11-08-2023, 01:45 AM
(This post was last modified: 11-08-2023, 01:48 AM by pythagorean.)

**Ancient Islamic Architects were frickin' idiots, and I got proof!**
Why didn't Allah tell them this?

Why do all of these ancient Islamic Penrose Tilings use more than one tile?

They could have used a single frickin' tile, for Allah's sake!

From the viewpoint of those of us who live in the 21st century, these ancient Islamic architects look like frickin' idiots!

How could anybody be so stupid?

This is mathematical proof that Allah was a frickin' idiot!

(Prove me wrong!)

Mathematicians Discovered a New 13-Sided Shape That Can Do Remarkable Things

https://www.yahoo.com/lifestyle/mathematicians-discovered-13-sided-shape-160000554.html

An aperiodic monotile

David Smith, Joseph Samuel Myers, Craig S. Kaplan, Chaim Goodman-Strauss

A longstanding open problem asks for an aperiodic monotile, also known as an "einstein": a shape that admits tilings of the plane, but never periodic tilings. We answer this problem for topological disk tiles by exhibiting a continuum of combinatorially equivalent aperiodic polygons. We first show that a representative example, the "hat" polykite, can form clusters called "metatiles", for which substitution rules can be defined. Because the metatiles admit tilings of the plane, so too does the hat. We then prove that generic members of our continuum of polygons are aperiodic, through a new kind of geometric incommensurability argument. Separately, we give a combinatorial, computer-assisted proof that the hat must form hierarchical -- and hence aperiodic -- tilings.

https://arxiv.org/abs/2303.10798

PDF: 2303.10798.pdf (arxiv.org)

Aperiodic monotile animation

An aperiodic monotile

David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss, 2023

In our paper we present a single shape called "the hat" that tiles the plane aperiodically. That shape turns out to be just one point in a broader continuum of related shapes. In this video we move smoothly through the space of all these shapes. The second half of the video is the reverse of the first; the whole thing is meant to be watched in a loop. The shapes shown at the start (and end), 1/4 position (and 3/4) and 1/2 position are degenerate cases that can also tile periodically. All other shapes are aperiodic.

Full details about the paper are available at https://cs.uwaterloo.ca/~csk/hat/.

Quasicrystals in Medieval Islamic Architecture, Harvard Physics Colloquium Lecture

Why do all of these ancient Islamic Penrose Tilings use more than one tile?

They could have used a single frickin' tile, for Allah's sake!

From the viewpoint of those of us who live in the 21st century, these ancient Islamic architects look like frickin' idiots!

How could anybody be so stupid?

This is mathematical proof that Allah was a frickin' idiot!

(Prove me wrong!)

Mathematicians Discovered a New 13-Sided Shape That Can Do Remarkable Things

https://www.yahoo.com/lifestyle/mathematicians-discovered-13-sided-shape-160000554.html

An aperiodic monotile

David Smith, Joseph Samuel Myers, Craig S. Kaplan, Chaim Goodman-Strauss

A longstanding open problem asks for an aperiodic monotile, also known as an "einstein": a shape that admits tilings of the plane, but never periodic tilings. We answer this problem for topological disk tiles by exhibiting a continuum of combinatorially equivalent aperiodic polygons. We first show that a representative example, the "hat" polykite, can form clusters called "metatiles", for which substitution rules can be defined. Because the metatiles admit tilings of the plane, so too does the hat. We then prove that generic members of our continuum of polygons are aperiodic, through a new kind of geometric incommensurability argument. Separately, we give a combinatorial, computer-assisted proof that the hat must form hierarchical -- and hence aperiodic -- tilings.

https://arxiv.org/abs/2303.10798

PDF: 2303.10798.pdf (arxiv.org)

Aperiodic monotile animation

An aperiodic monotile

David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss, 2023

In our paper we present a single shape called "the hat" that tiles the plane aperiodically. That shape turns out to be just one point in a broader continuum of related shapes. In this video we move smoothly through the space of all these shapes. The second half of the video is the reverse of the first; the whole thing is meant to be watched in a loop. The shapes shown at the start (and end), 1/4 position (and 3/4) and 1/2 position are degenerate cases that can also tile periodically. All other shapes are aperiodic.

Full details about the paper are available at https://cs.uwaterloo.ca/~csk/hat/.

Quasicrystals in Medieval Islamic Architecture, Harvard Physics Colloquium Lecture

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