| wdswan@erols.com Dear Yon and all, I believe that you would find, by
reviewing the various sites on symmetry, that a tessellation is created by
a perimeter (i. e., a shape) and not by the contents of that perimeter. So
if you have a field full of rectangles all of the same dimensions, you
have a tessellation. And it doesn't matter what is within those
rectangles. The tessellation rug in this salon contains, in the diamonds
of the "S" shape, some devices that are roughly boxes with crosses on them
and the triangles have smaller devices. All of this filler is irrelevant.
The "S" shape creates the tessellation. In your example of the primary and
secondary guls, the rectangles (consisting of a primary gul and the
quarters of the surrounding four secondary guls) do indeed form a
tessellation. In the case of primary guls in which lines pass through the
centers, one could have the secondary gul at the center of the rectangle.
The repetition of that rectangle over the field would fill the plane and
you would have the field of a Turkmen carpet. On that basis, you could
take any rug with a repeating design, draw a rectangle around the
repeating element, translate that rectangle all over the plane and say
that you have a tessellation. But in all cases it would be a tessellation
of rectangles, not of the contents. Take, for example, a Senneh rug filled
with rows of identical repeating botehs. Drawing rectangles around the
botehs will not convert the botehs into a tessellating shape; it will only
create a tessellation of rectangles that happen to contain botehs. The
fact that the device within the rectangle (or whatever tessellating shape
you chose) is symmetrical makes no difference. A Bakhtiyari garden carpet
could be an example of a tessellation of rectangles (containing
asymmetrical devices) in which the contents are not repeated and do not
correspond to the contents of another rectangle. Turkmen guls are created
through the application of other principles of symmetry and they may be
symmetrical in two parts or four parts. Reflection and/or rotation will
result in the creation of the whole gul from one part. I believe that
Escher has, in some instances, used the term tessellation in conjunction
with artistic works in which the contents of the shape align with
corresponding contents of other rectangles, but that is where art may
differ from mathematics. Perhaps it is helpful to think of a tessellation
(as least as we may confine it to discussions about carpet designs) as a
grid. A tessellation is a type of grid that may have very irregular lines
but each segment of the grid is identical in shape to all the other
segments. Escher's winged horse or his lizards are good examples of
tessellations created by highly irregular lines. Best,
Wendel |
I took
the liberty to change the perspective on the Belouch design. Your
interpretation on the Alhambra design is correct in your perception. Only
I saw flowers with stars. So I did not see your tessellation which is in
my perception: a leaf and a star; the star beïng the result of the setting
of the leafs. The two together forming the next tessellation. So it's a
tessellation within a tessellation. Pffjuh! So if a tessellation is the
product of one design creating another design and we draw the line-stop! -
the Alhambra isn't a tessellation. But...if we extract the whole flower
with all it's leaves the result is the star and it's a tessellation. 
Here's another mind-switching
rugdesign. This Bokhara is a tessellation of the star producing in it's
setting a "Mother Goddess with two animals at her side". 
In the perseption of that
cult it's a correct tessellation. But only because they see the Mother
Goddess with the breasts, feet up in the air and the two animals at her
side. So a tessellation is the product of cultural programming, and
therefore it's a mind-switching(left-right) experience for us. Best
regards, Vincent
But
I'll stop now. Afraid people will lose intrest. Thank you for this very
beautiful, never ending subject, it did trigger the correct handle (POINT)
(Which in tessellation produces a _______ line) Best regards,
Vincent
If I want to make a tessellation of a simple form, so everything
fits in perfectly and has exactly the same dimension, and in between
distance/space, so it's the perfect repetition, (Pfffjuh) of one design,
I'll divide in four parts: In order to get C, I make B1:The product of A.
In order to get it perfect I pick up the centre of B. Now it's a square
and easy to make. I do not think the mosaïc artist made a realistic
design, he knew what it would look like finished. But in order to get it
perfect he too had to divide it in four, mirrored etc. As you see in the
"Soumack Bird" design in the salon, the design has originaly been divided
in squares that had triangles within them. Put all the squares into the
field and change the colours of the triangles. I do not think the artist
made an, out of the blue, S shape, for the S's had to fit in perfectly. I
hope you can agree with this. The only problem is: With a simple,
universal designs it isn't so hard. With more complex, culture related
designs, we can'tt "translate". Best regards, Vincent
The tessellating shape is the
zigzag that crosses the field, including the "C's" at the peaks.
Wendel
Best regards,
Vincent