**Tessellation**

A tessellation is created when a shape is repeated over and over again covering a plane without any gaps or overlaps.

**Polygons That Would Tessellate**

Triangles, squares and hexagons are the only regular shapes which tessellate by themselves. If you put many of these shapes together side-by-side, they form a tessellation. You can have other tessellations of regular shapes if you use more than one type of shape.

Polygons will tessellate because the sum of the angles at a point must be 360.

Regular and Semi-Regular

Regular tessellations have only one shape, as opposed to semi-regular, that have more than one shape on the plane.

Semi-regular Tessellations

Examples of combinations of regular shapes include triangles & hexagons, squares & triangles and hexagons, squares & triangles.

Examples of Tessellations

Examples of Tessellation Artists

The King of Tessellations

M.C. Esher was a graphic artist known for his art tessellations. He used geometric shapes and architecture for inspiration. He created visual riddles, playing with the impossible.

Another Tessellation Artist

Robert Fathauer, born in 1960, creates his tessellations using a computer. Robert has an interest in mathematics and art and was a great fan of Escher. Later in 1993 he founded his own company called Tessellations to produce tessellation puzzles and offer them for sale.

"If there's anything one can be certain of in this world it's mathematics. It's the one discipline where results can be proven to be true. At the same time, there is great beauty and elegance in mathematics. Conversely, art is the discipline where beauty is the traditional goal, but art also strives to get at deep truths. Both disciplines appeal to me for these reasons, and it seems natural to combine them." - Dr. Fathauer

Penrose Tiles

The Penrose tiles are a pair of shapes that tile the plane only aperiodically (when the markings are constrained to match at borders).

In strict Penrose tiling, the tiles must be placed in such a way that the colored markings agree; in particular, the two tiles may not be combined into a rhombus (Hurd).

Kinds of Penrose Tiles

Kepler Tilings

Some of the first modern work on tilings was done by Johannes Kepler. He also was one of the first mathematicians to treat star polygons as regular polygons.

P1 Tilings

Roger Penrose discovered a number of aperiodic tilings with pentagonal symmetry. This one bears an obvious kinship to Kepler's tilings. This version, showing the shapes of the six component tiles, also tiles periodically.

Darts and Kites

In 1974, Roger Penrose discovered an aperiodic tiling that uses only two shapes, nicknamed kites and darts. When people use the term "Penrose tiling", this is usually what they mean.

Cartwheel Pattern

The Cartwheel, is the most important Penrose Tiling. The purple region at the center is outlined by a decagon consisting of a kite and dart edge. Every point in every tiling is contained inside an identical decagon.

Penrose Rhombi

Penrose tilings can also be based on rhombi. The acute angles in the rhombi are 36 and 72 degrees. Coloring the rhombi as shown forces aperiodicity.

Integrating Math and Art

Using Tessellations

Canadian Math teacher Jill Britton uses Escher tessellations to help students learn the mathematics term, congruent.

http://1-ps.googleusercontent.com/h/www.incredibleart.org/lessons/middle/images/animpegasus.gif.pagespeed.ce.gqYEvB2wYI.gif

http://britton.disted.camosun.bc.ca/escher/pegasus.jpg

Sources

incredibleart.org

www4.uwsp.edu

Jill Britton said, while examining Escher's picture, Tessellation 105, "When the students study a pegasus in its parent square, they discover how Escher modified the square to obtain his creature. Each "bump" on the upper/lower side is compensated for by a congruent "hole" on the lower/upper side. The same is true of the left/right sides. Corresponding modifications are related by translation. The area of the parent square is maintained."

CONGRUENCE: In geometry, two figures or objects are congruent if they have the same shape and size, or if one has the same shape and size as the mirror image of the other.

This is an example of congruence. The two triangles on the left are congruent, while the third is similar to them. The last triangle is neither similar nor congruent to any of the others.

Start with creating a tessellation shape. Begin either with a square that has all 90° corners or a triangle with 60° angles.

Create a pattern design based on a tessellation you create.

Your tessellation should be a recognizable (not abstract) object - animals, birds, insects, fish, etc.

Trace your tessellation onto a drawing paper.

Draw the details inside each tessellation. You can incorporate negative space into

your piece.

THE ASSIGNMENT:

>To integrate math and art

>Develop problem solving skills

>Gain abstract & critical thinking skills

OBJECTIVE:

http://www.juliannakunstler.com/art1_tessellations.html#.UxQD-PRdWRJ

Congruent

Not Congruent

Essential Question:

How do you incorporate math into artistic expression?

As you watch this video, write down three things you learn about M.C. Escher.

youtube.com/watch?v=Kcc56fRtrKU

More formally, two sets of points are called congruent if, and only if, one can be transformed into the other by a translation, a rotation, or a reflection.