p-e-r-i-o-d-i-c
and
APERIODIC
T
E
S
S
E
L
L
A
T
- basic concepts -
© 2014
A n to n i o P e t i c o v
I
O N
When you fit
individual tiles together
with no g aps or overlaps
to fill a flat space like a
ceiling, wall, or floor,
you have a
tiling or tessellation.
You can imagine that
you can use a variety of
shapes to do this.
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The majority of the TILINGS
t h at w e k n o w a r e m a d e w i t h
SQUARE, TRIANGULAR or HEXAGONAL
shapes,
b u t n e v e r l i k e a r e g u l a r p e n ta g o n .
PERIODICAL TILING
We call Periodic Tiling when
the design formed by its tiles
is found repeated
in other regions
i n s i d e t h e a r e a t h at i t c o v e r s
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There are many other shapes
t h at w e c a n u s e
t o “ t i l e ” a p l a n e p e r i o d i c a l ly
u s i n g a g r e at va r i e t y o f
r e g u l a r polyhedra...
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. . . and IRREGULAR
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I n the construction of each of the three
examples of Periodic Tiling
p r e s e n t e d i n t h i s pa g e
T WO di f f e r e n t r e g u l a r p olyg on s
were used.
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T h e r e p e t i t i o n o f t h e t i l e s i n a t e s s e l l at i o n
c a n va r y a c c o r d i n g t o i t s
R O TAT I O N , T R A N S L AT I O N AND/OR R E F L E C T I O N .
R
IO
N
T
TIO
A
TA
T
R O TAT I O N
O
RO
N
A TILE SUFFERS R O TAT I O N
WHEN IT IS TURNED ON ITS AXIS.
E A C H R O TAT I O N H A S A C E N T E R A N D A A N G L E .
THE EXAMPLE OF ROTATION PRESENTED ABOVE IS A DETAIL OF THE MURAL
“ANTHROPOPHAGIC MOMENT WITH OSWALD DE ANDRADE”
INSTALLED AT THE REPUBLICA STATION OF THE SÃO PAULO METRÔ, in BRAZIL.
© 2014 Antonio Peticov
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© 2014 Antonio Peticov
The 90º ROTATION of this square tile enables the formation
of a great number of geometric combinations .
© 2014 Antonio Peticov
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HYDRAULIC TILE
With 90º turns and some combinations
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Lorenzo Borletti’s apartment in São Paulo
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Lorenzo Borletti’s apartment in São Paulo
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“Saint Gerome’s Meditation” - acrylic on canvas - 180 x 140 cm - ©2015 Antonio Peticov
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“A Espreita” - acrylic on canvas - 160 x 120 cm - ©2015 Antonio Peticov
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“DREAMING” 61,8 x 61,8 cm © 2014 Antonio Peticov
© 2014 Antonio Peticov
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-
BLUE
Porcelaine Tile with 20 cm side
With a 120º rotation
© 2014 Antonio Peticov
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REFLECTION
T h e R E F L E C T I O N o f a t i l e c r e at e s i t s
MIRROR IMAGE.
Each REFLECTION has its MIRROR LINE.
THE EXAMPLE OF REFLECTION PRESENTED ABOVE IS A DETAIL OF THE MURAL
“ANTHROPOPHAGIC MOMENT WITH OSWALD DE ANDRADE”
INSTALLED AT THE REPUBLICA STATION OF THE SÃO PAULO METRÔ, In 1990.
© 2014 Antonio Peticov
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selpmaxE Examples
fo of
NOITCELFER REFLECTION
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T R A N S L AT ION
T h e T R A N S L AT I O N o f a t i l e o c c u r s w h e n i t i s
m o v e d w i t h o u t r e f l e c t i o n o r r o tat i o n .
T h e T R A N S L AT I O N h a s D I R E C T I O N a n d D I S TA N C E .
M. C. Escher
C r e at e d t h i s “ T W O B I R D S ”
t h at c o m p o s e t h i s P e r i o d i c T i l i n g
in 1945
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ROTATION and TRANSLATION
of Three Tiles.
NON-PERIODIC TESSELLATION
WITH THE SAME
ISOSELES TRIANGLES
TYPICAL PERIODIC TESSELLATION
WITH ISOSELES TRIANGLES
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“versatile” Poligons
We call a poligon “versatile” when it enables a tessellation
both periodic and non-periodic
note that this polygon comes with
the possibility to be completely
“embraced” by other two pieces only,
similar in size to the original
a periodic Tessellation
a non-periodic Tessellation
by Heinz Voderberg
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“REPTILES”
The R E P T I L E S
are tiles that
group together
forming replicas
of themselves.
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NON- PERIODIC
or APE RI ODI C
T E S S E L L AT I O N
A NON-PERIODIC tessellation
is when the overall design
formed by its tiles
has a CENTRAL point
that expands
without EVER repeating
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“dart”
“kyte”
The most famous non-periodic tiles are the
“PENROSE TILES”
Coming from the pentagon, DARTS and KYTES have
its measures and proportions always in a “golden” relation with each other,
tiling only “ n o n p e r i o d i c a l ly ” .
The quantities of DARTS and KY TES used to cover a given area
are also in a golden proportion,
s o , i f 1 0 0 0 k y t e s a r e u s e d , 6 1 8 da r t s w i l l b e n e e d e d .
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These Poligons were discovered in 1974
by Sir Roger Penrose,
an English Mathematician and philosopher.
The same that, with his father, invented the
“endless Staircase”
used by M. C. Escher
on his print
“Ascending and Descending”.
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There are only SEVEN ways of assembling
DARTS and KYTES around a vertex
and the only ones with pentagonal symmetry are
“The Infinite Star Pattern”
and
“The Infinite Sun PAttern”.
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A n o t h e r Va r i at i o n o f t h e P E N R O S E T I L E
The LargeRhomb and the LongRhomb
Long Rhomb
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Large Rhomb
A
Rhomb
is
the
same
as
a
Lozenge
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the figure above is called
“Quasitiler”
and uses a SQUARE
together with
the Long Rhomb
and the Large Rhomb
in order to build
this beautiful
aperiodic tesselation
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the marks drawn in the faces of these
Long Rhombs and Large Rhombs
contribute in the formation of an
ever changing geometric design
that expands infinitely
Hen 1
Hen 2
Adaptation of the Penrose Tiles DART and KYTE
in two “CHICKEN” shapes that tile aperiodicaly
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Project for a mural using multiple layers
of DARTS and KYTES
with different kinds of wood
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Some examples of
non periodic
tessellation
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Some examples of
periodic tessellation
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BIBLIOGRAPHY
© 2 0 1 4 Antonio Peticov
www.antoniopeticov.com.br
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