Three-dimensional art galleries

Solution 1:

I've actually printed the example J.M. mentions from my Art Gallery book on a 3D printer. :-)

The interior consists of many nearly cubical cells, each surrounded by beams above, below, left, right, front, back. Each "beam" derives from an indentation. The cells are not closed--there are cracks because the beams just miss one another. If you imagine standing in one of those cubical cells, you cannot see far, and certainly you cannot see a vertex.

Incidentally, it was discovered by William Thurston independently and at about the same time as Raimund Seidel. I agree that T.S. Michael's book is a great source here.

Solution 2:

There's a book by T S Michael, How to Guard an Art Gallery. Only one chapter of the book is actually about guarding art galleries, but that chapter has a section on the three-dimensional case and diagrams of the Octoplex and the Megaplex that you might find helpful.

EDIT: If you have access to The College Mathematics Journal, Michael has a paper, Guards, Galleries, Fortresses, and the Octoplex, Vol. 42, No. 3 (May 2011) (pp. 191-200).

EDIT2: Here's the description of the Octoplex from the book (but the picture in the book is worth 1,000 words). Start with a $20\times20\times20$ cube. Remove a rectangular channel 12 units wide and 6 units deep from the center of the front face (the channel runs from the top of the cube to the bottom). Remove an identical channel from the back face. Also make channels in the left and right faces, going from the front of the cube to the back, 6 units wide and 3 units deep. Finally make channels in the top and bottom faces, running left to right, 6 units by 6 units. What's left is the Octoplex: eight $4\times7\times7$ theaters connected to each other and to a central lobby by passageways one unit wide. And the claim is that even if you post a guard at each of the 56 corners there is a small region at the center of the Octoplex that no one is guarding.

EDIT3 by Rahul Narain: Here is a picture of the Octoplex.

Solution 3:

I didn't find the Wikipedia image you linked to difficult to understand, so perhaps it would help you if I just explained it a little.

Here's how I see it. Start with a rhombicuboctahedron, which has the same topology as the given figure. We will be manipulating the six of its faces which are axis-aligned squares. Consider the top square face. Clearly, the center of the polyhedron can see all of its vertices. Now take the face and elongate it in the left-right direction. If you stretch it enough, eventually its vertices will get hidden behind the squares on the left and right. A slice through the middle would look like this:

Now you do the same for the bottom face. The rest of the faces get the same treatment, except in the two orthogonal directions. In the end, you've hidden the vertices of all the axis-aligned faces behind each other. And since every vertex of the polyhedron is a vertex of some axis-aligned face, you're done.

From the inside, each triple of adjacent axis-aligned faces form three mutually orthogonal rectangles that are blocking each other's vertices. Kind of like this, where one vertex of each rectangle is hidden.

In the real thing, the other vertices will be hidden by other rectangles, but it's hard to depict them all simultaneously without a full 360° panoramic display.