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Aztec Diamond

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70 pointsluu19 comments

There's a beautiful video by Mathologer about this subject, giving some intuition on why the outside areas are "frozen"

https://www.youtube.com/watch?v=Yy7Q8IWNfHM

I'd love to read a few paragraphs on the subject, but a 50 minute video to gain some insight into math trivia is out of the question.

Plenty of reading in their video description.

The intuition is one can construct larger tilings from smaller ones using a weird trick in a way that prefers frozen corners.

I wrote some open source code a few years ago that introduces a simple way to convert LU factorizations into a Determinantal Point Process sampler and used large Aztec diamonds as the example

https://gitlab.com/hodge_star/catamari

If people are interested in playing with those, I made a website a while ago where you can generate Aztec Diamond (and other shapes): https://sites.uclouvain.be/aztecdiamond/

Why is it called Aztec though?

Aztec pyramids have a characteristic stair-stepped design, which when viewed from atop probably looks reminiscent of a tiling. A similar story probably applies to Aztec Codes

https://en.wikipedia.org/wiki/Aztec_Code

That's true of all or most Mesoamerican pyramids as far as I know, not just Aztec ones. (Mayan, Teotihuacano, etc.)

I suppose the Aztecs are the most famous Mesoamerican group in the popular imagination, though.

> I suppose the Aztecs are the most famous Mesoamerican group in the popular imagination, though.

Maybe, but they're not famous for pyramids. I would always say Maya if the prompt was American pyramids.

There is a map in counterstrike named Aztec with a stepped pyramid. I always think of mortarless marvels like macchu Picchu when I think of Mayan architecture. - just an anecdote

But Machu Picchu is an Inca city. Here's a Maya city:

https://en.wikipedia.org/wiki/Chichen_Itza

This is true, and thanks for reminding me about chichen itza, but I just wanted to emphasize how blurry the cultural lines are to the people around me - at least where I am in the Appalachias. My city has Mayans with Mayan restaurants. Not too long ago though, a kid told me he had thought the Mayans all died out, like the Romans he said.

Sorry, this is all kinda tangential. That kind of blurriness is something I'm super interested in.

Yeah, it seems to be a relatively common (false) belief that the Mayans died out or even mysteriously disappeared. In reality their “classical” civilization declined and was eventually completely conquered by Spain, but as you point out, the people are still around. There are still millions of ethnic Mayans, who still speak Mayan languages, primarily in Guatemala and to a lesser extent in Mexico.

The Incas were separated from the Mayans by thousands of miles of jungle (and their civilization reached its height hundreds of years after the Mayan decline). It’s unlikely they had much direct contact, if any.

For what it’s worth the Romans didn’t die out either, again their civilization declined but their cultural and biological descendants are alive and well in the form of Italians, Spaniards, etc.

So what you're saying is the Mayans were conquered by the Romans?

“Not exactly, no…”

TIL the Mayans were actually conquered by the Romans!

“‹gives up›”

I disagree, the Pyramid of the Sun is the third largest pyramid after two at Giza, is near Mexico City, and has been well known for longer than the impressive Maya pyramids, which were buried in jungle and remain fairly far from population centers.

The Pyramid of the Sun and the other pyramids of Teotihuacan actually predate the Aztecs by over a thousand years - https://en.wikipedia.org/wiki/Teotihuacan

That's true. It's still the case that the Aztec are famous for it, and for other pyramids, most notably the Templo Mayor, which was destroyed (and good riddance) to build Mexico City's cathedral.

"Acshually, the Pyramid of the Sun is pre-Aztec!" is a bit like insisting the Coliseum isn't Italian. It's not that it's wrong, just autistic and annoying.

FWIW, I agree with the other poster in that I don’t associate Teotihuacán with the Aztecs, and I think your rudeness is misplaced.

The analogy to the Coliseum is imprecise, because the Italians are the descendants of the ancient Romans, rather than a new group who migrated in later. A better analogy would be saying that the Hagia Sophia isn’t Turkish, and indeed I’d argue that it isn’t.

My contribution: https://halcanary.org/mathapplets/toadshuffle/#64

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