The Use of Repeating Patterns to TeachHyperbolic Geometry Concepts

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/

1

Hyperbolic Geometry — Unfamiliar

Reasons:

• Euclidean geometry seems to describe our world— no need to negate the parallel axiom.

• There is no smooth, distance-preserving embed-ding of it in Euclidean 3-space - unlike sphericalgeometry.

2

Outline

• History

• Models and Repeating Patterns

• M.C. Escher’s Patterns

• Analysis of Patterns

• Other Patterns

• Conclusions and Future Work

3

The Three “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.

4

Hyperbolic Geometry DiscoveredIndependently in Early 1800’s by:

• Bolyai Janos

• Carl Friedrich Gauss (did not publish results)

• Nikolas Ivanovich Lobachevsky

5

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.

6

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.

7

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.

8

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.

9

The van Hiele Levels of GeometricReasoning

1. Visualization - identifying different shapes

2. Analysis - measurement, classification

3. Informal Deduction - making and testing hypothe-ses

4. Deduction - construct (Euclidean) proofs

5. Rigor - work with different axiom systems

10

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.

11

One Model: The Beltrami-Klein Model ofHyperbolic Geometry

• 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.

12

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.

13

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: the hyperbolic measure of an angleis the same as its Euclidean measure. This im-plies that copies of a motif in a repeating patternretained their same approximate shape.

14

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.

15

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.

16

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.

17

Examples of Symmetries — Escher’sCircleLimit I

18

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.

19

Escher’s Hyperbolic Patterns —CircleLimit IV

• Only motif Escher rendered in all three classicalgeometries

• Is on display at the Portland Art Museum now.

• Have now seenCircle Limit I, Circle Limit III, andCircle Limit IV — Circle Limit II is below.

20

Analyzing Hyprebolic Patterns

• There are five kinds of hyperbolic isometries:

– Reflection– Rotation– Translation– Glide-Reflection– Parabolic Isometry: “rotation about a point at

infinity” (does not appear in Escher’s patterns)

• The first three are easiest to spot.

• As in Euclidean geometry, glide-reflections are some-times hard to spot.

21

Some Symmetries ofCircle Limit IV

• Reflection lines shown in blue

• 90◦ rotation centers shown in red

22

Escher’sCircle Limit III Revisited

• Backbone lines areequidistant curves

• Analogous to lines of latitude in spherical geome-try

23

Equidistant Curves in Circle Limit III

• Meeting points of left fins and noses are vertices ofa tessellation by regular octagons meeting three ata vertex.

• Red backbone lines are equidistant from the greenhyperbolic line that goes through the centers of theblue zigzag formed by sides of octagons.

24

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.

25

Escher’sCircle Limit II Pattern (3-colorsymmetry)

Escher’s least known Circle Limit pattern.

26

Other Hyperbolic Patterns

A Butterfly Pattern with Seven Butterfliesat the Center (in the Art Exhibit)

27

A Circle Limit III-like Pattern with FiveFish at the Center

28

A Butterfly Pattern with Six Butterflies atthe Center Center

29

A Butterfly Pattern with Three Butterfliesat the Center Center

30

A Butterfly Pattern with Five Butterflies atthe Center Center

31

Conclusions and Future Work

• We can use repeating hyperbolic patterns to gaininsight into the properties of hyperbolic geometry.

• We are working on a portable program that otherscan use to create repeating hyperbolic patterns.

32