Guarding a Koch Fractal Art Gallery
ABSTRACT
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 guards-to-side ratio. We compute this ratio for the cases of shallow and deep versions of the 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 guards-to-side ratio. We compute this ratio for the cases of shallow and deep versions of the Koch fractal art gallery.
Cite this paper
L. Cassell and W. Fuller, "Guarding a Koch Fractal Art Gallery," Open Journal of Discrete Mathematics, Vol. 2 No. 4, 2012, pp. 134-137. doi: 10.4236/ojdm.2012.24026.
L. Cassell and W. Fuller, "Guarding a Koch Fractal Art Gallery," Open Journal of Discrete Mathematics, Vol. 2 No. 4, 2012, pp. 134-137. doi: 10.4236/ojdm.2012.24026.
References
[1] J. O’Rourke, “Art Gallery Theorems and Algorithms,” Oxford University Press, Oxford, 1987. [2] V. Chvátal, “A Combinatorial Theorem in Plane Geometry,” Journal of Combinatorial Theory, Series B, Vol. 18, No. 1, 1975, pp. 39-41. doi:10.1016/0095-8956(75)90061-1 [3] B. B. Mandelbrot, “The Fractal Geometry of Nature,” W. H. Freeman, New York, 1983. [4] R. E. Mickens, “Difference Equations: Theory and Applications,” 2nd Edition, Van Nostrand Reinhold, New York, 1990.
[1] J. O’Rourke, “Art Gallery Theorems and Algorithms,” Oxford University Press, Oxford, 1987. [2] V. Chvátal, “A Combinatorial Theorem in Plane Geometry,” Journal of Combinatorial Theory, Series B, Vol. 18, No. 1, 1975, pp. 39-41. doi:10.1016/0095-8956(75)90061-1 [3] B. B. Mandelbrot, “The Fractal Geometry of Nature,” W. H. Freeman, New York, 1983. [4] R. E. Mickens, “Difference Equations: Theory and Applications,” 2nd Edition, Van Nostrand Reinhold, New York, 1990.