TEKS(7.8) Geometry and spatial reasoning. The student uses geometry to model and describe the physical

world.

The student is expected to:

(A) sketch three-dimensional figures when given the top, side, and front views;(B) make a net (two-dimensional model) of the surface area of a three-dimensional figure; and(C) use geometric concepts and properties to solve problems in fields such as art and architecture.

(7.13) Underlying processes and mathematical tools. The student applies Grade 7 mathematics to solve problems connected to everyday experiences, investigations in other disciplines, and activities in and outside of school.

The student is expected to:

(A) identify and apply mathematics to everyday experiences, to activities in and outside of school, with other disciplines, and with other mathematical topics

TEKS(8.6) Geometry and spatial reasoning. The student uses transformational geometry to develop spatial sense.

The student is expected to:

(A) generate similar figures using dilations including enlargements and reductions; and(B) graph dilations, reflections, and translations on a coordinate plane.

(8.7) Geometry and spatial reasoning. The student uses geometry to model and describe the physical world.

The student is expected to:

(A) draw three-dimensional figures from different perspectives;(B) use geometric concepts and properties to solve problems in fields such as art and architecture;

(8.14) Underlying processes and mathematical tools. The student applies Grade 8 mathematics to solve problems connected to everyday experiences, investigations in other disciplines, and activities in and outside of school.

The student is expected to:

(A) identify and apply mathematics to everyday experiences, to activities in and outside of school, with other disciplines, and with other mathematical topics

NCTM StandardsUse visualization, spatial reasoning, and geometric modeling to solve problems

Draw geometric objects with specified properties, such as side lengths or angle measures; use geometric models to represent and explain numerical and algebraic relationships;

Recognize and apply geometric ideas and relationships in areas outside the mathematics classroom, such as art, science, and everyday life.

Instructional programs from prekindergarten through grade 12 should enable all students to—create and use representations to organize, record, and communicate mathematical ideas;select, apply, and translate among mathematical representations to solve problems;use representations to model and interpret physical, social, and mathematical phenomena.

Instructional programs from prekindergarten through grade 12 should enable all students to—recognize and use connections among mathematical ideas;understand how mathematical ideas interconnect and build on one another to produce a coherent whole; recognize and apply mathematics in contexts outside of mathematics.

• A tessellation or tilling of the plane is a collection of plane figures that fills the plane with no overlaps and no gaps.

• Tessellation comes from the Latin word “tessera” which means a small cube.

• These cubes made up “tessellata” the mosaic pictures forming floors and tiling in Roman buildings.

Tessellations

History of Tessellations• They can be traced all the way back to the Sumerian

civilization (about 4000 B.C.) in which the walls of homes and temples were decorated by designs of tessellations constructed from slabs of hardened clay.

• Not only did these tessellations provide decoration but they also became part of the structure of the buildings.

• Since then, tessellations have been found in many of the artistic elements of wide-ranging cultures including the Egyptians, Moors, Romans, Persians, Greek, Byzatine, Arabic, Japanese, and Chinese.

Examples of tessellations from different cultures

Chinese

IslamicItalian

KoreanPersian Egyptian

•The study of tessellations is concerned with the use of multiple identical, none overlapping copies of a certain figure to cover the Euclidean plane (a flat surface unbounded in all directions).

• The first known attempt of tessellations was the tiling in the Alhambra in Spain. It was laid out by the Moors in the 14th century.

•They were made of colored tiles forming patterns that were symmetrical, geometrical, and beautiful.

• Some were not tessellations because they didn't cover a surface with a repetitive design without gaps or overlaps.

This is a picture of the Alhambra tiling.

Tessellations

• A Dutch graphic artist• Regarded as the 'Father' of modern

tessellations.• Born in Holland on June17th, 1898 and was

enrolled in the “School for Architecture and Decorative Arts” in Harlem where he studied until 1922.

• Much of his work is based on the ancient on the periodic designs of ancient Moorish mosaics, Moors of Alhambra, Spain.

M.C. Escher

The Alhambra Palace - Granada, Spain

There are tessellated patterns on the lower portions of the walls. This room, like in many Islamic buildings, is perfectly symmetrical (exhibiting reflective symmetry) as its left and right sides are identical. This was the inspiration of M.C. Escher’s work.

Escher produced '8 heads' in 1922 - a hint of things to come. You can immediately see 4 different heads but the others are not apparent until the picture is turned upside-down

8-heads 8-heads turned

M.C. Escher

Famous tessellations

Drawing Hands

Relativity

Sky and Water

Content that is useful to understanding tessellations

• Polygons: regular and irregular• Vertex• Angle• Degree• Interior angles• Symmetry• Transformation: translation, reflection, rotation, and

glide reflections• Direction• Magnitute

Regular & Irregular Tessellations

• A regular tessellation is a highly symmetric tessellation made up of congruent regular polygons. – There are only 3 regular tessellations: those made up of equilateral

triangles, squares, and hexagons.

• A semiregular tessellation uses more than one regular polygon and has the same polygon arrangement at each vertex.– There are 8 of these.

• A demiregular tessellation uses more than one regular polygon and has two or three different polygon arrangements.

• An edge-to-edge tessellation is even less regular: the only requirement is that adjacent tiles only share full sides (For example, no tile shares a partial side with any other tile.)

What tessellates?

• Not all polygons will tessellate– Qualifications for tessellations:

• Equal side lengths• Equal internal angles• The measure of an internal angle must be an exact divisor of 360.

Why is this?• Based on this definition, the following polygons are the only regular

polygons that will tessellate:– Triangle– Square– Hexagon

Non-regular polygons• There are some polygons that don’t meet the previous conditions, thus we

call them non-regular polygons.• The reason these tessellate is when you put these objects back to back

they form either a triangle, square, or hexagon which we know already tessellates.

• Examples:– Kites– Diamonds– Parallelograms– Rectangles

Principles of Tessellations

• There are 4 ways of how a diagram can be “mapped onto” itself– Translation– Rotation– Reflection – Glide reflection

Translation

• Translation is the moving of a pattern over a certain distance, such that it coincides and cover the underlying pattern again.

http://www.math.nus.edu.sg/aslaksen/gem-projects/maa/0203-2-03-Escher/translation.swf

Rotation

• Rotation similarly is the rotation of a pattern at a fixed origin and fixed angle such that it covers the underlying pattern also.

http://www.math.nus.edu.sg/aslaksen/gem-projects/maa/0203-2-03-Escher/rotation.swf

Reflection

• Reflection is the mapping of a pattern by “mirroring” the image with respect to an axis of reflection. The image is therefore a mirror image of the original pattern.

http://www.math.nus.edu.sg/aslaksen/gem-projects/maa/0203-2-03-Escher/reflection.swf

Glide Reflection

• Glide reflection is basically a combination of translation and reflection of the diagram

http://www.math.nus.edu.sg/aslaksen/gem-projects/maa/0203-2-03-Escher/glidereflection.swf

Creative Tessellations from Polygons

http://fc23.deviantart.com/fs5/f/2004/340/5/e/bunnytess.swf

Real World Examples

Activity

• Using the regular polygons, create your own tessellation.

• You can create a regular, semiregular, or demiregular tessellation.

Activity Examples

Integrating Tessellations

Works Cited• Chapin, S. H. Math Matters: Understanding the Math You Teach Grades K-6. California: Math Solutions Publications,

2000.

• “Chapter 111. Texas Essential Knowledge and Skills for Mathematics Subchapter B. Middle School” Texas Education Agency. 1 September 2006. Retrieved: 2 November 2008.

<http://www.tea.state.tx.us/rules/tac/chapter111/ch111b.html>.

• Flournoy, V. The Patchwork Quilt. New York: Dial Books for Young Readers, 1985.

• Hope, Martin. Integrating Mathematics Across the Curriculum. Arlington Heights, Ill: IRI/Skylight Training and Publisher, 1996.

• “Maurits Cornelius Escher.” Artinthepictures.com: An introduction to art history. 2008. Retrieved 4 November 2008. <http://www.artinthepicture.com/artists/MC_Escher/biography.html>.

• “Overview: Standards for Grades 6–8.” Principles and Standards for School Mathematics. NCTM: National Council of Teachers of Mathematics. 2004. Retrieved: 2 November 2008.

<http://standards.nctm.org/document/chapter6/index.htm>.

• “Principles of Tessellations.” The Mathematics behind the art of M.C. Escher. Retrieved 31 October 2008. <http://www.math.nus.edu.sg/aslaksen/gem-projects/maa/0203-2-03-Escher/main2.html#Principals>.

Works Cited• Taschen. Escher, M.C. London: Taschen, 2006.

• Tessellations: Escher and how to make your own. October 2003. Annal, David and Bareiss, Seth. Retrieved: 2 November 2008. <http://tessellations.org/index.htm>.

• Totally Tessellated. 1998. Bhushan, A., Kay, K., & Williams, E. Retrieved: 2 November 2008. <http://library.thinkquest.org/16661/history.html>.

• “What is a Tessellation?” The Math Forum. Drexel University. 2002. Retrieved 2 November 2008. <http://mathforum.org/sum95/suzanne/whattess.html>.

• Ziring, Neal. “M.C. Escher Brief Biography.” Escher Pages. 27 January 2003. Retrieved 4 November 2008. <http://users.erols.com/ziring/escher_bio.htm>.

Pictures CitedSlide 1 – Background:

Artist: Carlos Gershenson <http://hawmkoonstormbringer.deviantart.com/art/Fractal-Tessellation-89159878>

Slide 5 – Roman Catholic Bull tilinghttp://tessellations.org/tess-what.htm

Slide 7 – Various Culture Tessellations

http://library.thinkquest.org/16661/gallery/index.html Slide 8 – Alhambra Tiling

Taschen. Escher, M.C. London: Taschen, 2006. Page 49.

Slide 10 – Alhambra Palacehttp://library.thinkquest.org/16661/gallery/26.html

Slide 11 – Escher’s “8 Heads”

http://www.tessellations.org/tess-escher4.htm

Slide 12 – Escher DrawingsEscher’s “Drawing Hands.” Taschen. Escher, M.C. London: Taschen, 2006. Page 174.Escher’s “Relativity.” Taschen. Escher, M.C. London: Taschen, 2006. Page 168.Escher’s “Sky and Water.” Taschen. Escher, M.C. London: Taschen, 2006. Page 56.

Slide 14 – Types of Tessellations<http://library.thinkquest.org/16661/of.regular.polygons/index.html?tqskip1=1&tqtime=0709>.

Slide 15 – Regular Tessellations<http://mathforum.org/sum95/suzanne/whattess.html>.

Slide 16 – Non-Regular Tessellations<http://mathforum.org/sum95/suzanne/whattess.html>.

Pictures CitedSlide 18 – Translation

Translation Flash Movie – <http://www.math.nus.edu.sg/aslaksen/gem-projects/maa/0203-2-03-Escher/main2.html#Principals>.Escher’s “Flying Horses.” - Taschen. Escher, M.C. London: Taschen, 2006. Page 113.

Slide 19 – Rotation

Rotation Flash Movie – <http://www.math.nus.edu.sg/aslaksen/gem-projects/maa/0203-2-03-Escher/main2.html#Principals>.Escher’s “Humans.” – <http://www.math.nus.edu.sg/aslaksen/gem-projects/maa/0203-2-03-Escher/main3.html>.

Slide 20 – Reflection

Reflection Flash Movie – <http://www.math.nus.edu.sg/aslaksen/gem-projects/maa/0203-2-03-Escher/main2.html#Principals>. Slide 21 - Glide Reflection

Glide Reflection Flash Movie – <http://www.math.nus.edu.sg/aslaksen/gem-projects/maa/0203-2-03-Escher/main2.html#Principals>.Escher’s “Horseman.” - - Taschen. Escher, M.C. London: Taschen, 2006. Page 103.

Slide 22 – Rabbit Tessellation

Rabbit Tessellation Flash Movie - <http://bardaux.deviantart.com/art/Rabbit-Tesselation-12930458>. Slide 23 – Real World Examples

Honeycomb - <http://www.prsd.k12.pa.us/hs/tessellations%20revised.ppt>.Fish Scales - <theartofnature.org/id20.html>.Chain Link Fence - <houstonchainlink.com/>.Soccer ball - <www.barrowga.org/rec/soccer/u6.html>.

Slide 25 – Activity Examples

SemiRegular Pictures – <http://commons.wikimedia.org/wiki/Special:Search?search=Tiling+Semiregular&go=Go>.DemiRegular Pictures - <http://www.prsd.k12.pa.us/hs/tessellations%20revised.ppt>.

Slide 26 – Integrating Tessellations

Venn Diagram - Hope, Martin. Integrating Mathematics Across the Curriculum. Arlington Heights, Ill: IRI/Skylight Training and Publisher, 1996. Chapter 5.