
We Are All Scutoids: A Brand-New Shape, Explained - Tomte
https://www.newyorker.com/elements/lab-notes/we-are-all-scutoids-a-brand-new-shape-explained
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empath75
[https://youtu.be/2_NZ1ql8B8Y](https://youtu.be/2_NZ1ql8B8Y) A nice video that
explains it.

Basically you have a pentagon and hexagon parallel with each other, and
connect 4 of the corners straight down and the fifth, you split halfway down
to join up with the last two corners of the hexagon.

It’s a prismatoid that they just gave a new name too.

~~~
mirimir
Thanks. Your sentence is better than anything in the article :)

And, as a former biologist, I gotta say, "huh?". I mean, epithelial cells
aren't ideal geometric solids. They're messy and fluid. So faces can be
pentagonal, hexagonal or whatever.

~~~
mojomark
Concur, this comment explained the shape quite simply. From what I see, this
is a simply a special case octohedron.

Nevertheless, I've been working on an optimal 3D solid packing problem for a
hydrodynamic application, and I'm totally going to try this cell shape out in
my model to see if I can find any improvements.

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romwell
Interesting article, however this line is obnoxious:

>What matters is that mathematicians had never before conceived of the
scutoid, much less given it a name.

Of course this shape has been conceived of before; it's just an example of a
_polyhedron_. Specifically, scutoids seem to be nothing more than truncated
prisms[1]! But "truncated prism" doesn't have the same ring to it, does it?

It's like saying that mathematicians didn't "discover" a pair of pants[2]
before topology was invented.

What happened is that the researches realized that one can model the shapes of
the cells with just _a few_ polyhedral classes - and one of the classes didn't
have a special name yet because we didn't know what's interesting or special
about it. Chop some vertices off a prism, so what? The special part comes from
the optimization constraints inspired by biology.

[1][https://en.wikipedia.org/wiki/Truncation_(geometry)](https://en.wikipedia.org/wiki/Truncation_\(geometry\))

[2][https://en.wikipedia.org/wiki/Pair_of_pants_(mathematics)](https://en.wikipedia.org/wiki/Pair_of_pants_\(mathematics\))

~~~
kjeetgill
I had a similar eyeroll but I was wrong.

It's not a polyhedron.

Polygedrons have planar faces. The scutoids here have curved faces on one of
"pentagonal" the "walls". The 5 points for that wall aren't nessecarily
coplanar.

An interesting property of that face is that it compliments another scutoid.
See either YouTube link elsewhere.

From Wikipedia:

Scutoids are not necessarily convex, and lateral faces are not necessarily
planar, so several scutoids can pack together to fill all the space between
the two parallel surfaces.

~~~
romwell
Sure, make it "up to diffeomorphism". The shapes you get fit in a larger class
of piecewise-smooth closed surfaces, but all the interesting information is
encoded in the polyhedron.

Note that the other shapes they're dealing with - e.g. prisms - also don't
have planar faces (but are defined as having such on Wikipedia).

The point is that mathematical definitions are quite malleable, and depend on
the context.

"Lop a vertex off a prism and deform a bit" is hardly a radical notion _in
itself_.

As for combining these shapes - the term for that is _tiling_ or
_tessellation_. You can tesselate the space with all kinds of shapes. The
"faces match" is a property of objects a tessellation, not of any given object
in itself.

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dang
Related recent discussion:
[https://news.ycombinator.com/item?id=17632720](https://news.ycombinator.com/item?id=17632720)

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kjeetgill
> Buceta was keen to send me a diagram of a scutoid, to explain what one looks
> like, but I pressed him for a verbal description.

-_-

It's an interesting article and a good story! I like how the math informed the
later biological investigations but this was just trying to avoid diagrams to
add mystique instead of clarity.

