean distances toward the bounding circle inboth models.

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Repeating Patterns

• A repeating pattern in any of the three classical ge-ometries is a pattern made up of congruent copiesof a basic subpattern ormotif.

• The copies of the motif are related bysymmetries— isometries (congruences) of the pattern whichmap one motif copy onto another.

• A fish is a motif for Escher’s Circle Limit III pat-tern below if color is disregarded.

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Symmetries in Hyperbolic Geometry

• A reflection is one possible symmetry of a pattern.

• In the Poincare disk model, reflection across a hy-perbolic line is represented by inversion in the or-thogonal circular arc representing that hyperbolicline.

• As in Euclidean geometry, any hyperbolic isom-etry, and thus any hyperbolic symmetry, can bebuilt from one, two, or three reflections.

• For example in either Euclidean or hyperbolic ge-ometry, one can obtain a rotation by applying twosuccessive reflections across intersecting lines, theangle of rotation being twice the angle of intersec-tion.

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Examples of Symmetries — Escher’sCircleLimit I

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Repeating Patterns and HyperbolicGeometry

• Repeating patterns and their symmetry are nec-essary to understand the hyperbolic nature of thedifferent models of hyperbolic geometry.

• For example, a circle containing a few chords mayjust be a Euclidean construction and not representanything hyperbolic in the Beltrami-Klein model.

• On the other hand, if there is a repeating patternwithin the circle whose motifs get smaller as oneapproaches the circle, then we may be able to in-terpret it as a hyperbolic pattern.

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Symmetry, Discovery, Aesthetics, and NewKnowledge

• Probably the most important example of hyper-bolic symmetry leading to something new was Es-cher’s discovery of the Poincar disk model as asolution to his long-sought “circle limit” problem:how to construct a repeating pattern whose motifsget infinitely small toward the edge of a boundingcircle.

• In a “circle limit” pattern, the complete infinitepattern could be captured within a finite area, un-like the “point limit” and “line limit” patterns Es-cher had previously designed.

• The Canadian mathematician H.S.M. Coxeter sentEscher a reprint of a paper containing such a pat-tern of curvilinear triangles.

• Escher recounted that “gave me quite a shock” sincethe apparent visual symmetries of the pattern in-stantly showed him the solution to his problem.

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Coxeter’s Pattern that Inspired Escher

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Escher Inspiring Coxeter and Dunham

• In turn, Escher’s “Circle Limit” patterns inspiredCoxeter to analyze them.

• Coxeter determined the symmetry groups of Es-cher’s Circle Limit II, Circle Limit III, and CircleLimit IV as [3+, 8], (4, 3, 3), and [4+, 6] respectively.

• Later, Dunham, inspired by both Escher and Cox-eter determined the symmetry group ofCircle LimitI to be cmm3,2, a special case ofcmmp

2,q2

(which isgenerated by reflections across the sides of a rhom-bus with angles2π/p and 2π/q, and a 180 degreerotation about its center). Whenp = q = 4, we getthe Euclidean “wallpaper” group cmm = cmm2,2.

• Coxeter also determined that backbone arcs ofCir-cle Limit III made an angle ofω with the boundingcircle, wherecos(ω) =

√

3√

2−4

8

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Escher’sCircle Limit III — Note BackboneLines

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Escher and Coxeter Inspiring Dunham

• Several years ago I was inspired by Escher and aneditor of the Mathematical Intelligencer to conceiveof a generalization of Escher’sCircle Limit III pat-tern.

• Allowing p fish to meet at right fin tips, q fish tomeet at left fin tips, and r fish to meet nose-to-nose, I denote the resulting pattern(p, q, r), soCir-cle Limit III would be called (4, 3, 3) in this nota-tion.

• Luns Tee and Dunham also determined that back-bone arcs of of a(p, q, r) pattern made an angle ofω with the bounding circle, where

cos(ω) =sin( π

2r) (cos(π

p) − cos(π

q))

√

cos(πp)2 + cos(π

q)2 + cos(π

r)2 + 2 cos(π

p) cos(π

q) cos(π

r) − 1)

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A (5,3,3) Pattern

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Reinterpretation of Geometric Knowledge

• Lobachevsky and probably Gauss viewed hyper-bolic geometry in isolation.

• Bolyai also considered elliptic geometry as anotherpossible non-Euclidean geometry, thus becomingthe first to consider the three classical geometries,Euclidean, hyperbolic, and elliptic, as a related setdistinguished by their parallel properties.

• The classical geometries also have constant curva-ture: zero curvature for the Euclidean plane, pos-itive curvature for elliptic geometry, and negativecurvature for hyperbolic geometry.

• Gauss considered more general 2-dimensional sur-faces in which the curvature could also vary frompoint to point, so the classical geometries were thenjust special cases.

• In the mid-1800’s, Riemann extended the conceptof surfaces to n-dimensional manifolds, so surfacesin turn became just a special case of manifolds (ofdimension two).

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Relative Consistency of Euclidean andHyperbolic Geometry

• Beltrami’s interpretation of his model of hyper-bolic geometry as a set of Euclidean constructs provedthat hyperbolic geometry was just as consistent asEuclidean geometry.

• Conversely, it was shown later that horospheresin 3-dimensional hyperbolic space could be inter-preted as having the same structure as the Euclideanplane, thus proving that the geometries were equallyconsistent.

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Color Symmetry

• In the early 1960’s the theory of patterns with n-color symmetry was being developed for n largerthan two (the theory of 2-color symmetry had beendeveloped in the 1930’s).

• Previously, Escher had created many patterns withregular use of color. These patterns could now bereinterpreted in light of the new theory of colorsymmetry.

• Escher’s patternsCircle Limit II and Circle LimitIII exhibit 3- and 4-color symmetry respectively.

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Escher’sCircle Limit II Pattern (3-colorsymmetry)

Escher’s least known Circle Limit pattern.

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Escher’sCircle Limit IV Pattern

• Only motif Escher rendered in all three classicalgeometries

• This pattern inspired Australian polymath TonyBomford to create hyperbolic hooked rugs.

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Integration of Knowledge to CreateRepeating Hyperbolic Patterns

• Repeating hyperbolic patterns were first createda little more than 100 years ago. These were ab-stract patterns like the pattern of curvilinear tri-angles used by Coxeter that inspired Escher.

• Creating these patterns mostly required the knowl-edge of Euclidean constructions.

• To create aesthetically pleasing hyperbolic patternsit is necessary to have good artistic knowledge also.

• Escher became the first person to create such pat-terns because he had both the mathematical andthe artistic knowledge.

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Creating Hyperbolic Patterns Using aComputer

• Creating repeating hyperbolic patterns by hand istedious and time consuming, since unlike Euclideanrepeating patterns in which the motifs are all thesame size, the motifs vary in size and must be cre-ated individually.

• About 30 years ago it occurred to me that comput-ers could be used to draw hyperbolic patterns butthey had to be programmed to do so.

• Thus a third discipline, computer science, neededto be integrated into the process.

• My students and I were successful in this endeavor,not only re-creating Escher’s circle limit patterns,but other patterns as well.

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Another Hyperbolic Pattern

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Creating More Hyperbolic Art UsingComputers

• The computer programs that my students and Idesigned have also been used to draw templatesthat could be used by artists using other media.

• Irene Rousseau was the first artist to take advan-tage of hyperbolic pattern templates, creating sev-eral glass mosaic patterns.

• Later Mary Williams made a quilt with a hyper-bolic pattern, inspired by one of my computer draw-ings.

• Currently I am working with a hooked rug makerto create a new hyperbolic pattern related to thosemade by the Australian rug maker Tony Bomford.

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Tony Bomford’s Rug Number 17

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Integration of the Cerebral Hemispheres inCreating Patterns

• The process of creating repeating hyperbolic pat-terns via computer requires both left-brain and right-brain thinking.

• The left-brain functions of logic and mathematicsare necessary to create the computer programs todraw the patterns.

• The computer program also makes fundamentaluse of the 3-dimensional Weierstrass model of hy-perbolic geometry.

• Thus the right-brain functions of geometric insight,artistic appreciation, and 3D conceptualization arealso needed.

• Consequently, the creation of these patterns requiresan integrated and balanced use of both the left-brain and right-brain hemispheres.

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Future Work

• Extend the classification of families of Escher pat-terns such as his “Angels and Devils” patterns (ineach of the three classical geometries), and the pro-posed(p, q, r) generalizations ofCircle Limit III.

• Find an algorithm for computing the minimum num-ber of colors needed for a pattern, given the defini-tion of its symmetry group (ignoring color).

• Continue to inspire other artists to create hyper-bolic patterns in their media.

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