Shape #7: Cupola
Cupolas are the shape of the week
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.