Hello dear John Rigby,

Here is my approach of the "Karatay pattern", and some of my flight
of fancy with the Penrose tillings.

*Use the arrows to go upwards or backwards. If you click between the two
arrows it open some shorcuts.*

One more time i apologie for my bad english.

So,

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 city 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...

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 a 10-pointed star takes
place at the center.

Now we have to fill the vacuum.

New pentagones in a pentagonal symmetrie.

And...

New pentagones in a pentagonal symmetrie.

And...

... why not another pentagone at the center ?

Good, but the lines cannot stay like that !

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 !

Ah! some pentagones are overlapping !

Natural resolution...

Now we can symmetrize.

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 concentric pentagones.

Maybe not very original, but certainly the most natural.

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 ?

I s it finished now ?

Well, with your permission my dear John, 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 than Autocad.

Because this software is much more efficient for colouring than Autocad.

Symmetrization...

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.

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 interlaces.

*(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.

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 ?

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...

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.

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.

Another configuration, with an unexpected 10-pointed star !

Another point of view. There is still a symmetrical structure...

... but here, there is no more symmerty at all !

A closer look. I use a programm from Denis Gracias, which is used for studies
in quasicrystals

Another zellij-like decoration of the Penrose tiles, more moroccan style ( "Arabesques",
p. 287).

That one is the same with interlaces. With a center of symmetry...

... or without any global symmetry

More distance...

Here, we use zellij decoration and Penrose tilling in connection with an architectural
structure. Of course, in that structure the golden number can be find everywhere.

The pattern on the floor...

Implantation of the bungallow...

A view from the top...

... and a perspective.

Here i use a Penrose tilling as a map for a 5-fold symmetrie muqarnas structure

It is just a draft, with very simple modeling of the muqarnas
units.

You know the Dover printing of that book. But do you know the original edition
?

i have one copy... it's a long story, someone gave it to me...

So i decided to have a bookbinding. I did the layout and some of the best craftsman (friends of mines) made it.

My layout is a kind of meeting between the contemporary maths and the traditional art.

i have one copy... it's a long story, someone gave it to me...

So i decided to have a bookbinding. I did the layout and some of the best craftsman (friends of mines) made it.

My layout is a kind of meeting between the contemporary maths and the traditional art.

There is a Penrose tiling running all around the box (this book is made of individual
plates, not sewed together except the section of explanations.

Technic : incisions.

Technic : incisions.

The traditional pattern is hidden under the flap.

Technic : mosaic of leather incrusted.

Oppening the box...

On the top is the text section, with some explanations.

The plates are under :

10 colored plates and 190 drawings.

The text section.

The first colored plate: an amazing work.

Coloured plates and the first technical plate, in 1, sometimes 2 colors.

A close-up on my fridj. Some years ago my friend Marc Pelletier (USA)
used my decoration of Penrose tiles to make a magnetic game.

I also did decoration for the 3 tiles of non periodical paving with local 7-fold symmetry.

Of course, i use a lot the local 8-fold symmetry ! (see my paper "Play with infinity")...

Bye bye, John !

I also did decoration for the 3 tiles of non periodical paving with local 7-fold symmetry.

Of course, i use a lot the local 8-fold symmetry ! (see my paper "Play with infinity")...

Bye bye, John !