DYNAMIC VISUALIZATIONOF PLANAR HYPERBOLIC SYMMETRY

VIA THE KLEIN MODEL

David E. Flesner

August 2, 2009

Symmetry Festival 2009Budapest

Gettysburg College

dflesner@gettysburg.eduwww.gettysburg.edu/~dflesner

Gettysburg, Pennsylvania, USA

Outline

I. IntroductionA. Klein ModelB. Trichotomy for pairs of distinct lines

II. Curves associated with line pencilsA. Circles

B. HorocyclesC. Equidistant curves

III. Isometries associated with pairs of distinct lines

A. RotationsB. Parallel displacementsC. Translations

IV. Some discrete hyperbolic symmetryA. Rosette groupsB. Horosette groups

C. Frieze groups

V. Triangle reflection groups

VI. Tiling the hyperbolic plane

A. Triangle group tilings

B. Regular tilings

C. Semiregular tilings

www.gettysburg.edu/~dflesner

via Reflections

m

P

x

X

u

s t

Line x through point P intersects line m in point X.

Lines s and t through point P are sensed parallel to line m in opposite directions along m.

Line u through point P isultraparallel to line m.

Figure 1: Trichotomy for pairs of distinct lines

m

P

uLine u through point P isultraparallel to line m.

w Line w is the unique common perpendicular to ultraparallel lines u and m.

Curves associated with line pencils

Circles

Horocycles

Equidistant curves

O

RP

Q

X

Angle OPQ = 65.52 °Angle OQP = 65.52 °

OR = 0.85OX = 0.85

Figure 2: Circle

R

r

P

QP'

Q'

Angle P'PQ = 74.39 °Angle Q'QP = 74.39 °

Figure 3: Horocycle

m M

R PQ

Q'

S T

Angle SPQ = 82.66 °Angle TQP = 82.66 °

Angle SPQ' = 24.15 °Angle TQ'P = 24.15 °

Angle MSP = 90.00 °Angle MTQ = 90.00 °

X

X'

U

Angle MUX = 90.00 °

XU = 0.93X'U = 0.93

VRV = 0.93

Figure 4: Equidistant curve

Isometries associated with pairs of distinct lines viaReflections

Rotations

Parallel displacements

Translations

N

m

P

P'

M

PM = 0.62

P'M = 0.62

Angle PMN = 90.00 °

X

X'

Figure 5: Reflection

O

M

m

N

n

P

P"P'

Angle MON = 26.26 °

Angle POP' = 52.52 °2 x MON = 52.52 °

RX

X'Angle XOX' = 52.52 °

OP = 1.77

OP' = 1.77

Figure 6: Rotation

M m

n

P

P"

P'

U

V

Angle UPP' = 43.06 °

Angle VP'P = 43.06 °

R

X

X'

Figure 7: Parallel displacement

a

mn

PP" P'

Q

Q'M N

R X

X'

Angle PQQ' = 90.00 °Angle P'Q'Q = 90.00 °

MN = 0.94

QQ' = 1.882 x MN = 1.88

PP' = 2.29

PQ = 1.26P'Q' = 1.26

Figure 8: Translation

Some discrete Hyperbolic Symmetry

Rosette groups

Horosette groups

Frieze groups

O

n

0.5P

Rosette Group 3m

Flag Height:m

angle m,n = 60.00 °

Figure 9: Rosette group 3m

0 .50

Q

P

ML

S

Flag Height

Horosette Group Dihedral

m

n

Figure 10: Horosette group

0 .50

1 .00

ABM D

Flag Height

Translation DistanceN

Q

P

C

Frieze Group pma2

m

n

Figure 11: Frieze group pma2

Triangle reflection groups

Tiling the hyperbolic plane

Triangle group tilings

Regular tilings

Semiregular tilings

Triangle Reflection Groups

1/p + 1/q + 1/r = 1 1/p + 1/q + 1/r < 1

1 < p <= q <= r <= inf.

Euclidean Hyperbolic

p = 180/A q = 180/B r = 180/C

2 3 6

2 4 4

Integer Solutions:

3 3 3

2 3 7

2 4 5

anything larger.

Use the values shown and

3 3 4

A

B

C

WLOG, assume that

Figure 12: Triangle reflection groups

P

Q

Triangle Group T(2, 3, 7)

Figure 13: Triangle group T(2,3,7)

P

Q

Triangle Group T(2, 4, 5)

Figure 14: Triangle group T(2,4,5)

P

Q Triangle Group T(2, 3, inf)

Figure 15: Triangle group T(2,3,inf)

P

Q Regular Tiling {3,8}

Figure 16: Regular tiling {3,8}

P

Q

Semiregular Tiling {6,6,7}

Symmetry Group T(2,3,7)

Figure 17: Semiregular tiling {6,6,7}

Some References:

W.P. Thurston, Three-Dimensional Geometry and Topology, Vol 1, 1997.

H.E. Wolfe, Introduction to Non-Euclidean Geometry, 1945.

M.J. Greenberg, Euclidean and Non-Euclidean Geometry, 2008.

J.M. Laborde and F. Bellemain, Cabri Geometry II, Ver. 1.0 MS Windows, 1998.

Cabri figure files and Power Point slide show files are on my website:

www.gettysburg.edu/~dflesner

My e-mail: dflesner@gettysburg.edu