New Pentagon Discovery

[Paul Kranz]

Just When You Thought All of the Tile Pentagons Have Been Discovered...

http://www.theguardian.com/science/alexs-adventures-in-numberland/2015/aug/10/attack-on-the-pentagon-results-in-discovery-of-new-mathematical-tile

Very high regards,

 

Paul sends...

ReviveTheFlame.com

[Ken G. Brown]

Great! Now, can you use pentagonal tiling to tile the sphere? I somehow think it could be possible...

Ken,

from my iPhone

On Aug 12, 2015, at 19:18, Robert Clark <clark.rob...@gmail.com> wrote:

Nice to know that there are still more discoveries out there waiting to be found.

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[Blair Wolfram]

It already has a name, dodecahedron

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[Ken G. Brown]

More, and smaller pieces.

Ken,

from my iPhone

[TaffGoch]

Nice quip, Blair! I audibly laughed. (I refuse to use "LOL.")

I find the article odd, in that it didn't include (in it's images) the mirror-symmetrical "Cairo" tiling - one of the oldest-known planar pentagonal tilings:

(The Cairo tiling edge-lengths are equal, by-the-way.)

Some examples came close to Cairo tiling, but they're not quite symmetrical:

Ken, I've tried spherically-tiling the Cairo pattern. It doesn't work.

-Taff

[Paul Kranz]

Taff:

I think you should discover the next pentagon!

Paul sends...

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[TaffGoch]

Ken,

I've found only one pentagonal tiling that works on a sphere -- the "floret" petal:

(Recall previous postings regarding these duals of snub tessellations?)

-Taff

[Ken G. Brown]

Nice.

It looks like there are two sizes of pentagon, is that correct?

How many of each per sphere?

Ken

On Aug 13, 2015, at 20:05, TaffGoch <taff...@gmail.com> wrote:

Ken,

I've found only one pentagonal tiling that works on a sphere -- the "floret" petal:

<Floret dual; {2,2}.png>

(Recall previous postings regarding these duals of snub tessellations?)

-Taff

[Adrian Rossiter]

Hi Taff

On Thu, 13 Aug 2015, TaffGoch wrote:
> I've found only one pentagonal tiling that works on a sphere -- the
> "floret" petal:

The transformation that creates this pattern from another
tiling is related to the "isohedral pentagonal transform"

http://loki3.com/poly/transforms.html#penta

Projecting any of these onto a sphere produces a tiling with
one kind of pentagon.

The transformation joins rotating sets of spokes from three
symmtry axes to make a hexagon, but if one of the axes has
order two then its two spokes are in a line, producing a
pentagon.

In the plane there are two posibilities involving two
axes with order greater than two, and one of order 2.
These produce the fourth and fifth tilings on the top line
of the image you posted of the 15 tilings.

There is also a plane symmetry group with three kinds of
3-fold axis, which produces a hexagon. However, if you
arrange the spokes of two of these to point at each other
then it produces a pentagon. This produces the third tiling
on the top line.

Adrian.
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Adrian Rossiter
adr...@antiprism.com
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[TaffGoch]

Adrian,

Thanks for the link. I had not seen that topic. I suspect such methodology was the basis of the article in the original post. Still, I don't see how the authors would categorically-state that there are only "15 pentagonal tilings" -- leaving out the Cairo and floret tilings. Perhaps, I should re-read the article for caveats.

-Taff

[TaffGoch]

Ken,

As with triangular geodesic tessellations, appearance is deceiving. In this particular example {2,2}, there are 12 different pentagonal "petals." This image depicts the icosahedral-face symmetry. Each colored region contains 12 unique petals:

Here's our (lengthly) 2012 discussion of a slightly-different tessellation {1,3}:

https://groups.google.com/forum/#!msg/geodesichelp/sXqHpqkQ8SI/s_Kk04eJDkQJ

Here's my SketchUp collection of 11 different floret 3D models:

https://3dwarehouse.sketchup.com/collection.html?id=a026cd04804def0d86509ade26108092

-Taff

[Paul Kranz]

Taff:

Naturally, I appreciate the beauty of the pic, but The Guardian article argues that tile pentagons have to all be the same and when edge-matched have to provide complete coverage. Are they mistaken?

Paul sends...

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[TaffGoch]

Paul,

The Guardian authors are referring to planar tiling, not spherical, so, yes, their criteria are correct.

-Taff

[Ken G. Brown]

Ah, ok. 

It would be neat if the different size pieces had different color. Wonder what that would look like?

Ken,

from my iPhone

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[TaffGoch]

Ken,

Um, okay, well, it doesn't help perception much....

-Taff

[Ken G. Brown]

Very interesting but as you say doesn't help so much. I was hoping there might be a unique combination of all the different sized shapes that would be the same everywhere. 

Thx. 

Ken,

from my iPhone

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[Hector Alfredo Hernández Hdez.]

How many spherical  tilig are knowed ?

2015-08-15 20:13 GMT-07:00 Robert Clark <clark.rob...@gmail.com>:

Beutiful! It looks like a stained glass window (only curved) :)

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[Adrian Rossiter]

Hi Taff

The first five are classes of tilings. If you look at the three
I posted then the axes are fixed, but if you consider the green
spokes you can rotate them or vary their length (in type 3 the
rotation also fixes the length), and the other two sets of spokes
will follow.

In this way, the Cairo pentagonal tiling is type 4, and the floret
tiling is type 5.

The Wolfram Alpha applet uses different parameters, but I think
these are the values that produce these two tilings within their
classes

http://wolfr.am/6suXthPD
http://wolfr.am/6suEZ~4s

[Adrian Rossiter]

Hi Taff

On Sun, 16 Aug 2015, Adrian Rossiter wrote:
> The first five are classes of tilings. If you look at the three

The article specifically said this, but from the linked page

http://www.mathpuzzle.com/tilepent.html

it looks like others are also classes.

The 15th, from its description in the article, appears to be
a single pentagon.