At the begining it's always drawing by hand. Free hand on my little notebook
(A6 side, white paper), while having a cofee in a nice bar (Paris is a very
good citie for bars).
Looking at your Figure 1 on the table.
I use my black fountain pen, no rule, no eraser. So it is a nice feeling
when all the lines turn to fit together.
Of course, i do not try to draw the interlaces (is it better to say "braids"
? ), which i consider as an embellishment.
Ok, it is a perfectible drawing, but enough for going on working.
Then i use my computer...
I like very much Autocad for that kind of vectorial drawings.
So,
That story starts with the dance of 10 pentagons.
Why not another pentagon at the center ?
Yes, but each line have to find her sister...
That is.
Now, another one !
To avoid monotony we change the orientation inside.
And repeat the whole.
Here an idea happened : to use these symmetries and let an 10-pointed star take
place at the center.
Now we have to fill the vacuum.
New pentagones in a pentagonal symmetrie.
And...
... why not another pentagone at the center ?
Good, but the lines cannot stay like that !
As usual, they have to fit together. And that is the simplest, natural, way.
Going on in filling the space, with pentagones. It works very well (because
of the symmetries).
Ah! some pentagones are overlapping !
Natural resolution...
Now we can symmetrise.
After that is to draw the 10-pointed star on the corners.
The question is : how to choice the width of the petals (or branch, or rays
if you prefer to say) ?
i like the flowered stars
That
is a story of pentagones. So i decide to draw from 2 concentrics pentagones.
Maybe not very original, but certainly the most natural.
Like that, an then the same on the other side...
... that is !
I s it finished now ?
Well, with your permission i would like to have a "river" running
all along the pattern.
Here is the river. Yes, i have taken some liberties with geometry. I can't
denie. I confess. The worse is that i don't feel guilty for that !
I like rivers running in the starry gardens
And now, let us go to Adobe Illustrator !
Because this software is much more efficient for colouring that Autocad.
Symmetryzation...
Colors...
Colors...
That is fast done, just as an example. It is not so easy (you know ?) to set
up the colors. Computer give me a good help for that.
But i also like very much colouring by hand, on real paper.
Now changing the color of the background...
... and get the star nicely dressed with white enterlaces.
(i do not use the maximal width )
Now,
My point of view is morphogenesis more than calculation. Your Karatay pattern
come from the dance of pentagones. So, i would like to see which are the patterns
coming from that dance, before the "Karatay" one.
So, first is the dance of 10 pentagones.
As we saw before, if i put another pentagone at the midle, the lines goes on
and give way to that.
Like the fairy tales for childrens, this story like to use repetition...
...
pure repetition would be boring. So let us introduce some light variation.
One
more,
With 4 copies of the dance i can get a periodic pattern. But 2 pentagons are
overlapping.
No
problem !
So, here is a very simple pattern, a cousin of the "Karatay" one.
Could it be more simple ?
Yes !
Let us look at the symmetries.
Horizontal
axe,
Vertical axe,
so the pattern can be...
... reduced...
(here is no more symmetry axe, but a center of rotation)
To that very simple figure : just 5 lines !
Now, let us come back to the whole pattern...
... with another point of view :
So we can reduce it to a rhombus.
And you can recognize this rhombus as one of the 2 penrose tiles.
And if i imagine me playing with that tile...
... it could be something like that.
But there is a hole inside !
And the shape of this hole is the one of the second Penrose tile.
Let the lines running, and here is the simplest decoration of that tile.
So, now, each time i'm playing with the Penrose tiles...
I get a zellij pattern.
Here in a symmetrical (radial) configuration.
Each unit is a standard zellij unit, but the repartition is not so standard.
Now, another one !
Now, another one !
Now, another one !
Now, another one !
Now, another one !
Now, another one !
Now, another one !
Now, another one !
Now, another one !
Now, another one !
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