Analysis the mountain ridges formed by pouring salt on various polygons.

３－５　 Various shapesQuadratic Curve• There is a straight mountain ridge line on the center.• The mountain ridge line disappeared at a certain point.

Examination • When solving for the shortest distance, there is a range which contains no solution. So the mountain ridges disappeared

when p ＜ 1/2 minimum

when p ＞ 1/2 　　　　　　

Thus when p <1/2, the mountain ridges disappearedResult of Oval• There is a straight line different from the circle• Like the quadratic curve, the mountain ridges disappeared at a certain pointExamination of oval• Like the quadratic curve, there are no solutions when besides a certain point, so mountain ridges disappeared This is the side view of the straight line.• The curve line is related to the size of radius

５　 Application to Voronoi Diagram

　　 Analysis of Voronoi Diagrams using the Geometry of Salt Mountains 

 

 

Ritsumeikan High School Tomohiro Mimura , Kousuke Miyazaki, Koudai Murata

■ 　 Abstract 　 ■When salt is sprinkled on the board which is cut into a particular shape, it creates “salt mountains”. The lines created by the mountain ridges can express various characteristic. For example, the lines created on a triangular board make the bisectors of the angles and the point of intersection indicates the incenter. Therefore, when bisectors of angles, bisectors of vertical lines, and so on, are expressed by using the lines created by the salt mountain ridges, the various characteristics of plane geometry can be analyzed. In addition, we thought that this method could be applied to modeling compartmentalization of cells or problems regarding optimal location. Especially, in the case of modeling weighted Voronoi diagrams, we represent the size of the nucleus by the size of the hole, and hypothesize that by changing the size of the hole we can change the size of the cell.Finally, we verified whether the results matched the computer simulation. As a result, we found that Voronoi diagrams can be reproduced by salt mountains.Keywords: salt mountains, mountain ridges. Voronoi diagrams

■ 　 Motivation 　 ■We were interested in the handout given to us by our math teacher. It was about the analysis of various diagrams with salt mountains. We thought we could also analyze other diagrams by ourselves, so we started this study. Through analyzing many diagrams, we discovered that salt mountains can be applied to Voronoi diagrams and we furthered this study.

１　 Salt Mountain Geometrics 　Salt mountains were suggested by a book1 by Mr. Kurota. When salt is sprinkled on the board which is cut into a particular shape, it creates salt mountain and we analyzed it. We called it “Salt Mountain Geometrics”２　 Voronoi DiagramWhen 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. For example , the diagram which separates area by which points is the closest to eachof the three points is the Voronoi diagram. The lines which appear at the same distance between points are Voronoi diagram’s line. The point which is the intersection of the three lines is the point equidistant to all points

３－３　 Concave Quadrilaterals and Pentagons.Results• Angle bisectors appeared.• Curved lines appearedExaminationWe thought about why there were curved lines.If we call the vertex of the concave angle, A, and the base line, l, and find points which are equally close to point A and the line l, These points make a parabola like in this figure

We analyzed this using a numerical formula. If a is (0,0)y=-a, this curved line is

So it is parabola In the case of concave quadrilaterals and pentagons the mountain ridge line consists of angle bisectors and parabolas.

３－４　 Opening a Hole in the Board Result of Quadrilateral by opening hole.• It creates a curved linewhen a hole is opened in the board, 　• The line appeared at the same distance between the sides of the quadrilateral and the hole. So, applying the principle from concave polygons, we see it

creates a parabola Result of Opening Two Holes of the Same Size in the Board.• This picture shows the relationship between the two holes. The relationship of the edges and holes is same as the diagram• It creates the perpendicular bisector of the line segment connecting the centers of the two holes. Thus, it divides the two areas equally because of

opening two holes.

The Case with Holes not in the Center.• The mountain ridge was oval.• We analyzed this salt mountain 　 with GRAPES.As the result, when shifting holes from the center, it created oval shapes. In addition,The mountain ridges matched the oval from GRAPES simulation.

４　 Application of Salt mountains to Voronoi Diagrams.Voronoi Diagrams also divide any area with lines that are the same distance between points. We hypothesized that Voronoi diagrams could be made by salt mountains. 

４－２　 Compare with salt mountain and prediction of Voronoi Diagram.We did the experiment by opening holes at random places. The left side is the prediction by the program and the right one is the result of our experiment. The mountain ridges match the prediction from the program.

　　　

４－３　 Additively weighted Voronoi Diagrams

We tried to reproduce additively weighted Voronoi Diagrams. Additively weighted Voronoi Diagrams are an extension of Voronoi Diagrams. Each original point is the same weight in Voronoi Diagrams, but they are different in these diagrams.If p(i)’s weight is w(i)

d(x, p(i)) = d(p(i)) - w(i)

７ Future developments

• We want to analyze mountain ridge lines in various shapes. • We could reproduce additively weighted Voronoi Diagrams so we want to reproduce

multiplicatively weighted Voronoi Diagrams.• We want to be able to create the shape of the board to match any given mountain ridges.

■ 　 References 　■1. 「塩が教える幾何学」　　 　　　　　　　　黒田俊郎　　2. 「折り紙で学ぶなわばりの幾何学」　　 　　　　　　加藤渾一3. 「関数グラフソフト GRAPES 」 　　　　　　　　友田勝久

http://www.osaka-kyoiku.ac.jp/~tomodak/grapes/4. 「数学のいずみ」　　　　 　　　　　　　　早苗雅史5. 物理学教室　数理生物学研究室 R-JIRO 　　　　　 　中島久男

６　 Conclusion• We learned that mountain ridges appear where the distances to the nearest side is shared by two or more

sides.• If the radius of the opened holes are equal, the prediction of the program matches the mountain ridge

lines.• If the radius of the holes are different and replace this weight, the prediction of the program matches the

mountain ridges lines,• The diagram analyzed with numerical formulas matches the mountain ridges.• Voronoi Diagrams can reproduce various phenomenon in biology and physics.

３－１　 triangles Results• The mountain ridges appear at the same distance from edge to edge of the polygon.• The ridges lie along the bisectors of the angles and the intersection of the three lines forms the incenter of the triangle.

2 2

2

x ay

a

３－２　 quadrilaterals and pentagons.Results• Like the triangle, the bisector of each angle appeared

Examination of pentagon• In the pentagon, the bisector of each angle and the bisectors made by

extending each side also appear and J is incenter of this big triangle , H and J are the excenters of the fictitious triangles

Examination of Quadrilateral• The bisector of each angle appeared as well as another line which

is the same distance from AD to BC. F is the incenter of the big triangle and G is the excenter of the fictitious triangle.

• The quadrilaterals and pentagons which can be inscribed by a circle have their incenters and excenters lay on one point.

４－１　 Flowchart to make Voronoi DiagramWe made a program to compare salt mountains and Voronoi Diagrams. We used 十進 BASIC （白石和夫）。

Prediction of program Result of experiment Prediction of program Result of experiment

５－１　 The Problem of Separating School Districts

If there are four schools in some area, like this figure, each student wants to enter the nearest of the four schools. Then how should one separate the districts? The answer is this figure.

５－２　 The Configuration of Cells.

The configuration of the cells makes up a Voronoi Diagram.It is one example of Voronoi Diagrams in nature

epidermal cell of a plant （ taken in our school ）

４－４　　 Relationship of additively weighted Voronoi Diagrams and radii

Additively weighted Voronoi diagrams are made by subtracting weight from distance. We thought salt mountains could reproduce this by replacing weight with the radius of the hole. Applying the principle from changing the radius, we see it creates a curved line

■Multiplicatively Weighted Voronoi Diagrams■Voronoi Diagrams has two types. One is additively weighted Voronoi Diagrams and the other is multiplicatively weighted Voronoi Diagrams.Multiplicatively weighted Voronoi Diagrams are made by multiplying weight and distance, so we couldn’t devise a way to reproduce it by salt mountains. It is a big theme for the future.

５－３　 The Crystal Structure of MoleculesBy drawing all the lines connecting neighboring lattice pointsto each other and drawing the straight lines which divide those lines perpendicularly,

We can model crystal structures of molecules in this way.Result

• If the points are set like this, the structure can be represented by salt mountains, the mountain ridges appear like they are surrounding the holes.

Examination• The first one and second one are face centered cubic lattices. • The third one is a hexagonal close-packed structure. This means salt mountains can reproduce crystal structures of molecules.

4

1

2

21

)21()(2

2

2242222

pp

x

pxpxpxxd

We think this Voronoi Diagram can be applied to gravitational fields and magnetic field in nature.

ED ＝ EACE ＋ BE＝ CE ＋ EA ＋ AB＝ CE ＋ ED ＋ AB＝ CD ＋ AB＝（ big circle’s radius)＋（ small circle’s radius)＝ Constant

4

1 pd

d p

４－５　　 Simulation of mountain ridges Voronoi DiagramWhen analyzed by numerical formula, if we set the smaller circle’s radius to be the origin and bigger circle’s radius to be (a,0), then if the smaller circle’s radius is n and bigger circle’s is m, points create a ridge at the points which are the same distance from circumferences.So the numerical formula is

Like this. This mountain ridge appears at the points where the difference is constant with two centers so it is a hyperbola.

2 2 2 2( )x y n x a y m

Prediction of program Result of experiment Prediction of program Result of experiment

Start

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

Loop1From y0 to y1 about y

Loop2From x0 to x1 about x

Loop3From i=0 to num

L(i)=SQR((X-AX(i))^2+(Y-AY(i))^2)-r(i)

i=0NO

YES

MIN=L(i)ct=i

L(i)<MIN

NO

YES

MIN=L(i)ct=i

Give color which is ct 2 to point

SET POINT STYLE 1 PLOT POINTS: x,y

Loop3

Loop2

Loop1

Loop4From i=0 to num

Radius ｒ(i) middle(AX(i),AY(i))

such circle was drown

Loop4

End