# Voronoi Cones

- Author:
- Eva Bauer-Öppinger, Tom Walsh

- Topic:
- Cone

A Voronoi diagram is a partitioning of a plane into regions such that each region contains one generating point and every point in a given region is closer to its generating point than to any other. GeoGebra has a built-in Voronoi function, where you input a list of points and GeoGebra will generate the Voronoi boundaries. After reading recently about how Voronoi diagrams can be generated through the use of cones, I decided to make this worksheet to demonstrate the idea. Each point represents the vertex of a cone. Move the points around and watch the pattern morph. The cones, when viewed directly from above, show the Voronoi pattern. Use the checkbox to show the Voronoi boundaries generated by the GeoGebra Voronoi function. Slide the slider to see the cones collapse or grow.

## GAMING IDEAS

**1. Puzzle solving**

*material: Voronoi-Cone-Parts*Try to build the cone puzzle so that the cones fit exactly together. After finishing you can control, if the puzzle fits into the frame.

**2. Polygone throwing**

*material: Voronoi-Cone-Landscape, Polygons*Each player creates any polygon in GeoGebra and prints it 3 times. The aim is now to throw the polygons onto a cone tip from a distance defined together.

**3. Ring throwing**

*material: Voronoi-Cone-Landscape with differently coloured cones*Each player chooses a color and tries to throw his/her rings exclusively on cones of that color.

**4. Shortest way**

*material: Voronoi-Cone-Landscape, rope, sticky tape/dots*Imagine each one of you living in a hut on one of the cone tips. Mark your huts with the sticky dots. Now you want to meet on another tip. At which tip will your meeting take place? (shortest way) Check your result by measuring the path with the rope.

*Variation:*In the valley there is a coffee house at every junction. In which one will you meet? (After descending from your mountain, you'll only hike through the valleys.)

**5. Idea**

*material: Voronoi-Cone-Landscape with marked A and X, “solution rope”*Imagine you are a climber and you would like to climb up the mountain X. You start point is A. Thanks to your good equipment you can choose any route. Which would be the least strenuous (least altitude)? Check with the rope whether it is sufficient for your route. If not, there should be an even easier way.