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The Euclidean and hyperbolic geometry underlying M.C. Escher's regular division designs / Karen...
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The Euclidean and hyperbolic geometry underlying M.C. Escher's regular division designs / Karen Kleinschmidt Haxhi
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Description
Identifier
Thesis
1491
Author
Haxhi, Karen Kleinschmidt
Title
The
Euclidean
and
hyperbolic
geometry
underlying
M.C
.
Escher's
regular
division
designs
/
Karen
Kleinschmidt
Haxhi
Publisher
Central Connecticut State University
Date
1998
Resource Type
Master's Thesis
Notes
In this
paper
Euclidean
and
hyperbolic
discontinuous
isometries*
are
discussed
and
compared
. The
graphic
artist
M.C
.
Escher
(18981972)
produced
works
which
have for the
structure
points
,
lines
, and
angles
,
which
repeat
in the
plane
(Euclidean
and
hyperbolic)
at
regular
intervals
. These
regular
systems
of
points
,
lines
and
angles
can
be
described
with
mathematics


with
groups
of
discontinuous
isometries
and the
tilings
they
produce
. This
paper
is
therefore
also a
comparison
of
Escher's
Euclidean
and
hyperbolic
regular
division
(the
plane
is
divided
into
congruent
shapes
which
cover
the
plane)
designs
. The
isometries
of the
Euclidean
and the
hyperbolic
planes
are
discussed
empirically
.
Specific
isometries
are
described
and
diagrammed
in
thirteen
of
Escher's
Euclidean
designs
(Escher
made
dozens
of
regular
division
Euclidean
designs.)
and in
two
of
Escher's
four
"
Circle
Limit
"
pictures
,
which
are
hyperbolic
regular
division
designs
. In the
empirical
spirit
of this
paper
,
I
have also
provided
instructions
of how to
construct
with
ruler
and
compass
a
hyperbolic
tiling
. The
construction
of
Euclidean
tilings
are
straightforward
when
compared
to the
construction
of
hyperbolic
tilings
.
Euclidean
tilings
follow
directly
from the
diagrams
in this
paper
of the
Euclidean
isometry
groups
.
Indeed
since
childhood
we
have
all
stared
at
Euclidean
tile
patterns
on
walls
and
floors
and
can
produce
many
tilings
with
no
understanding
of the
underlying
mathematics
. The
creation
of
hyperbolic
tilings
is
not
so
straightforward
.
We
do
not
often
see
hyperbolic
patterns
, and
when
we
do
, as
when
looking
at an
Escher
"
Circle
Limit
"
print
,
it
seems
strange
that
similar
hyperbolic
shapes
in
our
model
(Escher
used
the
Poincare
disk
model)
of the
hyperbolic
plane
.
Additionally
,
our
Euclidean
notion
of
straight
lines
is
not
visually
apparent
in
Escher's
hyperbolic
designs
. In the
Poincare
disk
model
hyperbolic
lines
not
passing
through
the
center
of the
disk
are
represented
by
circular
arcs
orthogonal
to the
bounding
circle
of the
disk
. This
paper
is
designed
for the
secondary
school
mathematics
teacher
who
wants
to
gain
a
qualitative
understanding
of the
differences
between
the
Euclidean
and
hyperbolic
planes
, and their
tilings
,
without
the
abstract
notation
that
often
accompanies
such
discussions
.
Tilings
of the
Euclidean
plane
and the
hyperbolic
plane
may
then be
used
to
illustrate
,
visually
, the
differences
between
the
two
planes
. These
visual
differences
can
then be
used
to
illustrate
the
differences
in the
theorems
of the
Euclidean
and the
hyperbolic
planes
.
All
differences
between
the
two
planes
stem
from their
different
parallel
axioms
. The
introduction
of
hyperbolic
geometry
on the
secondary
school
level
may
serve
to
show
students
that the
proofs
they
do
in
class
may
not be
mere
formalities
to
show
something
that
is
visually
obvious
. In
our
models
of the
hyperbolic
plane
,
congruence
between
line
segments
,
polygons
,
circles
, etc.
may
not be
visually
obvious
.
Therefore
a
proof
written
to
prove
a
theorem
for the
hyperbolic
plane
may
give
a
student
a
clearer
understanding
of the
importance
of
applying
the
rules
of
logic
to a
set
of
axioms
and
theorems
to
prove
, or
disprove
, a
conjecture
they
may
have.
Copies
of
two
Escher
prints
below
will
visually
give
a
preview
of the
differences
between
Euclidean
designs/tilings
and
hyperbolic
designs/tilings
that will be
discussed
in this
paper
.
Figure
A has for its
structure
a
Euclidean
isometry
group
.
Figure
B
has for its
structure
a
hyperbolic
isometry
group
.
Subject
Geometry, Descriptive
Geometrical drawing
Hyperbolic spaces
Geometry, NonEuclidean
Department
Department of Mathematical Sciences
Advisor
McGowan, Jeffrey K
Type
Text
Digital Format
application/pdf
Language
eng
OCLC number
40251155
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