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| from Gómez-Gálvez et. al |
If you’ve seen the recent headlines of “Scientists Discover NEW SHAPE!,” then you’ve been touched by bad science reporting.
If you skimmed an article or two and came away thinking, so what the f*ck is a SCUTOID, or, why the f*ck is a scutoid, it’s also not really your fault. Though the new-not-new scutoid is a fairly simple 3D shape, explaining why it exists and what it does involves referencing cellular biology, tiling, geometric projection, and Voronoi diagrams; in passing, entomology; and optionally, Steiner trees.
Here we go.
The scutoid (‘skew-toyed’) is a naturally-occuring shape that cells commonly assume when in a curved layer. It is ubiqitous in nature and found in every multicellular organism. It is definitely NOT new. (It is, however, newly described and named, by Gómez-Gálvez et. al in their recent research article in Nature Communications.)
Cells in a flat layer are often hexagonal, as in a honeycomb. This is because the hexagon is the most efficient tilable flat shape, enclosing the most area using the least line length.
However, depending on their sizes and spacing, cells can take other shapes as well. One way to predict and model this is with Voronoi diagrams: a Voronoi diagram (or map) divides a space into different regions, each containing a central point (or seed). Each region consists of all points that are closer to its central seed than to any other seed. A simple example:
Everywhere inside the purple region is closer to its seed than to the other seeds.
If you use cell nuclei as seeds, a Voronoi diagram can predict the approximate shapes of cells:
Try it!—Seed your own Voronoi diagram here: http://alexbeutel.com/webgl/voronoi.html
Also see Jason Davies’ excellent United States of Voronoi (http://www.jasondavies.com/maps/voronoi/us-capitals/)
Now, cells in a flat layer will generally have the same shape on top and bottom surfaces, thus having the shape of a prism:
However, if the layer is curved into a cylinder—to form a blood vessel, for example—things get complicated. Because the outer surface of the cylinder has a greater surface area than the inner, the outer surface of the cells are stretched—though only in certain directions. Note here that the red line becomes longer on the outer surface, though the green line stays the same:
Because of this, an arrangement of four cells on the front of the above cylinder would have their centers change from inner to outer surfaces like this, with two centers shifting and two remaining:
If we generate Voronoi diagrams of the before and after, we get:
Notice what happens: the Voronoi diagrams change, and regions that were hexagonal become pentagonal, and vice versa. In other words, we can predict the shape of these cells would be a prism-like form with a hexagon at one end and pentagon on the other. And this shape is the SCUTOID!
Because one surface has 5 corners, and the other, 6, one of the lines connecting the former to latter must bifurcate, creating the distinctive Y-shape and resulting triangle. To the researchers, this arrangement around the triangle brought to mind a beetle carapace:
In which the center triangle is called the SCUTUM…and hence: SCUTOID.
So, in other words, the scutoid is a shape which cells adopt when a layer of them is bent into a cylinder. To visualize this: if you take two scutoids with triangles on their upper halves, and connect them triangle to triangle, it would make an inverted V-shape, with hexagons on top. This is the blue and green pair in the above Voronoi diagrams. Repeat with another pair with lower-half triangles, and you get a V-shape with pentagons on top—the red and aqua pair. Then slip the first pair down on top of the second, and you would get the completed 3D structure of the 4 cells, tightly packed.
Given the additional shapes that can be produced with Voronoi diagrams, we should also expect variations on the hexagon + pentagon bauplan, with heptagons and other surfaces included.
Now, a bonus: there is something in mathematics called the 4-house problem. Given 4 houses in a square, what is the minimum length of connections required to connect all 4 in a network? Here are three solutions:
The third, proven to be the shortest possible, is called a minimum Steiner tree. Not coincidentally, the Voronoi diagrams of our 4 cells contain central Steiner trees, in rotated orientations to each:
In their research article, Gómez-Gálvez et. al describe observing exactly these Steiner tree shapes on opposite surfaces of cell layers, in the salivary gland epithelial folds of Drosophila embryos, as well as in Zebrafish embryos:
To read more about Steiner trees and minimum connectionist networks, see the fabulous article “Steiner Trees on a Checkerboard” by Chung, Gardner, and Graham.
Accolades and congratulations to Gómez-Gálvez et. al for bringing this intriguing new-not-new shape to light!
If you skimmed an article or two and came away thinking, so what the f*ck is a SCUTOID, or, why the f*ck is a scutoid, it’s also not really your fault. Though the new-not-new scutoid is a fairly simple 3D shape, explaining why it exists and what it does involves referencing cellular biology, tiling, geometric projection, and Voronoi diagrams; in passing, entomology; and optionally, Steiner trees.
Here we go.
The scutoid (‘skew-toyed’) is a naturally-occuring shape that cells commonly assume when in a curved layer. It is ubiqitous in nature and found in every multicellular organism. It is definitely NOT new. (It is, however, newly described and named, by Gómez-Gálvez et. al in their recent research article in Nature Communications.)
Cells in a flat layer are often hexagonal, as in a honeycomb. This is because the hexagon is the most efficient tilable flat shape, enclosing the most area using the least line length.
Everywhere inside the purple region is closer to its seed than to the other seeds.
If you use cell nuclei as seeds, a Voronoi diagram can predict the approximate shapes of cells:
Try it!—Seed your own Voronoi diagram here: http://alexbeutel.com/webgl/voronoi.html
Also see Jason Davies’ excellent United States of Voronoi (http://www.jasondavies.com/maps/voronoi/us-capitals/)
Now, cells in a flat layer will generally have the same shape on top and bottom surfaces, thus having the shape of a prism:
 |
| from Gómez-Gálvez et. al |
However, if the layer is curved into a cylinder—to form a blood vessel, for example—things get complicated. Because the outer surface of the cylinder has a greater surface area than the inner, the outer surface of the cells are stretched—though only in certain directions. Note here that the red line becomes longer on the outer surface, though the green line stays the same:
Because of this, an arrangement of four cells on the front of the above cylinder would have their centers change from inner to outer surfaces like this, with two centers shifting and two remaining:
If we generate Voronoi diagrams of the before and after, we get:
Notice what happens: the Voronoi diagrams change, and regions that were hexagonal become pentagonal, and vice versa. In other words, we can predict the shape of these cells would be a prism-like form with a hexagon at one end and pentagon on the other. And this shape is the SCUTOID!
 |
| from Gómez-Gálvez et. al |
Because one surface has 5 corners, and the other, 6, one of the lines connecting the former to latter must bifurcate, creating the distinctive Y-shape and resulting triangle. To the researchers, this arrangement around the triangle brought to mind a beetle carapace:
 |
| from Gómez-Gálvez et. al |
In which the center triangle is called the SCUTUM…and hence: SCUTOID.
So, in other words, the scutoid is a shape which cells adopt when a layer of them is bent into a cylinder. To visualize this: if you take two scutoids with triangles on their upper halves, and connect them triangle to triangle, it would make an inverted V-shape, with hexagons on top. This is the blue and green pair in the above Voronoi diagrams. Repeat with another pair with lower-half triangles, and you get a V-shape with pentagons on top—the red and aqua pair. Then slip the first pair down on top of the second, and you would get the completed 3D structure of the 4 cells, tightly packed.
Given the additional shapes that can be produced with Voronoi diagrams, we should also expect variations on the hexagon + pentagon bauplan, with heptagons and other surfaces included.
Now, a bonus: there is something in mathematics called the 4-house problem. Given 4 houses in a square, what is the minimum length of connections required to connect all 4 in a network? Here are three solutions:
 |
| from Chung-Gardner-Graham |
The third, proven to be the shortest possible, is called a minimum Steiner tree. Not coincidentally, the Voronoi diagrams of our 4 cells contain central Steiner trees, in rotated orientations to each:
In their research article, Gómez-Gálvez et. al describe observing exactly these Steiner tree shapes on opposite surfaces of cell layers, in the salivary gland epithelial folds of Drosophila embryos, as well as in Zebrafish embryos:
To read more about Steiner trees and minimum connectionist networks, see the fabulous article “Steiner Trees on a Checkerboard” by Chung, Gardner, and Graham.
Accolades and congratulations to Gómez-Gálvez et. al for bringing this intriguing new-not-new shape to light!
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