Expo '67 in Montreal was a wonderland of architecture. One of the most striking structures at this world's fair was the U.S. pavilion, designed by R. Buckminster Fuller. This gigantic geodesic dome looked like a glistening bubble, barely tethered to the ground.
Looking out from inside the pavilion, you see row after row of the hexagonal units that make up the structure's skin. At the same time, you know that the skin cannot consist entirely of regular hexagons. Regular hexagons fit together to cover a flat surface. Where does the curvature come from?
The secret is in the pentagon. Twelve regular pentagons combine to form a regular solid known as a dodecahedron.
You can enlarge this structure and make it more spherical by adding a ring of regular hexagons around each pentagon. The first iteration produces a truncated icosahedron—the traditional pattern on a soccer ball. This structure has twelve pentagonal and twenty hexagonal faces.
Adding more rings of hexagons produces structures even closer to a sphere. Remarkably, each of these structures contains precisely twelve pentagons, and it is these pentagons that force the curvature. Fuller made such configurations the basis for his geodesic domes.
It's easy to spot the pentagons in small geodesic structures.
Finding the pentagons in structures as large as the U.S. pavilion, now the Biosphère, can be very difficult. I was at Expo '67 during the summer that the fair opened, and I visited Montreal several times since. I knew the pentagons were there, and I tried to find them, but for a long time I had very little success in locating even one.
My eye was continually fooled as it tried to make sense of the array of metal struts that define the structure, and all I typically saw was hexagons.
Added Note (7/24/2020)
You have to look closely, but the pentagons are there.
Photos by I. Peterson