Hyperbolic Ornaments Drawing in Non-Euclidean Crystallographic - - PowerPoint PPT Presentation

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Hyperbolic Ornaments Drawing in Non-Euclidean Crystallographic - - PowerPoint PPT Presentation

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Basics Program Hyperbolic Ornaments Drawing in Non-Euclidean Crystallographic Groups Martin von Gagern joint work with Jrgen Richter-Gebert Technische Universitt Mnchen Second International Congress on Mathematical Software, September 1

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Basics Program

Hyperbolic Ornaments

Drawing in Non-Euclidean Crystallographic Groups Martin von Gagern joint work with Jürgen Richter-Gebert

Technische Universität München

Second International Congress on Mathematical Software, September 1 2006

Martin von Gagern Hyperbolic Ornaments

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Basics Program

Educational Value

Martin von Gagern Hyperbolic Ornaments

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Basics Program

Escher

Martin von Gagern Hyperbolic Ornaments

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Basics Program

Hyperbolic Escher

Martin von Gagern Hyperbolic Ornaments

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Basics Program

Hyperbolic Escher

Martin von Gagern Hyperbolic Ornaments

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Basics Program Symmetries Hyperbolic Geometry

Outline

1

Basics Symmetries Hyperbolic Geometry

2

Program Intuitive Input Group Calculations Fast Drawing

Martin von Gagern Hyperbolic Ornaments

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Basics Program Symmetries Hyperbolic Geometry

Rigid Motions

Reflection Rotation Translation Glide Reflection Definition (Rigid Motion) Rigid Motions ( = Isometries) are the length-preserving mappings

  • f the plane onto itself.

Martin von Gagern Hyperbolic Ornaments

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Basics Program Symmetries Hyperbolic Geometry

Rigid Motions

Reflection Rotation Translation Glide Reflection Definition (Rigid Motion) Rigid Motions ( = Isometries) are the length-preserving mappings

  • f the plane onto itself.

Martin von Gagern Hyperbolic Ornaments

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Basics Program Symmetries Hyperbolic Geometry

Groups of Rigid Motions

  • Group E(2): all euclidean planar isometries
  • Discrete Subgroups

Definition (Discreteness) A group G is discrete if around every point P of the plane there is a neighborhood devoid of any images of P under the group operations. The discrete groups of rigid motions in the euclidean plane:

  • 17 Wallpaper Groups
  • 7 Frieze Groups
  • 2 kinds of Rosette Groups

Martin von Gagern Hyperbolic Ornaments

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Basics Program Symmetries Hyperbolic Geometry

Groups of Rigid Motions

  • Group E(2): all euclidean planar isometries
  • Discrete Subgroups

Definition (Discreteness) A group G is discrete if around every point P of the plane there is a neighborhood devoid of any images of P under the group operations. The discrete groups of rigid motions in the euclidean plane:

  • 17 Wallpaper Groups
  • 7 Frieze Groups
  • 2 kinds of Rosette Groups

Martin von Gagern Hyperbolic Ornaments

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Basics Program Symmetries Hyperbolic Geometry

Groups of Rigid Motions

  • Group E(2): all euclidean planar isometries
  • Discrete Subgroups

Definition (Discreteness) A group G is discrete if around every point P of the plane there is a neighborhood devoid of any images of P under the group operations. The discrete groups of rigid motions in the euclidean plane:

  • 17 Wallpaper Groups
  • 7 Frieze Groups
  • 2 kinds of Rosette Groups

Martin von Gagern Hyperbolic Ornaments

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Basics Program Symmetries Hyperbolic Geometry

Anatomy of the Hyperbolic Plane

Definition (Hyperbolic Axiom of Parallels) Given a point P outside a line ℓ there exist at least two lines through P that do not intersect ℓ.

  • Many facts of euclidean geometry don’t rely on the Axiom of

Parallels and are true in hyperbolic geometry as well.

  • The sum of angles in a triangle is less than π.
  • Lengths are absolute, scaling is not an automorphism.
  • Geometry of constant negative curvature.

Martin von Gagern Hyperbolic Ornaments

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Basics Program Symmetries Hyperbolic Geometry

Poincaré Disc Model

  • hyperbolic points:

inside of the unit circle

  • hyperbolic lines:

lines and circles perpendicular to the unit circle

  • hyperbolic angle:

identical to euclidean angle

  • hyperbolic distance:

changes with distance from center

Martin von Gagern Hyperbolic Ornaments

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Basics Program Symmetries Hyperbolic Geometry

Poincaré Disc Model

  • hyperbolic points:

inside of the unit circle

  • hyperbolic lines:

lines and circles perpendicular to the unit circle

  • hyperbolic angle:

identical to euclidean angle

  • hyperbolic distance:

changes with distance from center

Martin von Gagern Hyperbolic Ornaments

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Basics Program Symmetries Hyperbolic Geometry

Poincaré Disc Model

  • hyperbolic points:

inside of the unit circle

  • hyperbolic lines:

lines and circles perpendicular to the unit circle

  • hyperbolic angle:

identical to euclidean angle

  • hyperbolic distance:

changes with distance from center

Martin von Gagern Hyperbolic Ornaments

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Basics Program Symmetries Hyperbolic Geometry

Poincaré Disc Model

  • hyperbolic points:

inside of the unit circle

  • hyperbolic lines:

lines and circles perpendicular to the unit circle

  • hyperbolic angle:

identical to euclidean angle

  • hyperbolic distance:

changes with distance from center

Martin von Gagern Hyperbolic Ornaments

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Basics Program Symmetries Hyperbolic Geometry

Hyperbolic Rigid Motions

Reflection Rotation Translation Glide Reflection N.B.: translations now have only a single fixed line.

Martin von Gagern Hyperbolic Ornaments

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Basics Program Intuitive Input Group Calculations Fast Drawing

Outline

1

Basics Symmetries Hyperbolic Geometry

2

Program Intuitive Input Group Calculations Fast Drawing

Martin von Gagern Hyperbolic Ornaments

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Basics Program Intuitive Input Group Calculations Fast Drawing

Tilings by regular Polygons

  • Square
  • Triangular
  • Hexagonal

Martin von Gagern Hyperbolic Ornaments

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Basics Program Intuitive Input Group Calculations Fast Drawing

Tilings by regular Polygons

  • Square
  • Triangular
  • Hexagonal

Martin von Gagern Hyperbolic Ornaments

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Basics Program Intuitive Input Group Calculations Fast Drawing

From regular Polygons to Triangles

regular heptagons △(2, 3, 7) regular triangles angles 2π 3 angles π 2, π 3, π 7 angles 2π 7

Martin von Gagern Hyperbolic Ornaments

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Basics Program Intuitive Input Group Calculations Fast Drawing

From regular Polygons to Triangles

regular heptagons △(2, 3, 7) regular triangles angles 2π 3 angles π 2, π 3, π 7 angles 2π 7

Martin von Gagern Hyperbolic Ornaments

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Basics Program Intuitive Input Group Calculations Fast Drawing

From regular Polygons to Triangles

regular heptagons △(2, 3, 7) regular triangles angles 2π 3 angles π 2, π 3, π 7 angles 2π 7

Martin von Gagern Hyperbolic Ornaments

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Basics Program Intuitive Input Group Calculations Fast Drawing

General Tesselations

△(4, 6, 7) △(2, 5, ∞)

Martin von Gagern Hyperbolic Ornaments

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Basics Program Intuitive Input Group Calculations Fast Drawing

Why All Angles are Different

  • △(n, n, n) ⊂ △(2, 3, 2n)
  • △(n, 2n, 2n) ⊂ △(2, 4, 2n)
  • △(n, m, m) ⊂ △(2, m, 2n)

△(k, m, n) : π k + π m + π n < π

Martin von Gagern Hyperbolic Ornaments

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Basics Program Intuitive Input Group Calculations Fast Drawing

Why All Angles are Different

  • △(n, n, n) ⊂ △(2, 3, 2n)
  • △(n, 2n, 2n) ⊂ △(2, 4, 2n)
  • △(n, m, m) ⊂ △(2, m, 2n)

△(k, m, n) : π k + π m + π n < π

Martin von Gagern Hyperbolic Ornaments

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Basics Program Intuitive Input Group Calculations Fast Drawing

Why All Angles are Different

  • △(n, n, n) ⊂ △(2, 3, 2n)
  • △(n, 2n, 2n) ⊂ △(2, 4, 2n)
  • △(n, m, m) ⊂ △(2, m, 2n)

△(k, m, n) : π k + π m + π n < π

Martin von Gagern Hyperbolic Ornaments

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Basics Program Intuitive Input Group Calculations Fast Drawing

Algebraic Calculations

General triangle reflection group △(k, m, n)

  • Coxeter group (finitely represented group for GAP)
  • a, b, c | a2 = 1, b2 = 1, c2 = 1, (ab)k = 1, (ac)m = 1, (bc)n = 1
  • Subgroups with finite index are

non-euclidean crystallographic (N.E.C.) groups

  • Orientation preserving subgroups are Fuchsian

Martin von Gagern Hyperbolic Ornaments

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Basics Program Intuitive Input Group Calculations Fast Drawing

Algebraic Calculations

General triangle reflection group △(k, m, n)

  • Coxeter group (finitely represented group for GAP)
  • a, b, c | a2 = 1, b2 = 1, c2 = 1, (ab)k = 1, (ac)m = 1, (bc)n = 1
  • Subgroups with finite index are

non-euclidean crystallographic (N.E.C.) groups

  • Orientation preserving subgroups are Fuchsian

Martin von Gagern Hyperbolic Ornaments

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Basics Program Intuitive Input Group Calculations Fast Drawing

Algebraic Calculations

General triangle reflection group △(k, m, n)

  • Coxeter group (finitely represented group for GAP)
  • a, b, c | a2 = 1, b2 = 1, c2 = 1, (ab)k = 1, (ac)m = 1, (bc)n = 1
  • Subgroups with finite index are

non-euclidean crystallographic (N.E.C.) groups

  • Orientation preserving subgroups are Fuchsian

Martin von Gagern Hyperbolic Ornaments

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Basics Program Intuitive Input Group Calculations Fast Drawing

Algebraic Calculations

General triangle reflection group △(k, m, n)

  • Coxeter group (finitely represented group for GAP)
  • a, b, c | a2 = 1, b2 = 1, c2 = 1, (ab)k = 1, (ac)m = 1, (bc)n = 1
  • Subgroups with finite index are

non-euclidean crystallographic (N.E.C.) groups

  • Orientation preserving subgroups are Fuchsian

Martin von Gagern Hyperbolic Ornaments

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Basics Program Intuitive Input Group Calculations Fast Drawing

Algebraic Calculations

General triangle reflection group △(k, m, n)

  • Coxeter group (finitely represented group for GAP)
  • a, b, c | a2 = 1, b2 = 1, c2 = 1, (ab)k = 1, (ac)m = 1, (bc)n = 1
  • Subgroups with finite index are

non-euclidean crystallographic (N.E.C.) groups

  • Orientation preserving subgroups are Fuchsian

Martin von Gagern Hyperbolic Ornaments

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Basics Program Intuitive Input Group Calculations Fast Drawing

Group Generation

1 Generator entered by user 2 Add inverse operations 3 Find “all” combinations

  • Group representation
  • Orbit of centerpiece
  • Each element starts

a new domain

4 For all triangles that are

not yet part of any orbit

  • add triangle to

central domain

  • combine triangle with all

group elements to calculate its orbit, adding to domains

Martin von Gagern Hyperbolic Ornaments

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Basics Program Intuitive Input Group Calculations Fast Drawing

Group Generation

1 Generator entered by user 2 Add inverse operations 3 Find “all” combinations

  • Group representation
  • Orbit of centerpiece
  • Each element starts

a new domain

4 For all triangles that are

not yet part of any orbit

  • add triangle to

central domain

  • combine triangle with all

group elements to calculate its orbit, adding to domains

Martin von Gagern Hyperbolic Ornaments

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Basics Program Intuitive Input Group Calculations Fast Drawing

Group Generation

1 Generator entered by user 2 Add inverse operations 3 Find “all” combinations

  • Group representation
  • Orbit of centerpiece
  • Each element starts

a new domain

4 For all triangles that are

not yet part of any orbit

  • add triangle to

central domain

  • combine triangle with all

group elements to calculate its orbit, adding to domains

Martin von Gagern Hyperbolic Ornaments

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Basics Program Intuitive Input Group Calculations Fast Drawing

Group Generation

1 Generator entered by user 2 Add inverse operations 3 Find “all” combinations

  • Group representation
  • Orbit of centerpiece
  • Each element starts

a new domain

4 For all triangles that are

not yet part of any orbit

  • add triangle to

central domain

  • combine triangle with all

group elements to calculate its orbit, adding to domains

Martin von Gagern Hyperbolic Ornaments

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SLIDE 48

Basics Program Intuitive Input Group Calculations Fast Drawing

Group Generation

1 Generator entered by user 2 Add inverse operations 3 Find “all” combinations

  • Group representation
  • Orbit of centerpiece
  • Each element starts

a new domain

4 For all triangles that are

not yet part of any orbit

  • add triangle to

central domain

  • combine triangle with all

group elements to calculate its orbit, adding to domains

Martin von Gagern Hyperbolic Ornaments

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SLIDE 49

Basics Program Intuitive Input Group Calculations Fast Drawing

Group Generation

1 Generator entered by user 2 Add inverse operations 3 Find “all” combinations

  • Group representation
  • Orbit of centerpiece
  • Each element starts

a new domain

4 For all triangles that are

not yet part of any orbit

  • add triangle to

central domain

  • combine triangle with all

group elements to calculate its orbit, adding to domains

Martin von Gagern Hyperbolic Ornaments

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Basics Program Intuitive Input Group Calculations Fast Drawing

Group Visualization

Martin von Gagern Hyperbolic Ornaments

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Basics Program Intuitive Input Group Calculations Fast Drawing

Fast and Perfect Drawing

Fast draw smooth lines in real time Perfect image looks as correct as display hardware allows

Martin von Gagern Hyperbolic Ornaments

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Basics Program Intuitive Input Group Calculations Fast Drawing

Fast and Perfect Drawing

Fast draw smooth lines in real time Perfect image looks as correct as display hardware allows

Martin von Gagern Hyperbolic Ornaments

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Basics Program Intuitive Input Group Calculations Fast Drawing

Fast and Perfect Drawing

Fast draw smooth lines in real time Perfect image looks as correct as display hardware allows

Martin von Gagern Hyperbolic Ornaments

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Basics Program Intuitive Input Group Calculations Fast Drawing

Reverse Pixel Lookup

1 Scan convert triangles

Triangle preprocessing

2 Map into central domain

Group preprocessing

3 Update only changes

Realtime drawing

4 Supersampling

Antialiasing

Martin von Gagern Hyperbolic Ornaments

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Basics Program Intuitive Input Group Calculations Fast Drawing

Reverse Pixel Lookup

1 Scan convert triangles

Triangle preprocessing

2 Map into central domain

Group preprocessing

3 Update only changes

Realtime drawing

4 Supersampling

Antialiasing

Martin von Gagern Hyperbolic Ornaments

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Basics Program Intuitive Input Group Calculations Fast Drawing

Reverse Pixel Lookup

1 Scan convert triangles

Triangle preprocessing

2 Map into central domain

Group preprocessing

3 Update only changes

Realtime drawing

4 Supersampling

Antialiasing

Martin von Gagern Hyperbolic Ornaments

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Basics Program Intuitive Input Group Calculations Fast Drawing

Reverse Pixel Lookup

1 Scan convert triangles

Triangle preprocessing

2 Map into central domain

Group preprocessing

3 Update only changes

Realtime drawing

4 Supersampling

Antialiasing

Martin von Gagern Hyperbolic Ornaments

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Basics Program Intuitive Input Group Calculations Fast Drawing

Reverse Pixel Lookup

1 Scan convert triangles

Triangle preprocessing

2 Map into central domain

Group preprocessing

3 Update only changes

Realtime drawing

4 Supersampling

Antialiasing

Martin von Gagern Hyperbolic Ornaments

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SLIDE 59

Basics Program Intuitive Input Group Calculations Fast Drawing

Reverse Pixel Lookup

1 Scan convert triangles

Triangle preprocessing

2 Map into central domain

Group preprocessing

3 Update only changes

Realtime drawing

4 Supersampling

Antialiasing

Martin von Gagern Hyperbolic Ornaments

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SLIDE 60

Basics Program Intuitive Input Group Calculations Fast Drawing

Reverse Pixel Lookup

1 Scan convert triangles

Triangle preprocessing

2 Map into central domain

Group preprocessing

3 Update only changes

Realtime drawing

4 Supersampling

Antialiasing

Martin von Gagern Hyperbolic Ornaments

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SLIDE 61

Basics Program Intuitive Input Group Calculations Fast Drawing

Reverse Pixel Lookup

1 Scan convert triangles

Triangle preprocessing

2 Map into central domain

Group preprocessing

3 Update only changes

Realtime drawing

4 Supersampling

Antialiasing

Martin von Gagern Hyperbolic Ornaments

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