Tessellations with Java

# Tessellations with Java

### By Roseann Krane, Monte Vista HS, Danville, CA, and students: Jefferson Ng, Mike Carns, and Amber Bullington, and Sherida Hare, Bellport HS, Brookhaven, NY, editor

"Understanding symmetry is essential to the understanding of Escher art and understanding symmetry involves a familiarity with the movements that mathematicians call transformations." (Jill and Walter Britton)

Click to view the "Step by Step Tessellations" for a detailed explanation on tessellations. Click to print the "Step by Step Tessellations" for a detailed explanation on tessellations.

Vocabulary:

• Tessellation - A tessellation is a pattern of one or more shapes, completely covering a two-dimensional plane. Picture a puzzle with pieces that have the same shape and size that fit together perfectly as in a square. (A circle would not be a shape for a tessellation because circles cannot fit together in a puzzle.) A tessellation is a pattern that is repeated. If we take squares and layer them on the floor, we can cover the entire floor with them, without any space leftover.

• Plane - Imagine a plane to be a tabletop with no thickness that continues infinitely in all directions.

• Translation - If we move a figure to a new location by sliding it a fixed distance in a fixed direction, the motion is called a translation.

• Rotation - If we move a figure to a new location by turning it about a fixed axis, the motion is referred to as a rotation or a turn. Center of Rotation - The point or axis about which a figure is rotated is called the center of rotation. The angle through which the figure turns is called the angle of rotation.

• Reflection - If we move a figure to a new location by flipping it about a fixed line the motion is called a reflection or flip. The fixed line about which the figure is flipped is called the line of reflection. (It is called a reflection because if a mirror were placed along a line the transformed figure would coincide with the mirror image of the figure in its original location.)

• Glide Reflection - This final transformation combines the motions of reflection and translation to move a figure to its new location.
1. By flipping it about a fixed line.
2. By sliding it at a fixed distance in a direction parallel to that line.
3. The motion is called a glide reflection.

Click on the following for a detailed explanation on:

Student Exercises
1. Modify the paint function to create a tessellation of triangles.
2. Modify the paint function to create a tessellation of hexagons.
3. Modify the paint function to create a tessellation of octagons and squares.

### Links about tessellations: World of Escher Han's Computer Art Swarthmore Tessellation Tutorials Glide Reflections Tutorial Movies on Tessellations Swarthmore Discussions Thread Student Work with Tesselmania Student Work with Paint Programs

Lesson Plan

You can contact me at : roseann@krane.net
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