Really cool to see the convex polyhedra honeycomb's in the paper! (sci hub link in this thread) -- for any fans of the platonic solids check out the wiki on these [0] -- cubes are the only platonic solid (1 of 5) that tesselate 3D space but if you loosen the constraints to convex polyhedra (any polyhedra where every vertex lies on the surface of a sphere) you get a huge diversity of 'packing' patterns with interesting symmetries -- if you take cross-sections of any of these honeycombs you'll find 2D tesselations, many of which correspond to mosaic patterns used in indo-islamic architecture -- I find it fascinating that a thousand years later we started to find the molecular structure of crystals lined up with these old tessellations which explored all the available symmetry groups. Atoms can only arrange themselves in so many ways if they want to be free of any entropy-induced gaps in their structure.

These toys models are all well and good, but until they have a real theory of the Disc that includes detailed modeling of the gold-dense counter weight, elephants, and turtle then I just can't take this work seriously [0]

[0] RIP Terry Pratchett... I can't see anything like this without thinking of him, and missing his regular biting criticism yet still somehow optimistic novels.

> Unaware of this work, Domokos wrote a proof which pointed to cubes as the answer. He wanted to double-check, though, and he suspected that if an answer to the same problem already existed, it would be locked in an inscrutable volume by the German mathematicians Wolfgang Weil and Rolf Schneider, an 80-year-old titan in the field of geometry. Domokos is a professional mathematician, but even he found the text daunting.

> “I found someone who was willing to read that part of the book for me and translate it back into human language,” Domokos said. He found the theorem for any number of dimensions. That confirmed that cubes were indeed the 3D answer.

I'd like to know who this someone is, that's an unsung hero of science right there! So much existing knowledge is locked away in "daunting tomes"

The last bit about how none of the sampled rocks was actually a cube reminded about this article (https://www.thestar.com/news/insight/2016/01/16/when-us-air-...) on how when you design for averages you're actually designing for no one. They tried to make a scruple of the "average woman" based on average measurements, but no woman they found actually had the average measurements

In Minetest, Terrain is generated using Perlin noise (or Simplex noise?). Voronoy noise is used to put a landscape on top of it. The result is quite convincing. even if you use a bit crazy values for noise parameters, you can get arches and "floating islands" that remind of fantasy landscapes.

Any sufficiently complex process looks like randomness (that's what we colloquially call "luck"). Therefore randomness (PRNG, coherent noises) can look like a complex process with enough trickery.

I'm going to cite this next time I try to talk someone into using dual contour voxels :).

In addition to being a really cool result, this is a really approachable article with a lot of interesting ideas in the background information. I expect pop math education youtube is gonna have a field day.

Basically you're right for crystals -- the planes of fragmentation will be self-similar to the molecular structure. But most of the crust is not diamond and beryl, so this study is looking at the average structure of much larger conglomerates of stone and ice and mud -- not as much long-range symmetry there!

Quoting the paper to give an idea of the formation being modeled:

These patterns have been reproduced in experiments of mud and corn-starch cracks, model 2D fragmentation systems, where the following have been observed: Fast drying produces strong tension that drives the formation of primary (global) cracks that criss-cross the sample and make “X” junctions; slow drying allows the formation of secondary cracks that terminate at “T” junctions; and “T” junctions rearrange into “Y” junctions to either maximize energy release as cracks penetrate the bulk or during reopening–healing cycles from wetting/drying

I studied a metallurgy subject that focussed on crystals as a bit of fillin in my electrical engineering degree some 35 years ago. The thing is that large scale macro features like crystalline surfaces and grains are driven by atomic structure that was first deduced through x-ray imaging and then like.

How do they justify such claims of an abstract universal concept given most scientists hold the naive empiricist worldview in which only particulars exist?

So by the time you reached that sentence in the article you must have gone over the part where this is a result co-authored by a geophysicist, and that the paper has gone through peer review of at least three other geophysicists (whom all have been contacted to hear their opinion on the work, and they're very positive about it), and that the geology community in general seems excited to see what practical uses this model will have in practice.

But sure, the fact that this is a model that starts with making the least amount of assumptions[0] completely invalidates it as geology and makes it pure maths. Because as we all know, no useful model in physics was ever built up from the ground like that, only introducing complex caveats as required to make sense of empirical data as necessary.

[0] the simplest way that one can cut a shape is with a straight line (2D) or flat plane (3D). In this case no new concave shapes can be created unless a concave angle was a part of the initial shape. Ergo, if the starting shape is convex and we cut away parts randomly, then all shapes produced this way will be convex. If the initial shape has concave parts, then on average most shapes derived from it will be convex. This is even more true if we apply this "cutting away" recursively to the shapes that were cut away. This makes the assumption of shapes being (mostly) convex quite reasonable, unless there is a specific reason to expect concavity, like specific crystalline structures perhaps. Also, for the record, I just reasoned this out on the spot, it's not exactly difficult.

I appreciate the time you've taken to reply to my brief comment, though you admit it took little effort, assuming, as a sibling post here states, "spherical cows in a vacuum", or rather your own equivalents for the sake of reasoning. While I'll spare myself listing the several assumptions your post inserts on it's path to your conclusions, I would like to compliment you on your obvious passion for the subject.

I actually appreciated the explanation. Less so the heavy sarcasm. I'm glad it's boring, obvious cubes and not starfish or irreducible complexity, but not in the least surprised, and I still think atoms are concave.

I think what you're failing to see is that a sentence like "math; not geology" sounds incredibly condescending and snarky.

I don't know if that was your intent, but if not it might be worth reflecting on what you meant to say instead, and why you did not notice that what you wrote instead comes across so negative. Because then there would be no need for heavy sarcasm.

How much better would your experience of life on Earth be if HN were to stop posting articles with clickbait titles from Quanta magazine? Can you quantify the improvement?

## 33 Comments:

Really cool to see the convex polyhedra honeycomb's in the paper! (sci hub link in this thread) -- for any fans of the platonic solids check out the wiki on these [0] -- cubes are the only platonic solid (1 of 5) that tesselate 3D space but if you loosen the constraints to convex polyhedra (any polyhedra where every vertex lies on the surface of a sphere) you get a huge diversity of 'packing' patterns with interesting symmetries -- if you take cross-sections of any of these honeycombs you'll find 2D tesselations, many of which correspond to mosaic patterns used in indo-islamic architecture -- I find it fascinating that a thousand years later we started to find the molecular structure of crystals lined up with these old tessellations which explored all the available symmetry groups. Atoms can only arrange themselves in so many ways if they want to be free of any entropy-induced gaps in their structure.

[0] https://en.wikipedia.org/wiki/Convex_uniform_honeycomb

These toys models are all well and good, but until they have a real theory of the Disc that includes detailed modeling of the gold-dense counter weight, elephants, and turtle then I just can't take this work seriously [0]

[0] RIP Terry Pratchett... I can't see anything like this without thinking of him, and missing his regular biting criticism yet still somehow optimistic novels.

Sorry to disappoint but I think you'll find it's turtles all the way down!

This work came out a few months ago, and we discussed coverage in Science:

https://news.ycombinator.com/item?id=23984568

I'm not sure if the Quanta article adds any technical content, but it's certainly a nice read.

>

Unaware of this work, Domokos wrote a proof which pointed to cubes as the answer. He wanted to double-check, though, and he suspected that if an answer to the same problem already existed, it would be locked in an inscrutable volume by the German mathematicians Wolfgang Weil and Rolf Schneider, an 80-year-old titan in the field of geometry. Domokos is a professional mathematician, but even he found the text daunting.>

“I found someone who was willing to read that part of the book for me and translate it back into human language,” Domokos said. He found the theorem for any number of dimensions. That confirmed that cubes were indeed the 3D answer.I'd like to know who this someone is, that's an unsung hero of science right there! So much existing knowledge is locked away in "daunting tomes"

The humility of the mathematician (or at least, his portrayal as such) made this a fantastic read.

The last bit about how none of the sampled rocks was actually a cube reminded about this article (https://www.thestar.com/news/insight/2016/01/16/when-us-air-...) on how when you design for averages you're actually designing for no one. They tried to make a scruple of the "average woman" based on average measurements, but no woman they found actually had the average measurements

So... Minecraft is an accurate simulation of the real world?

Also Dwarf Fortress ;-)

In Minetest, Terrain is generated using Perlin noise (or Simplex noise?). Voronoy noise is used to put a landscape on top of it. The result is quite convincing. even if you use a bit crazy values for noise parameters, you can get arches and "floating islands" that remind of fantasy landscapes.

Any sufficiently complex process looks like randomness (that's what we colloquially call "luck"). Therefore randomness (PRNG, coherent noises) can look like a complex process with enough trickery.

Right mean (!!!!!), but not the right standard deviation.

I'm going to cite this next time I try to talk someone into using dual contour voxels :).

In addition to being a really cool result, this is a really approachable article with a lot of interesting ideas in the background information. I expect pop math education youtube is gonna have a field day.

Full paper: https://sci-hub.se/https://www.pnas.org/content/117/31/18178

Fascinating,

Would this give any insight into what type of cement/stone laying tile pattern should be use to mitigate future cracks?

I propose sand.

Sand with glue? But that will crack into cubes:)

Shouldn't the atomic crystalline pattern also play a role?

Basically you're right for crystals -- the planes of fragmentation will be self-similar to the molecular structure. But most of the crust is not diamond and beryl, so this study is looking at the average structure of much larger conglomerates of stone and ice and mud -- not as much long-range symmetry there!

Quoting the paper to give an idea of the formation being modeled:

These patterns have been reproduced in experiments of mud and corn-starch cracks, model 2D fragmentation systems, where the following have been observed: Fast drying produces strong tension that drives the formation of primary (global) cracks that criss-cross the sample and make “X” junctions; slow drying allows the formation of secondary cracks that terminate at “T” junctions; and “T” junctions rearrange into “Y” junctions to either maximize energy release as cracks penetrate the bulk or during reopening–healing cycles from wetting/drying

I studied a metallurgy subject that focussed on crystals as a bit of fillin in my electrical engineering degree some 35 years ago. The thing is that large scale macro features like crystalline surfaces and grains are driven by atomic structure that was first deduced through x-ray imaging and then like.

Would love to see this turned around into a generative model.

FTA:

Domokos argued, any rocks that broke randomly would crack into shapes that have, on average, six faces and eight vertices.“How in the hell does nature let this happen?”MFW we all live in Minecraft

How do they justify such claims of an abstract universal concept given most scientists hold the naive empiricist worldview in which only particulars exist?

> How do they justify such claims

Read the article

> given most scientists hold the naive empiricist worldview in which only particulars exist

What do you base your beliefs on? Intuition?

"...further suppose these shapes are all convex, with no indentations."

Math; not geology.

So by the time you reached that sentence in the article you must have gone over the part where this is a result co-authored by a geophysicist, and that the paper has gone through peer review of at least three other geophysicists (whom all have been contacted to hear their opinion on the work, and they're very positive about it), and that the geology community in general seems excited to see what practical uses this model will have in practice.

But sure, the fact that this is a model that starts with making the least amount of assumptions[0] completely invalidates it as geology and makes it pure maths. Because as we all know, no useful model in physics was ever built up from the ground like that, only introducing complex caveats as required to make sense of empirical data as necessary.

[0] the simplest way that one can cut a shape is with a straight line (2D) or flat plane (3D). In this case no new concave shapes can be created unless a concave angle was a part of the initial shape. Ergo, if the starting shape is convex and we cut away parts randomly, then all shapes produced this way will be convex. If the initial shape has concave parts, then

on averagemost shapes derived from it will be convex. This is even more true if we apply this "cutting away" recursively to the shapes that were cut away. This makes the assumption of shapes being (mostly) convex quite reasonable, unless there is aspecific reason to expect concavity, like specific crystalline structures perhaps. Also, for the record, I just reasoned this out on the spot, it's not exactly difficult.I appreciate the time you've taken to reply to my brief comment, though you admit it took little effort, assuming, as a sibling post here states, "spherical cows in a vacuum", or rather your own equivalents for the sake of reasoning. While I'll spare myself listing the several assumptions your post inserts on it's path to your conclusions, I would like to compliment you on your obvious passion for the subject.

I actually appreciated the explanation. Less so the heavy sarcasm. I'm glad it's boring, obvious cubes and not starfish or irreducible complexity, but not in the least surprised, and I still think atoms are concave.

I think what you're failing to see is that a sentence like

"math; not geology"sounds incredibly condescending and snarky.I don't know if that was your intent, but if not it might be worth reflecting on what you meant to say instead, and why you did not notice that what you wrote instead comes across so negative. Because then there would be no need for heavy sarcasm.

Spherical cows in a vaccuum

Please stop posting articles with clickbait titles from Quanta magazine.

How much better would your experience of life on Earth be if HN were to stop posting articles with clickbait titles from Quanta magazine? Can you quantify the improvement?