death_map_2009_ajou

death_map_2009_ajou, "The Death Map - Reading Nature's Geometry", a talk to the Mathematics Department at Ajou University, Suwon, Korea on 12 May 2009.

My talk was sponsored by Professor Hyung-Chun Lee. Despite the gruesome title, the subject was primarily Voronoi diagrams, with a little information on their properties and construction, and the creation of "centered" or "centroidal" Voronoi diagrams.

A plain text abstract of the talk is available as death_map_2009_ajou.txt.

• death_map_2009_ajou.tex
• death_map_2009_ajou.pdf

Reference:

1. The John Snow web site maintained by the UCLA School of Public Health at https://www.ph.ucla.edu/epi/snow.html.
2. Eldridge Adams,
   Territory Size and Shape in Fire Ants: A Model Based on Neighborhood Interactions,
   Ecology,
   Volume 79, Number 4, June 1998, pages 1125-1134.
3. Franz Aurenhammer,
   Voronoi diagrams - a study of a fundamental geometric data structure,
   ACM Computing Surveys,
   Volume 23, Number 3, September 1991, pages 345-405.
4. George Barlow,
   Hexagonal Territories,
   Animal Behavior,
   Volume 22, 1974, pages 876-878.
5. John Byers,
   Dirichlet Tessellation of Bark Beetle Spatial Attack Points,
   Journal of Animal Ecology,
   Volume 61, 1992, pages 759-768.
6. John Byers,
   Correct Calculation of Dirichlet Polygon Areas,
   Journal of Animal Ecology,
   Volume 65, 1996, pages 528-529.
7. Marc deBerg, Marc Krevald, Mark Overmars, Otfried Schwarzkopf,
   Computational Geometry,
   Springer, 2000,
   ISBN: 3-540-65620-0,
   LC: QA448.D38.C65.
8. Qiang Du, Vance Faber, Max Gunzburger,
   Centroidal Voronoi Tessellations: Applications and Algorithms,
   SIAM Review,
   Volume 41, Number 4, December 1999, pages 637-676.
9. Herbert Edelsbrunner,
   Geometry and Topology for Mesh Generation,
   Cambridge, 2001,
   ISBN: 0-521-79309-2,
   LC: QA377.E36.
10. Sandra Hempel,
    The Strange Case of the Broad Street Pump,
    University of California, 2007,
    ISBN13: 978-0520250499,
    LC: RA644.C3.H46.
11. Christian Icking, Rolf Klein, Peter Koellner, Lihong Ma,
    A Java Applet for the Dynamic Visualization of Voronoi Diagrams,
    https://www.pi6.fernuni-hagen.de/GeomLab/VoroGlide
12. Steven Johnson,
    The Ghost Map,
    Riverhead, 2006,
    ISBN-13: 978-1594489259,
    LC: RC133.G6.J64.
13. Lili Ju, Qiang Du, Max Gunzburger,
    Probabilistic methods for centroidal Voronoi tessellations and their parallel implementations,
    Parallel Computing,
    Volume 28, Number 10, October 2002, pages 1477-1500.
14. Atsuyuki Okabe, Barry Boots, Kokichi Sugihara, Sung Nok Chiu,
    Spatial Tessellations: Concepts and Applications of Voronoi Diagrams,
    Second Edition,
    Wiley, 2000,
    ISBN: 0-471-98635-6,
    LC: QA278.2.O36.
15. Joseph ORourke,
    Computational Geometry,
    Second Edition,
    Cambridge, 1998,
    ISBN: 0521649765,
    LC: QA448.D38.
16. Robert Renka,
    Algorithm 772: STRIPACK: Delaunay Triangulation and Voronoi Diagram on the Surface of a Sphere,
    ACM Transactions on Mathematical Software,
    Volume 23, Number 3, September 1997, pages 416-434.
17. Peter Vinten-Johansen, Howard Brody, Nigel Paneth, Stephen Rachman, Michael Rip,
    Cholera, Chloroform, and the Science of Medicine: A Life of John Snow,
    Oxford University Press, 2003,
    ISBN: 019513544X,
    LC: RA649.5.S66.S647.

The following files were used:

• causeway1.png, an image of the Giant's Causeway, showing how the hexagonal pillars seem to form a staircase into the ocean;
• causeway2.png, an image of the Giant's Causeway from above, emphasizing the roughly equal size and hexagonal shape of the "tiles", and the irregularity of the pattern;
• cholera_art.png, an imaginative piece of artwork showing the spread of cholera, which also illustrates the belief that it was spread through "bad air";
• cholera_spread.png, a map of the world, showing the spread of cholera during the seventh pandemic.
• cube.png, an image of a 3D cube, to illustrate Euler's Formula for bounded polyhedrons.
• cvt_movie2_0001.png, frame 1 of an animation of a CVT computation in the square. For the whole movie, refer to CVT_MOVIE2.
• cvt_movie2_0002.png, frame 2 of an animation of a CVT computation in the square.
• cvt_movie2_0010.png, frame 10 of an animation of a CVT computation in the square.
• cvt_movie2_0020.png, frame 20 of an animation of a CVT computation in the square.
• cvt_movie2_0040.png, frame 40 of an animation of a CVT computation in the square.
• cvt_movie2_0080.png, frame 80 of an animation of a CVT computation in the square.
• diamond_mountains.png, a "mountain" plot of the diamond points shows the generators as peaks, and the diagram lines as straight valleys, generated by VORONOI_MOUNTAINS.
• diamond_pixel_plot_l1.png, a pixel plot of the Delaunay triangulation of the diamond points, generated by VORONOI_PLOT, using the L1 distance. Some "mistakes" in the plot are actually caused by the fact that for the L1 distance, the boundary can consist of an area, rather than a line, that is, for certain generators, a relatively huge (and area filling) number of points can be equidistant from two generators. The plotting routine doesn't spot this, but its results suggest what is happening.
• diamond_pixel_plot_l2.png, a pixel plot of the Delaunay triangulation of the diamond points, generated by VORONOI_PLOT, using the L2 distance.
• diamond_points.png, a set of 9 points, whose Voronoi diagram includes a "diamond", generated using MATLAB.
• diamond_voronoi.png, the Voronoi diagram of the diamond points, generated using MATLAB.
• different_maps.png, shows that, without the correct organizing principle, the explanation for the cholera death data can be invisible.
• drying_mud.png, a photograph of drying mud shows a pattern of cracks which suggests a Voronoi pattern, except that the mud drying is a slow process, and the initial patterns seem to divide further as the process proceeds.
• fsu_logo.pdf, a logo.
• giraffe.png, a photograph of a giraffe shows the distinctive pattern of patches that have some features of a Voronoi diagram.
• golden_square.png, a map of the Golden Square area, where a highly localized outbreak of cholera occurred in 1854, from the John Snow archive and research companion site at https://www.epi.msu.edu/johnsnow/index.html.
• golden_square_detail.png, a detail of the Golden Square map, showing the suspected pump, and a dotted line around its "neighborhood", from the John Snow archive and research companion site at https://www.epi.msu.edu/johnsnow/index.html.
• hundred_points.png, a hundred points chosen randomly in the unit square, generated using MATLAB.
• hundred_voronoi.png, a plot of the Voronoi diagram of the hundred points, generated using MATLAB.
• john_snow.png, an image of Dr John Snow.
• pump_memorial.png, a photograph of the John Snow memorial in Broad Street, which is a replica of the pump he identified as the source of the epidemic.
• pumps_map.png, a map of the locations of pumps and deaths in the Golden Square cholera epidemic.
• pumpwomen.png, a drawing of the Broad Street Pump, suggesting how local women collected water in pumps for use at home.
• random_mountains.png, a "mountain" plot of a set of random points, tilted a bit, and with a banded color map to emphasize contours, generated by VORONOI_MOUNTAINS.
• star.png, the figure of a star, used to illustrate Euler's formula.
• tilapia_birth.png, a photograph of tilapia "giving birth", showing why they are called "mouthbreeders";
• tilapia_nest.png, a photograph of tilapia nests in a sandy bottom, from the paper by Barlow.