Ritsumeikan high school Mimura Tomohiro Miyazaki Kosuke Murata Kodai Mr,Kuroda suggest “the geometry of salt” When a lot of salt is poured on a board which is cut into a particular shape, it creates a “salt mountain. We named “ Geometry of salt mountain”. 3 4 When some points are put like this on a diagram, a Voronoi Diagram is the diagram which separates the areas closest to each point from the other points. 5 6 Same distance incenter 7 8 △ＡＢＥの傍心点 △ＡＢＥの内心点 9 10 12 The reason of appearing curve line is that there are different shortest line from a concave point A E line l F Point E is same distance to line ｌ and A There were curve lines. 13 15 ED＝EA CE＋BE ＝CE＋EA＋AB ＝CE＋ED＋AB ＝CD＋AB ＝（big circle’s radius） ＋（ small circle’s radius ） ＝Constant 16 17 p＞PQ p＜PQ 18 To solve d which is make up (0,p) on yaxis and Q on y=x2 d 2 x 2 ( x 2 p ) 2 x 4 (1 2 p ) x 2 p 2 1 2 p 1 x2 p 2 4 2 p p d 1 4 p 1 2 If p ＜1/2 , the minimum If p ＞１/2 , d dp p 1 4 Thus the mountain ridges are disappeared at p<1/2. 19 20 21 22 Start i=0 Set the range and domain (x0,y0)-(x1,y1) Set the number of point num Set the coordinates of point (AX,AY) Set the radius which is r of circle Set the color ct NO Loop3 YES Loop2 MIN=L(i) ct=i Loop1 From y0 to y1 about y Loop1 L(i)<MIN Loop2 From x0 to x1 about x Loop3 From i=0 to num L(i)=SQR((X-AX(i))^2 +(Y-AY(i))^2)-r(i) NO Loop4 From i=0 to num YES MIN=L(i) ct=i Give color which is ct 2 to point SET POINT STYLE 1 PLOT POINTS: x,y Radius ｒ(i) middle (AX(i),AY(i)) such circle was drown Loop4 End 23 24 25 Weighted Voronoi Diagrams are an extension of Voronoi Diagrams. d(x, p(i)) = d(p(i)) - w(i) 26 salt mountains could reproduce this by replacing weight with the radius of the hole . this mean weight = radius 27 29 30 31 If there are four schools in some area, like this figure, each student wants to enter the nearest of the four schools. 32 34 Mountain ridges appear where the distances to the nearest side is shared by two or more sides. The prediction of the program matches the mountain ridge lines and the additively weighted Voronoi Diagram also matches the program. Salt mountain can reproduce various phenomenon in biology and physics. 35 We want to analyze mountain ridge lines in various shapes. We could reproduce additively weighted Voronoi Diagrams so we research how to reproduce Multiplicatively weighted Voronoi Diagrams. We want to be able to create the shape of the board to match any given mountain ridges. 36 Toshiro Kuroda 塩が教える幾何学 塩が教える幾何学 折り紙で学ぶなわばりの幾何 Konichi Kato 折り紙で学ぶなわばりの幾何 Spring of Mathematics Masashi Sanae http://izumi-math.jp/sanae/MathTopic/gosin/gosin.htm Function Graphing Software GRAPES Katuhisa Tomoda http://www.osaka-kyoiku.ac.jp/~tomodak/grapes/ 38 Ritsumeikan High School Mr,Saname Msashi Ritumeikan University College of Science and Engineering Dr,Nakajima Hisao 39 40

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