This is a **cupola**:

It was defined by mathematician Norman W. Johnson in his 1966 paper titled “Convex Polyhedra with Regular Faces” in a list that attempted to catalog *all *the 3D (convex) solids whose faces are regular polygons. (There are 92.) In the middle of all this, he drops the cupola.

Here are some of its properties:

It has a top and a bottom base, and those bases are parallel.

The bottom base has twice as many sides as the top.

Each edge of the top base is attached to a square (or rectangle). In between each of those squares is a triangle.

Those triangles are always isosceles, sometimes equaliteral.

Those bases happen to be regular, but they don’t have to be.

Here are some questions I have about the cupola.

**Is a cube a cupola?**

In class I casually defined it as the thing you get when you attach squares to the sides of a polygon and then fill in the gaps between those squares with triangles. This immediately got me into trouble, as a kid asked if a **cube **is a cupola. And…I guess it is?

Then she asked what if you start with a cube and then open up one of those sides and make it longer…so the sides are 3 squares and a long rectangle. Is *that *a cupola?

~~But the weird thing is that in this case the bottom base would only have 6 sides, so it would violate the “always has twice as many sides as the top base” rule.~~ I don’t know???

(UPDATE: A kid in class explained how I misunderstood things and messed this up. That bottom base if it extends all the way down would not be a hexagon.)

**Is a tent a cupola? **

This is a triangular prism, but wikipedia flips it on its side and calls it a **digonal cupola**.

There are some problems here. That top base isn’t really a polygon. It’s a “digon,” i.e. what normally functioning adults would typically call a “line.” Also, does it preserve the “bottom base is twice the sides of the top base” rule? No it does not — 4 is not twice 1. THOUGH it does work if we count vertices, since 4 is twice 2.

Do we count this tent (triangular prism) as a cupola? Then how do we define cupola?

**When can you make a unit cupola?**

I made a little assignment for kids with cupolas, but then I had a moment of panic. Is the shape I defined—a triangular cupola whose sides are all 1—actually possible?

I quickly decided that it was. But then what about unit square cupolas?

We have this great toy Geofix at school and so I used it to start building. The square unit cupola was really easy to make. I had *some *trouble getting the pieces to fit with a pentagon on top. But then I was very much sure that the hexagon would *not *work.

A colleague walked in and identified the problem right away. She smooshed the hexagonal cupola down flat—this is a known tessellation pattern. The hexagonal unit cupola only exists when its height is equal to 0.

But can I prove that it won't work for anything above the regular hexagon? I think I probably can, but I’m not sure exactly how.1

### Who decided to define the rotunda?

I mean, it was Norman Johnson. OK, cupolas, cute enough. But Johnson got around to the shape you make when you swap out those squares for pentagons and he called them—**rotundas**?

That’s too cute, I don’t like it. He should have called them **pentagonalcupolaloids **or something dumb and not-cute like that.

But, anyway, he did, and while rotundas are most definitely *not *interesting enough to be the shape of the week, cupolas are. So, congrats cupolas. Long may you reign! For a week!

I think it could pop out of the height formula given on the wikipedia page. Also a Johnson solid is a 3D shape where each face is a regular polygon, and wikipedia says that only the triangular, square, and pentagonal cupolas are Johnson solids. So…I guess that’s another way to talk about all this.