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Comment by jpfed | Hacker News Reader

It was obviously designed to be a cunning way of limiting the expansion of the military-industrial complex; they can't just keep building adjacent pentagons because they don't tesselate.

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I had always assumed it was the single exposed face of a mostly subterranean dodecahedron.

That is correct, but the subterranean dodecahedron itself is merely the portal that the interdimensional lizard beings use to move between our world and the 4-dimensional hyperdodecahedron.

Your comment made me check if it actually was possible, and what the platonic solids in higher dimensions were.

http://math.ucr.edu/home/baez/platonic.html

https://en.wikipedia.org/wiki/120-cell

There's also this fantastic Numberphile: "Perfect Shapes in Higher Dimensions"

https://www.youtube.com/watch?v=2s4TqVAbfz4

At last, I've found the end of the internet!

But they do in 3 dimensions.

Is the future of the Pentagon the Dodecahedron?

The Pentagon might actually just be the top face of the massive subterranean Dodecahedron.

They would be building adjacent hexagons, to follow the curvature of the Earth. There would be at most 11 other pentagons.

For those who don't get this - it is a clever reference to Euler's Twelve Pentagon theorem : "any closed polyhedron composed no polygons other than hexagons and pentagons and having no holes must have exactly 12 pentagons."

Perhaps the military was a fan of Roger Penrose, and really did want to expand. Or perhaps they intended to tile the entire world with a geodesic grid.

They can just keep adding concentric rings, and the military-industrial complex expands exponentially with each new one.

(Pedantic/educational): quadratically (ignoring the bending of spacetime due to the mass of the military-industrial complex)

That's also assuming that the thickness of each ring is constant. If they increased the thickness by a certain percentage relative to the previous band, then you'd have exponential growth.

But the existing rings are equal-width, and wider rings do not make sense (the buildings would get less and less daylight)

Fair point. I guess there's another assumption, that the rings are built at a constant rate. In fact, if we make the typical assumption that the economy grows exponentially (which might hold for all but the longest term models), it could be possible to build newer rings at an exponentially faster speed by constructing more and more of them simultaneously.

even more pedantically: sub-quadratically since it's being built on a positively curved manifold. As the Pentagon approaches the size of a hemisphere, growth will stop and then reverse.

If they wanted to add concentric rings, then they screwed up big-time by sticking the thing next to a river.

Though with the military budget, rerouting the river shouldn't be that hard to do. But the river also constitutes the boundary between Virginia and DC, so there may be political problems with that.

I don't see the problem with including the occasional courtyard among adjacent pentagons, though.

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