The Art Gallery Theorem
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- Francine Lindsey
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1 Monthly Math Hour at UW May 18, 2014
2
3 Victor Klee, formerly of UW
4 Klee s question posed to V. Chvátal Given the floor plan of a weirdly shaped art gallery having N straight sides, how many guards will we need to post, in the worst case, so that every bit of wall is visible to a guard? Can one do it with N/3 guards?
5 Klee s question posed to V. Chvátal Given the floor plan of a weirdly shaped art gallery having N straight sides, how many guards will we need to post, in the worst case, so that every bit of wall is visible to a guard? Can one do it with N/3 guards?
6 Vasek Chvátal: Yes, I can prove that!
7 Steve Fisk: OK, but I have a proof from THE BOOK!
8 Our weirdly-shaped art museum: The Weisman
9 The floor plan
10 Who designed that thing?
11 Does Frank Gehry ever design reasonable things?
12 They can get crazier!
13 Enough fooling around let s understand Klee s question!
14 A convex gallery needs only one guard
15 A convex gallery needs only one guard
16 A star-shaped gallery needs only one guard
17 A star-shaped gallery needs only one guard
18 A 4-sided gallery needs only one guard
19 A 4-sided gallery needs only one guard
20 A 5-sided gallery needs only one guard
21 A 5-sided gallery needs only one guard
22 A 6-sided gallery might need two guards
23 N-sided galleries might need N/3 guards: the comb
24 Klee s question for Chvátal Question (V. Klee, 1973) How many guards does an N-sided gallery need? Is the comb the worst case? Theorem (V. Chvátal, 1973, shortly thereafter) Yes, the combs achieve the worst case: every N-sided gallery needs at most N/3 guards. (Of course, you can still have star-shaped galleries with a huge number of sides N, but they ll only need one guard.)
25 Klee s question for Chvátal Question (V. Klee, 1973) How many guards does an N-sided gallery need? Is the comb the worst case? Theorem (V. Chvátal, 1973, shortly thereafter) Yes, the combs achieve the worst case: every N-sided gallery needs at most N/3 guards. (Of course, you can still have star-shaped galleries with a huge number of sides N, but they ll only need one guard.)
26 What if N isn t divisible by 3?
27 Steve Fisk s wonderful 1978 proof appears in this book
28 ... by Gunter Ziegler...
29 ... and Martin Aigner
30 ... aided and inspired by Paul Erdős
31 Guess which museum appears on page 231? How to guard a museum Chapter 35 Here is an appealing problem which was raised by Victor Klee in Suppose the manager of a museum wants to make sure that at all times every point of the museum is watched by a guard. The guards are stationed at fixed posts, but they are able to turn around. How many guards are needed? We picture the walls of the museum as a polygon consisting of n sides. Of course, if the polygon is convex, thenoneguardisenough.infact,the guard may be stationed at any point of the museum. But, in general, the walls of the museum may have the shape of any closed polygon. Aconvexexhibitionhall Consider a comb-shaped museum with n =3m walls, as depicted on the right. It is easy to see that this requires at least m = n 3 guards. In fact, there are n walls. Now notice that the point 1 can only be observed by a guard stationed in the shaded triangle containing 1, and similarly for the other points 2, 3,...,m.Sinceallthesetrianglesaredisjointweconclude that at least m guards are needed. But m guards are also enough, since they can be placed at the top lines of the triangles. By cutting off one or two... walls at the end, we conclude that for any n there is an n-walled museum which requires n m 3 guards. Areallifeartgallery...
32 Fisk s proof from THE BOOK that N/3 guards suffice
33 First triangulate the gallery without new vertices
34 Then 3-color its vertices
35 The least popular color gets used at most N/3 times
36 Post guards near the least popular color vertices
37 How to triangulate without new vertices? Induct!
38 How to triangulate without new vertices? Induct!
39 How to 3-color the vertices? Induct!
40 One can always glue the colorings back together
41 How to get the red dividing line to start inducting? The flashlight argument!
42 The flashlight argument Starting at a vertex X, shine a flashlight along the wall to an adjacent vertex Y, and swing it in an arc until you first hit another vertex Z. Then either XZ or YZ works as the red dividing line.
43 How good is Fisk s method for convex galleries?
44 It depends on the triangulation
45 A couple of variations one might wonder about Three dimensional galleries? Only right-angled walls in two dimensions?
46 3D explains why we worried about how to triangulate!
47 Schoenhardt s (1928) untriangulable sphere in 3D!
48 Existence of such examples makes the 3-D theory harder.
49 What about when the walls meet only at right angles? J. Kahn, M. Klawe, and D. Kleitman proved this result in a 1983 paper titled Traditional galleries require fewer watchmen Theorem For right-angled galleries with N sides, N/4 guards suffice. One might guess how they feel about Frank Gehry.
50 Right-angled combs again achieve the worst case
51 Does the Art Gallery Theorem have real applications? Not directly that I know. But related ideas from the areas of discrete geometry and combinatorics get used in designing algorithms for searching terrains, robot-motion planning, motorized vacuum cleaners (!)
52 The set of triangulations of a polygon is interesting!
53 A polyhedron beloved to me: the associahedron
54 Thanks for listening! Bibliography Norman Do s Mathellaneous article on Art Gallery Theorems, Australian Math. Society Gazette, Nov Art Gallery Theorems, by J. O Rourke Proofs from THE BOOK, by M. Aigner and G. Ziegler Triangulations: Structures for algorithms and applications, by J. De Loera, J. Rambau and F. Santos.
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