Tessellation ProjectTessellation Project

Created by Jim GeisthardtCreated by Jim Geisthardt

Graphic images created with The Geometer’s SketchpadGraphic images created with The Geometer’s Sketchpad

Student images produced by hand.Student images produced by hand.

All M. C. Escher works (C) Cordon Art, Baarn, the All M. C. Escher works (C) Cordon Art, Baarn, the Netherlands. All rights reserved. Used by permission.Netherlands. All rights reserved. Used by permission.

Before the lesson begins, we’ll take Before the lesson begins, we’ll take a look at some student work that a look at some student work that

was created using these techniques was created using these techniques and some works by the artist and some works by the artist

M.C. Escher.M.C. Escher.

Student Art SectionStudent Art Section

Faces of the Lady by Liz KirkFaces of the Lady by Liz Kirk

Dog PoundDog Pound

byby

Darrin Darrin BrueningBruening

R2D2 by Eric BrotenR2D2 by Eric Broten

Motorcycle Guy by Kevin HerwigMotorcycle Guy by Kevin Herwig

Dogs by Tim VandenbergDogs by Tim Vandenberg

Birds by Bonnie BueschelBirds by Bonnie Bueschel

Fish by BeckyFish by Becky FranklinFranklin

M.C. Escher SectionM.C. Escher Section

LizardsLizards

Fish and BoatsFish and Boats

ButterfliesButterflies

Sun and MoonSun and Moon

Let’s Get StartedLet’s Get Started

Tessellating a parallelogramTessellating a parallelogramby translating opposite sidesby translating opposite sides

• Start with any kind of Start with any kind of parallelogram, here I parallelogram, here I used a rectangleused a rectangle

• Carefully line up Carefully line up points on a pair of points on a pair of opposite sidesopposite sides

1

2

• Cut out sections from Cut out sections from side 1, tape to side 2side 1, tape to side 2

• Cut out sections from Cut out sections from side 2, tape to side 1side 2, tape to side 1

• Make sure that you Make sure that you line up the locations line up the locations exactly or the shape exactly or the shape you make will not tile!you make will not tile!

1

2

• Repeat the process with the other 2 sides

43

• Try to imagine what your figure looks like!

• Give it some color and Be Creative!

• To tile your shape, start near the center of the paper and trace your shape.

• Slide your shape so the edges match and trace again.

• Fill the page, draw in your details, and color!

Tessellating a parallelogram by Tessellating a parallelogram by rotation about midpoints and rotation about midpoints and translation of opposite sidestranslation of opposite sides

• Start with any size Start with any size parallelogram.parallelogram.

• Locate the midpoints Locate the midpoints of one pair of opposite of one pair of opposite sides.sides.

• Cut out a section from Cut out a section from one side of the one side of the midpoint.midpoint.

• Rotate the shape 180° Rotate the shape 180° about the midpoint about the midpoint and tape it in place.and tape it in place.

• Repeat the process on Repeat the process on the opposite side.the opposite side.

• Cut out a section, Cut out a section, rotate 180° about rotate 180° about midpoint and tape in midpoint and tape in placeplace..

• As in the previous As in the previous example, locate example, locate matching points on the matching points on the 2 unchanged sides,2 unchanged sides,

• Cut out shapes,Cut out shapes,• And tape in place on And tape in place on

the opposite side.the opposite side.

• Try to decide what it is that Try to decide what it is that you’ve created.you’ve created.

• Add some detailsAdd some details• and get ready to tessellate.and get ready to tessellate.• Rotate the shape 180° Rotate the shape 180°

about point K and also about point K and also about point I.about point I.

• Translate the shape by Translate the shape by sliding along the vector ABsliding along the vector ABI

K

A'

A

B

C

Tessellating any quadrilateral by Tessellating any quadrilateral by rotation about midpointsrotation about midpoints

• Start with any shape Start with any shape quadrilateralquadrilateral

• Locate the midpoint of Locate the midpoint of all four sidesall four sides

• Cut out a section from Cut out a section from either side of the either side of the midpoint.midpoint.

• Rotate the section Rotate the section 180° about the 180° about the midpoint,midpoint,

• And tape into place.And tape into place.

• Repeat the process Repeat the process carefully with the next carefully with the next side...side...

• Repeat the process Repeat the process with the two with the two remaining sides.remaining sides.

• To make the shape To make the shape tessellate…tessellate…

• Rotate each side about Rotate each side about its midpoint.its midpoint.

• Start in the center of Start in the center of the page and the page and

• ROTATE!ROTATE!

The End...

Or just the beginning!