Moroccan Geometry

[Paul Kranz]

I took this picture in the Marrakesh Theater, Busch Gardens, Tampa, FL. Have any of you seen this geometry projected onto a sphere? I am thinking it has a dodecahedronal pattern.

Paul sends...

 

[TaffGoch]

Hiya, Paul,

Moroccan tiling can be fitted to icosa/dodeca polyhedral solids, as long as they exhibit six-way radial symmetry. The same is true for 5-way radial symmetry, in many cases. At first, I thought mirror symmetry was also required, but my first example, below, demonstrates that supposition was fallacious.

("Cairo" tiling, initially, looks like 5-way symmetry, but it is 4-way radial symmetry. For a while, I've been trying to get it to fit a polyhedral solid, with no luck.)

What I have had "luck" with, over the last couple of years...

Star Weave:

http://sketchup.google.com/3dwarehouse/details?mid=dab9d4b020a3212894d3b6d776085e61

Temari Ball:

http://sketchup.google.com/3dwarehouse/details?mid=b897af205cc0f30d27f3401e25bea5a3

Arabian Tessellation:

http://sketchup.google.com/3dwarehouse/details?mid=9690b790eba2f9b020c2e31efdb183b1

It looks like the "Temari Ball" geometry comes closest to the design in the image you posted -- not spot on, but similar.

-Taff

[TaffGoch]

This one qualifies, I suppose, although it doesn't as-strongly resemble Moroccan/Arabic tiling (to me, anyway) :

Geodesic Weave:

http://sketchup.google.com/3dwarehouse/details?mid=59a2e66e771eeb8a99716478f2af0fca

-Taff

[Paul Kranz]

Taff:

 

Thank you for your speedy reply. I love your drawings!!

 

Paul sends...

[Thomas Rainwater]

Taff should publish a book FULL of these drawings! I know that I'd buy a copy!

-Tommy

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thomas.r...@gmail.com

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[biagiodicarlo]

The second copy will be for me

-Biagio

Il giorno 08/dic/2013, alle ore 02:53, Thomas Rainwater <thomas.r...@gmail.com> ha scritto:

Taff should publish a book FULL of these drawings! I know that I'd buy a copy!

-Tommy

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On Sat, Dec 7, 2013 at 5:46 PM, Paul Kranz <pa...@revivetheflame.com> wrote:

Taff:

 

Thank you for your speedy reply. I love your drawings!!

 

Paul sends...

 

On Sat, Dec 7, 2013 at 6:40 PM, TaffGoch <taff...@gmail.com> wrote:

This one qualifies, I suppose, although it doesn't as-strongly resemble Moroccan/Arabic tiling (to me, anyway) :

Geodesic Weave:

http://sketchup.google.com/3dwarehouse/details?mid=59a2e66e771eeb8a99716478f2af0fca

<Geodesic Weave.jpg>

-Taff

[Paul Kranz]

Surprise, surprise, surprise. It's a four-frequency dodecahedron!
 

Paul sends...

[homespun]

Question:  We have been talking about projecting this Morrocan pattern onto a sphere, but what is it as a plane tiling?  Is it a true aperiodic Penrose pattern?  I am only just now beginning to investigate Penrose Tilings for the first time, having seen him last week at a lecture here in San Antonio at Palo Alto College.  So I am printing out a bunch of them to fool around with.

                             Dan

[TaffGoch]

Paul,

More-accurately described as a 2-frequency triacontahedron, perhaps? (Putting each ten-pointed star at a vertex.)

Here's the tiling pattern. I'm sure everyone can recognize the pentagonal symmetry:

...and here's how it would appear, mapped onto a dodecahedron:

The ten-pointed stars "turn into" nine-pointed stars, at the vertices. Subdividing the dodecahedron, into a 2v triacontahedron, would produce a more-spherical geodesic, of course. Higher frequencies (3 or greater) would "round" out the polyhedron, even more.

-Taff

[TaffGoch]

Here's the 2v triacontahedron (also known by other polyhedron nomenclature):

[Paul Kranz]

Taff:

 

Can you project the Moroccan pattern onto a sphere?

 

Paul sends...

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Very high regards,

 

Paul sends...

ReviveTheFlame.com

[TaffGoch]

Paul,

Rather than "project" the 2D pattern image onto a sphere, which severely distorts straight lines (into curved lines,) I constructed planes, to intersect the "sphere," aligned with the endpoints in the pattern (endpoints, where a line takes a "jog.")

This produces geodesic arcs, more-closely approximating the straight-line-segments appearance of the 2D pattern:

The pentagonal stars get stretched, a little, but that's due to the 2D-to-3D translation.

Note that SketchUp doesn't really do curves or spheres -- these are approximated by short line segments. This "sphere" is a 16v, Class-II (aka, triacontahedron) tessellation.

-Taff

[Blair Wolfram]

Here's another 16v sphere, Order In Chaos by Fr. Magnus Wenninger.

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[TaffGoch]

Blair,

Father Magnus was kind enough to send me a note of appreciation, regarding my 3D-modeling of his sculpture.

http://sketchup.google.com/3dwarehouse/details?mid=342449af889f191527f3401e25bea5a3

http://taffgoch.deviantart.com/art/Order-in-Chaos-214480976

-Taff

[Paul Kranz]

Taff:

 

Simply beautiful! Thank you.

 

Paul sends...

 

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[TaffGoch]

"Beefed up," enhanced tiling + triacon geodesic edge-shading modified to produce "smooth" sphere:

[Paul Kranz]

Very Beefy!

On Wed, Dec 11, 2013 at 9:01 PM, TaffGoch <taff...@gmail.com> wrote:

"Beefed up," enhanced tiling + triacon geodesic edge-shading modified to produce "smooth" sphere:

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[TaffGoch]

To model one of my previously-referenced examples, I 3D-wrapped this pattern...

...onto a sphere, to produce:

 

(Hexagonal patterns are relatively easy.)

For further explorations (both Paul and Dan,) these pattern images can be researched, using the term, "zillij"

www.google.com/search?as_st=y&tbm=isch&sa=1&q=zillij

-Taff

[TaffGoch]

I "zoomed out" the same pattern image depicted in the previous post, to produce a new "zillij" sphere (also based on an icosahedron,) following these steps:

 1)  "Trace" pattern on icosahedron faces,

 2)  Explode vertices and intersections to unit sphere, and

 3)  Truncate radial lines, to produce lattice.

Paul, these are the steps that I employ, going from 2D-to-3D. and isn't conceptually-difficult, but it does require careful, meticulous attention to precision, while modeling. SketchUp provides for this precision, as long as you know a few SketchUp tool "tricks." 

I like this one better (more apparent complexity,) and will likely post to the 3D Warehouse.

-Taff

[homespun]

Taff, please post link to final product.

          Dan

----- Original Message -----

From: TaffGoch

To: Geodesic Help Group

Sent: Monday, December 16, 2013 7:43 PM

Subject: Re: Moroccan Geometry

 

[TaffGoch]

Dan,

I have posted it, here: http://sketchup.google.com/3dwarehouse/details?mid=41d1ad4b10b8b628bd158aa30fff4e9a

(You can "turn off" the black background, by opening the window for "fog" settings.)

Here's the pattern (which is included in the model, in a separate "layer"):

[Paul Kranz]

A-maze-ing! I am happy to contribute.

 

Paul sends...

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[TaffGoch]

I "tinkered" a bit, to re-align arc-segments, to better follow parallel, lesser-circle paths. It's much improved, but not quite spot on.

It'll do....

http://taffgoch.deviantart.com/art/Zillij-421487757

-Taff

[TaffGoch]

Paul,

I found a description of how the design (from your original post) is produced, within a PDF from 

Craig Kaplan, "Computer Graphics and Geometrical Ornamental Design"

-Taff

[Paul Kranz]

Taff:

Very cool! Thanks for the info.

Pal sends...

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[Entopii Morcillist]

Dear Taff,

hoping you are well.. am not a physicist nor a mathematician but really admire your work. My reason for writing is that how can your moroccan design can be applied to producing a real-life spherical product... if I was to use 0,9mm bamboo plywood using clips to connect the each piece together to have a spherical end-product.. something like your "Order in Chaos"

Thanks and regards,

Cengiz