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Duals
 

1. Definition
2. Examples
3. Hands-On Activities
Definition
The centroid of a polygon is simply the center of balance for the polygon. In regular polygons, the centroid is the same distance from all vertices and is also the same distance from all sides.

Centroids of Some Regular Polygons

Centroids of some regular polygons

 

We can create a new tessellation from an existing one by joining the centroids of adjacent polygons. The polygons that are made from all of these joinings is called the dual of the original tessellation.

How to apply the dual technique (animated)

How to apply the dual technique

 

1. Definition 2. Examples 3. Hands-On Activities

Examples
Here is an example of the dual technique.

Example (Step 1)

A tessellation of quadrilaterals
Example (Step 2)

Centroids of adjacent polygons connected; the original tessellation is shown in red
Example (Step 3)

This is the dual with the original tessellation removed

 

Here is a second example of the dual technique.

Example (Step 1)

The demiregular tessellation 3.3.3.4.4 / 3.4.6.4
Example (Step 2)

Centroids of adjacent polygons connected; the original tessellation is shown in light blue
Example (Step 3)

This is the dual with the original tessellation removed

 

Here is a third example of the dual technique.

Example (Step 1)

The demiregular tessellation 3.3.4.3.4 / 3.3.4.12 / 3.4.3.12
Example (Step 2)

Centroids of adjacent polygons connected; the original tessellation is shown in gray
Example (Step 3)

This is the dual with the original tessellation removed

 

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TemplatesAdditionally, after having tried the hands-on activities, you can proceed to the templates page where you can practice creating duals out of regular, semiregular, demiregular tessellations.

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