Figure 1: Trichotomy for pairs of distinct lines.
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Transcript of Figure 1: Trichotomy for pairs of distinct lines.
DYNAMIC VISUALIZATIONOF PLANAR HYPERBOLIC SYMMETRY
VIA THE KLEIN MODEL
David E. Flesner
August 2, 2009
Symmetry Festival 2009Budapest
Gettysburg College
[email protected]/~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.
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
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
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
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
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: [email protected]