The Use of Repeating Patterns to Teach Hyperbolic Geometry ...ddunham/mfesttlk09.pdf · The Use of...
Transcript of The Use of Repeating Patterns to Teach Hyperbolic Geometry ...ddunham/mfesttlk09.pdf · The Use of...
The Use of Repeating Patterns to TeachHyperbolic Geometry Concepts
Douglas DunhamDepartment of Computer ScienceUniversity of Minnesota, DuluthDuluth, MN 55812-3036, USA
E-mail: [email protected] Site: http://www.d.umn.edu/˜ddunham/
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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.
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Outline
• History
• Models and Repeating Patterns
• M.C. Escher’s Patterns
• Analysis of Patterns
• Other Patterns
• Conclusions and Future Work
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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.
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Hyperbolic Geometry DiscoveredIndependently in Early 1800’s by:
• Bolyai Janos
• Carl Friedrich Gauss (did not publish results)
• Nikolas Ivanovich Lobachevsky
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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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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
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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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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.
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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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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.
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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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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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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.
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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.
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Some Symmetries ofCircle Limit IV
• Reflection lines shown in blue
• 90◦ rotation centers shown in red
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Escher’sCircle Limit III Revisited
• Backbone lines areequidistant curves
• Analogous to lines of latitude in spherical geome-try
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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.
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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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Other Hyperbolic Patterns
A Butterfly Pattern with Seven Butterfliesat the Center (in the Art Exhibit)
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