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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.

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 8y ago
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 8y ago
  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.

kjeetgill 8y ago

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 8y ago
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.