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Copy of Art of M.C. Escher

Exploration of the Math behind tessellations by M.C. Escher
by

Sarah Kitchen

on 9 October 2013

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Transcript of Copy of Art of M.C. Escher

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?
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