Square Octagon Tessellation

One of the objectives of GeoGebra Applet Central this year is to update the output files of my GeoGebra Tutorial Series. The previous outputs were still in version 3.2. The applet below is the output of GeoGebra Tutorial 10 - Vectors and Tessellation. I just changed the color of the vectors to make it more visible. In this tutorial, the Vector between Two Points tool is used to translate objects and create a tessellation.

This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com The Step by Step instructions on how to create the applet above can be found here.

Square Rectangle Tessellation

The applet below is another tessellation. It is composed of rectangles and equilateral triangles. It is one of the demiregular tessellations.
This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com

Hexagon Triangle Tessellation

Move Points A and B to explore the figure. Explain why the figure tessellates. This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com The tessellation above is one of the 8 semi-regular tessellations. Semi regular tessellations are regular tessellations of the plane by two or more convex regular polygons such that the same polygons in the same order surround each polygon vertex are called semiregular tessellations, or sometimes Archimedean tessellations.

Weisstein, Eric W. "Semiregular Tessellation." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/SemiregularTessellation.html

Tessellation 2: Square, Hexagon, Triangle

The figure below shows that a plane can be tessellated using 3 regular polygons: squares, hexagons, and triangles. This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com

Polygons and Tessellation

Move the slider and observe what happens.

This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com

1. Which of the regular polygons tessellate? Can you explain why they do?
2. As n increases can we can find a regular polygons that tessellate? 
• If yes, how many of them can be found? Are they finite?
• If no, what is the maximum number of side of a polygon that can tessellate? Why?
3. Make a conjecture about your observations above. 

Download GGB File

If you want to know more about tessellation, you may want to check out the following articles that I have written:

• Tessellation: The Mathematics of Tiling
• Regular Tessellations: Why only three of them?