As I’ve mentioned on more than one occasion, a polyhedron made of regular pentagons and hexagons always has exactly twelve pentagons. If it had none, it would be completely flat and would cover the plane.

The theorem is actually more general than that: any polyhedron made of pentagons and hexagons, in which exactly three faces met at each vertex, must contain exactly twelve pentagons.

The dual of this theorem is that in any polyhedron made of triangles in which each vertex is either 5- or 6-valent, there are exactly 12 5-valent vertices.

In this picture we see Matt Parker the Stand-up Mathematician pointing out one of the 12 such vertices in a large deltahedron.

Thanks to Robin Houston for the tip.

Many fruits have pentagonal cross-sections. Shown above is the most famous example, the *carambola* or “starfruit”.

Okra, botanically a fruit, is famous for its pentagonal shape when sliced, although some large okra fruit have six or even seven sides.

At right is *babaco*, a hybrid variety of the mountain papaya:

But it’s not only exotic tropical fruits that contain a fivefold symmetry. Tomorrow’s post will feature a much more familiar pentagonal fruit.

Fort McHenry, in Baltimore, is a five-pointed example of a star fort. With the widespread adoption of cannon, the high vertical walls of the typical medieval castle became a liability, an unmissable artillery target that cannonballs would soon reduce to rubble. The design that evolved to replace these had lower, sloping walls that could absorb the effect of a cannonball, and protruding points, called *bastions*, from which defenders armed with rifles could defend the walls from attacking soldiers. Fort McHenry was built between 1794 and 1803.

Here’s another example, Fort Bourtange in the Netherlands:

Not all star forts were five-pointed. Earlier examples often had four points. As the design evolved, it became increasingly elaborate, with multiple levels of embankments and six, seven, nine, or even more points.

This pentagonal furniture, designed by Thomas Tritsch, is modular. The tabletops and cushions are all interchangeable.

Tritsch also makes hexagonal bases that match the pentagonal ones. I think I see a hexagonal interloper in the upper picture. The collection is manufactured by Quinze & Milan.

Sometimes when planning posts for this blog, I get silly ideas. This time it was “pentagonal mittens”.

“Shoot,” I said, “that doesn’t even make any sense. Of course there are no pentagonal mittens.”

But I did the search anyway.

Crocheted by Fluxx.

The 120-cell is the four-dimensional analogue of the dodecahedron. It is a polytope, which is the four-dimensional analogue of a polyhedron, as a polyhedron is the three-dimensional analogue of a polygon. A polygon has one-dimensional edges and zero-dimensional vertices, and always the same number of each. For example, a pentagon has 5 vertices and 5 edges. A polyhedron has, in addition to edges and vertices, two-dimensional faces. For example, a dodecahedron has 20 vertices, 30 edges, and 12 pentagonal faces.

In addition to vertices, faces, and edges, a polytope has three-dimensional cells, and the polytope known as the 120-cell, depicted above, has 120 cells, each of which is a regular dodecahedron. 120 dodecahedra have 12·120 = 1440 pentagons between them, but because each pentagonal face is shared between two of the dodecahedra, the 120-cell has only 1440/2 = 720 pentagonal faces. Similarly, the 120-cell has 1200 edges and 600 vertices.

Just as a polyhedron can’t be faithfully depicted on a two-dimensional sheet of paper, a polytope can’t be faithfully depicted in our three-dimensional space. The dodecahedron at right has been “squashed” into two dimensions; it’s something like what you would see if you made a wireframe dodecahedron and projected its shadow onto a piece of paper.

The sculpture above, designed by Marc Pelletier, a geometer, depicts an analogous projection of the 120-cell into a three-dimensional space. Pelletier is one of the designers of the geometry toy Zometool. The sculpture is five feet across and made of stainless steel. It hangs in the lobby of the Fields Insitute for Research in Mathematical Sciences in Toronto.

Source: The Mathematical Tourist.

My search for pentagonal hats also turned up this item, which isn’t exactly a hat, but Ms. Van Den is wearing it on her head, and I’ll take what I can get.

Jess Van Den of Epheriell Designs was crocheting a granny square, which normally has a *D*_{4} symmetry, as its name implies. It begins with a center that has four groups of connected stitches. But she slipped up and began with five groups instead.

**Addendum**: Ms. Van Den did eventually turn the pentagonal object into a hat. Picture here.

In yesterday’s post, I discussed the pentagonal hexecontahedron. This spectacular lamp, designed by Tom Dixon, is a copper hexecontahedron.

Check out Dixon’s web site to see the beautiful shadows it makes.