Guarding a Koch Fractal Art Gallery

This article presents a generalization of the standard art gallery problem to the case where the sides of the gallery are continuous curves which are limits of polygonal arcs. The allowable limiting processes for such generalized art galleries are defined. We construct an art gallery in which one side is the Koch fractal and the other sides are three sides of a rectangle. The appropriate measure of coverage by guards is not the total number of guards but, rather, the guardsto-side ratio. We compute this ratio for the cases of shallow and deep versions of the Koch fractal art gallery.


Introduction
O'Rourke [1] describes how at a geometry conference in 1973 at Stanford University Victor Klee extemporaneously gave to Vasek Chvátal what has become known as the classical art gallery problem: Determine the minimum number of guards sufficient to cover the interior of any art gallery with walls.n To make the statement of the problem more precise we introduce some notation and definitions.We say that a set in the plane is a polygon if is compact, connected, and simply-connected and if the boundary of is a polygonal Jordan curve.A point A in sees or covers another point B in if the line segment connecting the two points is contained in of points in is a set of guards, or watchmen, if for each point B in there is a point A in such that A sees B. We define  to be the smallest cardinality of any set of guards for  .For any natural number n greater than 2, let P(n) be the set of polygons that have exactly n vertices, and set    :     = max g n G P   n .With these definitions Klee's problem takes the form: For each n, find g(n).
In 1975 Chvátal [2] published a proof that . If we define the guard-to-side ratio,

 
gsr  , of a polygon as

 
G  divided by n, then the last inequality is equivalent to   1 3 gsr   .It is the purpose of this article to generalize the art gallery problem to galleries whose walls are continuous curves.In Section 2 we indicate how the concept of guard-to-side ratio may be extended to a certain class of art galleries whose walls are limits of polygons.In Section 3 the Koch fractal art gallery is defined as a sequence of approximant art galleries and auxiliary notations are introduced.In Section 4 a system of recursion relations is obtained for the minimum number of watchmen needed to guard the   approximant gallery.In Section 5 we calculate the guard-to-side ratio for shallow and deep art galleries in which one of whose walls is the Koch fractal.

Generalized Art Galleries
Consider a sequence 1 2 3 , , ,     , of polygons such that the vertex set of each polygon is strictly contained in the vertex set of the next.Assume also that all the polygons are contained in a compact region of the plane.If the limit of the boundaries of the polygons is a Jordan curve with a connected interior, then we denote the union of the Jordan curve and its interior as   and write limn n      .We call   a generalized art gallery.We also define the guards-to-side ratio, or the gsr, of

A Koch Fractal Art Gallery
We consider a rectangle R = ABCD with the vertices labelled consecutively in a clockwise direction.For ease in visualization suppose that edge AB is a horizontal line segment and is the upper horizontal side of R. We perform a "basic process" on edge AB (See Figure 1): Let points E and F on AB divide AB into thirds.Construct an equilateral triangle EGF with base EF on AB and such that G lies outside of R. Set 1 K to be the polygon with boundary AEGFBCDA.
For ease in refering to parts of the boundary of a polygon, we will adopt the following notation.When 1 and are two vertices of a polygon  , by we will mean the polygonal path along K is the generator of the Koch curve [3].
We now proceed to define a sequence n K of approximant art galleries inductively.For each K  is constructed by performing the basic process on each edge of arc(A, B), always choosing new vertices to lie outside of n K .Since the vertex set of each Koch approximant gallery is strictly contained in the vertex set of the next one in the sequence, it makes sense to say for example that vertices A and G belong to each n K for .In n 1 n  K we call arc(A, B) the front edge of the art gallery and will denote it by n .Because the vertices of each n L K are fixed in all subsequent approximant art galleries, the limit of the sequence is welldefined.We label this limit K  and call it a Koch fractal art gallery.
To facilitate the derivation of a system of difference equations for the minimum number of watchmen needed to guard the approximant art galleries, we introduce notations for some of the geometrical features of the approximant art galleries.These notations are illustrated in Figure 2. The front edge may be decomposed into the union of three separate arcs:   side CD is the central line of symmetry of n K .If the side BC in n K is sufficiently shorter than side AB, no guard placed on n  will be able to see completely any 1 in either of the wing galleries.We'll call such a n P K "shallow" and will denote it by

Recursion Relations for the Koch Approximant Art Galleries
Let .Using the empirically determined value of as an initial condition gives that .
k , for , one central guard placed sufficiently close to side CD will be able to see all the 1 side galleries in . The number of such side galleries is .Once the 1 side galleries in the wings have been covered, the remaining guards can be placed as in .Since for any , we have for To obtain a for n we first note that in each   K the central arc in the front edge is the union of two congruent arcs: n .Since each of and is similar to  ,

E G arc ,
 and since these side galleries open onto the wide central region bounded by n , we expect that the number of guards just sufficient to guard each arc should be close to . This gives finally that .
This latter recursion relation, when the expression for . We now have a coupled system of difference equations for and v , namely, for In addition, in view of the empirical data, we may take as an initial condition .The system of difference equations we have obtained is amenable to standard techniques.Substitution of the expression for This is a first order equation of the form 1 n n n n q y p y    .The general solution is given by Mickens [4] as In the case at hand for , 0,

Conclusion
In this article we have presented a generalization of the standard art gallery problem to the case where the sides of the gallery are continuous curves which are limits of polygonal arcs.In such cases the appropriate measure of coverage by guards is not the total number of guards but, rather, the guards-to-side ratio.This ratio has been computed for both shallow and deep versions of a Koch fractal art gallery and has been found to be 1 18 .Obtaining a formula for the number of watchmen needed to guard approximant art galleries in the cases intermediate between the shallow and the deep limits is an open question.However, since the gsr's of the Koch fractal art

Figure 1 .
Figure 1.The basic process in forming the first Koch fractal approximant K 1 .

Figure 2 .
Figure 2. Structure of the Koch approximant gallery K n .L n is the front edge of the gallery and is the union of arcs M n , P n , and Q n .
the side BC in n K is sufficiently long in comparison to side AB, then there is a point on n  from which a guard is able to see all of the side galleries off of the main hall.We'll call such a 1 P n K "deep" and will denote it by D n K .
 .Certainly the role of the guard closest to side CD in 1 D n k  is covered by the central guards on n  . of each side gallery's innermost watchman is covered by the central guards.Hence the count 1 1 and n = arc(F, B).Both n V n The "main hall" of n V n  drawn through G in n K and perpendicular to without too much difficulty, for larger n the optimal arrangement of guards becomes less clear.Our strategy will be to position those guards with the largest fields of vision first.Such guards will certainly lie on n n , n