It may not be immediately obvious to folks outside of geoscience, but the main way something like this is useful is as a measure/metric to compare things. Looking at the number of faces of fractured pieces isn't normally something we do often in geology.
Sure, the pieces average 6 faces when materials are relatively homogenous and iostropic (i.e. no preferential direction to break in and no free surface nearby). However, as they note in the article, this isn't always the case. Things like mud flats and other cases with very anisotropic materials and/or free surfaces nearby don't fracture with the same average.
This is a good example of a potential metric that could be used to give some clues about overall material behavior even if all you have are the broken remains.
Fractal dimension is also pretty esoteric. However, it's somewhat widely used in geoscience, even though what we're measuring isn't _actually_ fractal. It's still a very useful comparative metric, though, because it lets us measure how complex an interface or surface is quantitatively and scale-independent.
It reminds me of how we use measures like the VIX in finance; not because markets are actually log-normal, but because having a standardized way to compare "choppiness" across different periods is incredibly useful. I like your fractal dimension example too. Even if real coastlines aren't truly fractal, being able to say "this coastline has dimension 1.3 vs 1.7" gives you meaningful information about erosion patterns, wave energy, and rock composition. The cube metric could work similarly for forensic geology.
Plato envisioned Earth’s building blocks as cubes, a shape rarely found in nature. The solar system is littered, however, with distorted polyhedra—shards of rock and ice produced by ubiquitous fragmentation. We apply the theory of convex mosaics to show that the average geometry of natural two-dimensional (2D) fragments, from mud cracks to Earth’s tectonic plates, has two attractors: “Platonic” quadrangles and “Voronoi” hexagons. In three dimensions (3D), the Platonic attractor is dominant: Remarkably, the average shape of natural rock fragments is cuboid. When viewed through the lens of convex mosaics, natural fragments are indeed geometric shadows of Plato’s forms. Simulations show that generic binary breakup drives all mosaics toward the Platonic attractor, explaining the ubiquity of cuboid averages. Deviations from binary fracture produce more exotic patterns that are genetically linked to the formative stress field. We compute the universal pattern generator establishing this link, for 2D and 3D fragmentation.
Voronoi diagrams I see have few if any hexagons (use your favorite mathematical reference or image search). Is that idea that if the points are distributed equidistant in 'alternating' ranks [0], then the diagram is hexagons?
Also, what is "binary breakup" and "binary fracture"?
[0] Alternating ranks: I mean something like the following (is there a better name?):
Group theory and crystallograpy without either word? I suppose I can look at this as an extension of group theory to glassy and partial.domains, but it doesn't appear to offer much more.
Columnar basalt formation has been understood for a long time, I really don't understand what this explained that wasn't already known?
It doesn't even explain it particularly well. The reference to a taco cart was unexpected; it wasn't necessary, but an interesting literary device, I suppose. I just feel a good editor could have made it a better explainer. It's all over the place.
> Years ago, Domokos had won renown by proving the existence of the Gömböc, a curious three-dimensional shape that swivels into an upright resting position no matter how you push it.
all/most materials can be foamed, ALL foam bubbles aproximate a very specific shape where each intersection is at the SAME angle, which is not 90°
so at minimum there are two "universal geometrys", and yes there are natural rock foams.
This is really cool thinking. The fundamental concept I got out of it, was fracturing something means that it can fit together again, so there is a constraint. Of course, but cool. Thanks!
It may not be immediately obvious to folks outside of geoscience, but the main way something like this is useful is as a measure/metric to compare things. Looking at the number of faces of fractured pieces isn't normally something we do often in geology.
Sure, the pieces average 6 faces when materials are relatively homogenous and iostropic (i.e. no preferential direction to break in and no free surface nearby). However, as they note in the article, this isn't always the case. Things like mud flats and other cases with very anisotropic materials and/or free surfaces nearby don't fracture with the same average.
This is a good example of a potential metric that could be used to give some clues about overall material behavior even if all you have are the broken remains.
Fractal dimension is also pretty esoteric. However, it's somewhat widely used in geoscience, even though what we're measuring isn't _actually_ fractal. It's still a very useful comparative metric, though, because it lets us measure how complex an interface or surface is quantitatively and scale-independent.
It reminds me of how we use measures like the VIX in finance; not because markets are actually log-normal, but because having a standardized way to compare "choppiness" across different periods is incredibly useful. I like your fractal dimension example too. Even if real coastlines aren't truly fractal, being able to say "this coastline has dimension 1.3 vs 1.7" gives you meaningful information about erosion patterns, wave energy, and rock composition. The cube metric could work similarly for forensic geology.
nerdy ahh jit needs to sybau
The paper is here:
Gábor Domokos, Douglas J. Jerolmack. Plato’s cube and the natural geometry of fragmentation. PNAS (2020)
https://www.pnas.org/doi/10.1073/pnas.2001037117
Abstract:
Plato envisioned Earth’s building blocks as cubes, a shape rarely found in nature. The solar system is littered, however, with distorted polyhedra—shards of rock and ice produced by ubiquitous fragmentation. We apply the theory of convex mosaics to show that the average geometry of natural two-dimensional (2D) fragments, from mud cracks to Earth’s tectonic plates, has two attractors: “Platonic” quadrangles and “Voronoi” hexagons. In three dimensions (3D), the Platonic attractor is dominant: Remarkably, the average shape of natural rock fragments is cuboid. When viewed through the lens of convex mosaics, natural fragments are indeed geometric shadows of Plato’s forms. Simulations show that generic binary breakup drives all mosaics toward the Platonic attractor, explaining the ubiquity of cuboid averages. Deviations from binary fracture produce more exotic patterns that are genetically linked to the formative stress field. We compute the universal pattern generator establishing this link, for 2D and 3D fragmentation.
Voronoi diagrams I see have few if any hexagons (use your favorite mathematical reference or image search). Is that idea that if the points are distributed equidistant in 'alternating' ranks [0], then the diagram is hexagons?
Also, what is "binary breakup" and "binary fracture"?
[0] Alternating ranks: I mean something like the following (is there a better name?):
The dots need to be the vertices of equilateral triangles for the Voronoi diagram to be hexagons, the above is a rectangular grid rotated 45 degrees.
You can overlay a regular hexagonal tessellation over a regular triangular tessellation to see this.
In context, binary breakup and binary fracture apppear to mean a splitting ofa whole into two parts along a given line or plane
Group theory and crystallograpy without either word? I suppose I can look at this as an extension of group theory to glassy and partial.domains, but it doesn't appear to offer much more.
Columnar basalt formation has been understood for a long time, I really don't understand what this explained that wasn't already known?
It doesn't even explain it particularly well. The reference to a taco cart was unexpected; it wasn't necessary, but an interesting literary device, I suppose. I just feel a good editor could have made it a better explainer. It's all over the place.
Why do distinct systems end up having the same look?
I am often surprised to discover different systems that arrive at the same shapes.
You will find it in nature, but also within.
N/A
Organizational charts.
Patterns of traffic.
Ways to Fail.
Despite different inputs and histories they all yield the same outcome.
I use a mental model of sorts.
Local regulations and limits.
Consistent application of weight.
In the end, only a handful of stable shapes endure.
I'm curious now.
Are we overestimating the uniqueness of our systems?
What design patterns to use in your product?
Have you taken note of shapes showing up in different domains time and again? ~
> Years ago, Domokos had won renown by proving the existence of the Gömböc, a curious three-dimensional shape that swivels into an upright resting position no matter how you push it.
Some researchers are just incredible achievers.
all/most materials can be foamed, ALL foam bubbles aproximate a very specific shape where each intersection is at the SAME angle, which is not 90° so at minimum there are two "universal geometrys", and yes there are natural rock foams.
This is really cool thinking. The fundamental concept I got out of it, was fracturing something means that it can fit together again, so there is a constraint. Of course, but cool. Thanks!
Was this from a second chance pool ? If so I am very happy.
yo bean headed ahhh needs to release a site like this but its really geometry dash NOW or else...
We could be in a Minecraft indeed ^^
(2020) Title: Scientists Uncover the Universal Geometry of Geology
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