**My Hypothesis**

**Math in the Art of M.C. Escher**

**Alyssa Mora**

How do you make a Tessellation?

What math is used

behind

tessellations?

Bio:

Maurits Cornelis Escher born: 1898 in Leeuwarden, the Netherlands.

The youngest of four children son of a civil engineer.

He went to the School for Architecture and Decorative Arts in Haarlem.

He told his father he would rather study graphic art instead of achitecture

After school, he and his wife, Jetta Umiker (1942) settled in Rome.

For the next 11 years Escher traveled Italy working on sketches

Died: 1972

Tessellation:

A covering of a plane by geometric shapes (tiles) so there are no empty spaces or overlapped tiles.

Types of Tessellations:

Monohedral -- Only use one kind of tile.

Regular -- Regular polygons and vertices only meet vertices

Semi-Regular -- Two or more types of regular polygons, vertices meet vertices, same configurations

Examples:

Monohedral

Regular

Semi-Regular

Math Component:

**Tessellations by Triangles**

This works because all angles (A,B,C) come together and make 180 degrees (a straight line)

It could tesselate a plane just like the Monohedral example from the beginning.

Before moving on to Quadrilateral Tessellations we can prove that they work by dividing it into two triangles.

This image proves the following equation:

How did Escher do Triangular Tessellations?

This piece is called

"Liberation"

first printed in 1955.

M. C. Escher's tessellation style for Liberation isn't one that follows the definition of a tessellation. There is no completely covered plane and there are empty spaces.

Tessellations by

Quadrilaterals

Take the quadrilateral

ABCD

Rotate it around the midpoint (between A&B)

Repeat

Turning the one quadrilateral into....

A tessellated plane!

How did Escher

do a quadrilateral tessellation?

A tessellation is constructed by tiling images of a regular polygon across axes of reflection and about vertices of the pre-image.

Vertex

Axis of reflection

This piece is called

Escher's tessellation style in this piece starts distorted from the start.

You can see that the original shape to each the fish and the bird are quadrilaterals.

They are simply distorted in each level of the tessellation.

The birds become to water to the fish.

The fish become the sky to the birds.

"Sky and Water I"

Events

Research Tessellations by M.C. Escher

Take tessellations by M.C. Escher and break them down one by one.

Explain the math and system of rotation behind each

Research the math behind reflections, and rotations with regular poygons

First

Then

Data & Analysis

180

180

What does this mean?

Any right/equilateral triangle will tessellate

Any quadrilateral will tessellate

See the transitions that Escher creates in Metamorphose

Metamorphose also ends in the same way that it began!

Take two pieces from Escher that show:

Triangular Tessellation

Quadrilateral Tessellation

Analyze the pieces shape by shape.

90

90

Conclusion:

Tessellations are made

by

Rotating an image about a changing midpoint

Symmetry

NOT by being mirrored off of a line of reflection!

Reflection

Create my own

"Escher-esque"

Tessellation

First,

a protractor, pencil, and paper

Next,

An equilateral triangle

Start to distort the shape

Cut it out, and trace it onto your paper

Rotate the cut out piece

Trace the new side into the bottom

Same equilateral, "Escher'd out"

Courtesy of

Jessica Juarez

Class of 2012

Trace the third side onto the triangle

Now you have your FINAL shape!

Courtesy of

Cynthia Tran

Class of 2012

Get to Tessellatin'!!

Starting with a plain tessellated plane

you can change from this.....

to this!

Bibliography

Student, SLU. "Math and the Art of M. C. Escher." - EscherMath. St. Louis University, 1 May 2012. Web. 04 May 2012. <http://mathcs.slu.edu/escher/index.php/Math_and_the_Art_of_M._C._Escher>.

Wray, Link. "How to Make an Escher-esque Tessellation." YouTube. YouTube, 26 Mar. 2009. Web. 04 May 2012. <http://www.youtube.com/watch?v=212XC1zfxXY>.

Stanchinsky, Stan. "M.C. Escher Documentary." YouTube. YouTube, 19 Apr. 2011. Web. 04 May 2012. <http://www.youtube.com/watch?v=1d5blV9RDgM>.

Free, Wikipedia. "Tessellation." Wikipedia. Wikimedia Foundation, 05 Feb. 2012. Web. 04 May 2012. <http://en.wikipedia.org/wiki/Tessellation>.

Acknowledgments

Miss "Josay" Johnson &

Mr. K

Mr. Paradise

Mr. Burch & Mr. Melino

Mr. Graham and my small group

Mrs. Redmond

Mr. de la Cruz

My Sister & Mama :)

Favianna "Voltron" Oropeza

"Order is repetition of units,

Chaos is multiplicity without rhythm"

-M. C. Escher

"I have embarked on this geometric problem again and again over the years, trying to throw light on different aspects each time. I cannot imagine what my life would be like if this problem had never occurred to me; one might say that I am head over heels in love with it, and I still don't know why."

-- M. C. Escher

**May 4th 2012**

Any Questions?