Transformations, symmetries, and tiling

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Guide to Transformations, Symmetries, & Tiling A parents guide to helping their children with Geometric transformations, symmetries, and tiling

Tiling (and Escher-like designs)

Video about Escher designs Maurits Cornelius Escher (1898-1972) was a Dutch artist. He was famous for his bird designs he created. He would first start with a parallelogram and replace an edge of the parallelogram with a curve and translate this curve to the opposite side of the parallelogram, and then on the opposite parallel sides, make a curve and translate it to the other side.

The video gives an overview about Escher and the mathematics behind his artwork.

Tiles and Tiling Definition: Tiles are simple closed curves together with its interior. Tiling is the result of individual tiles covering a shape placed so their interior never overlaps. Tiling is also called Tesselations.

Tiling with Congruent Polygonal Shapes:

A plane cannot be tiled by any convex tile with seven or more sides. Here are the ways a plane can be tiled:

1. Any triangular tile

2. Any quadrilateral tile- convex or not

3. Certain pentagonal tiles (those with two parallel sides) 4. Certain hexagonal tiles (those with two opposite parallel of the same length)

Example of tiling with congruent polygons

"Imperfect Congruence." Http://gruze.org/tilings/durer. Drupal, n.d. Web.

Semiregular Tiling:

An edge to edge tiling with more than one regular polygon and identical vertex figures. The same types of polygons must surround each vertex, and they must occur in the same order. There are 18 ways of forming a vertex using 2 or more regular polygons, but only 8 ways complete a full semiregular tiling.

Example of semiregular tiling

https://upload.wikimedia.org/wikipedia/commons/thumb/c/c6/Tiling_Semiregular_3-4-6-4_Small_Rhombitrihexagonal.svg/2000px-Tiling_Semiregular_3-4-6-4_Small_Rhombitrihexagonal.svg.png

Regular Tiling: There are only 3 shapes that allow regular tiling: 1. equilateral triangles 2. squares 3. regular hexagon

An arrangement of nonoverlapping polygonal tiles forming around a vertex figure. The measure of the interior angles must add up to be 360 degrees. (i.e. a square has 90 degree angles so, 90 + 90 + 90 + 90 = 360). An equation can be used (n = # of sides): (n - 2)x180/n So, if you use a pentagon (5 sides), you would get (5 - 2)x180/5 = 108 degrees, the degree of vertex for each pentagon, but if you add 108 degrees even 3 times, you get 324 degrees, not leaving room for 2 more additions of 108 degrees to complete the 5 sides/vertices meeting, meaning there is a significant overlapping, making a pentagon not possible for regular tiling.

Regular hexogon tiling

https://upload.wikimedia.org/wikipedia/commons/thumb/5/53/Tiling_Regular_6-3_Hexagonal.svg/270px-Tiling_Regular_6-3_Hexagonal.svg.png

Square tiling

http://math.ucr.edu/home/baez/dodecahedron/square_tiling.png

Equilateral triangle tiling

http://i.stack.imgur.com/0vFd0.png

Transformational Geometry

Video explaining 4 types of Rigid Motion

Lee, Florence. "Geometric Transformations Presentation." YouTube. YouTube, 14 June 2012. Web. 09 July 2015.

Rigid Motion: Every pair of points P and Q moves to P' and Q' in a way that leaves the distance unchanged; that is, PQ = P'Q'. This does not allow for shrinking or stretching. There are 4 basic types. Rigid Motion is also called isometry, meaning "same measure"

4. Glide-reflection or Glide:

A combination of both a translation or slide and reflection or flip. Slides a fixed distance and reflects over a determined line.

Glide-Reflection Example

Kelvinsong. "File:Glide Reflection.svg." Wikipedia. Wikimedia Foundation, n.d. Web. 09 July 2015

3. Reflection or Flip:

All points P are transformed to the opposite side P', but the same distancd away from a determined line.

Reflection/Flip Example

2. Rotation or Turn:

One point on the plane, called the turn center or the center of rotation, is fixed. The remaining points turn about the center of rotation through the same number of degrees- the Turn Angle or Angle of Rotation.

Rotation/Turn Example

Center of Rotation. Digital image. North Texas Institute for Educators on the Visual Arts. N.p., n.d. Web.

1. Translation or Slide:

All points of the plane are moved the same distance in the same direction. An arrow drawn from a point P to its image point P': the direction of the slide is the direction of the arrow, and the distance moved is the length of the arrow. The arrow is known as the slide arrow or translation vector.

Translation/Slide Example

Translation or Slide. Digital image. Emathematics.net. N.p., n.d. Web.

Definition: Every point P is moved from a starting position to a final position P'.

Resources

"Border Design." Artlandia. N.p., n.d. Web. 09 July 2015.

Center of Rotation. Digital image. North Texas Institute for Educators on the Visual Arts. N.p., n.d. Web.

DeTemple, and Millman. "Chapter 11: Transformations, Symmetries, and Tilings." Mathematical Reasoning for Elementary Teachers Plus New Mymathlab With Pearson Etext Access Card. By Long. 7th ed. N.p.: Pearson College Div, 2014. N. pag. Print.

"Imperfect Congruence." Http://gruze.org/tilings/durer. Drupal, n.d. Web.

Kelvinsong. "File:Glide Reflection.svg." Wikipedia. Wikimedia Foundation, n.d. Web. 09 July 2015

Lee, Florence. "Geometric Transformations Presentation." YouTube. YouTube, 14 June 2012. Web. 09 July 2015.

"Reflection Symmetry." CK-12 Foundation. N.p., n.d. Web. 09 July 2015.

"Rotational Symmetry." Hotmath. N.p., n.d. Web. 09 July 2015.

Translation or Slide. Digital image. Emathematics.net. N.p., n.d. Web.

http://i.stack.imgur.com/0vFd0.png

http://math.ucr.edu/home/baez/dodecahedron/square_tiling.png

https://upload.wikimedia.org/wikipedia/commons/thumb/5/53/Tiling_Regular_6-3_Hexagonal.svg/270px-Tiling_Regular_6-3_Hexagonal.svg.png

https://upload.wikimedia.org/wikipedia/commons/thumb/c/c6/Tiling_Semiregular_3-4-6-4_Small_Rhombitrihexagonal.svg/2000px-Tiling_Semiregular_3-4-6-4_Small_Rhombitrihexagonal.svg.png

https://s-media-cache-ak0.pinimg.com/236x/ff/b9/83/ffb983ae1d7dd59607c7196bd6a2407b.jpg

Symmetry

Video about Symmetry

This video is geared for kids. It's a song about the different symmentries. It would be great for parents to use while working with their kids. The videos that were for adults were very long, dry, and too indepth for refreshers for parents to help their child.

Periodic Patterns:

These have at least one translation symmetry pattern. Common in motifs such as wallpapers, decorative brick walls, printed and woven fabrics, ribbons, and ceiling borders (also called friezes) in older buildings. There are 2 types of Periodic Patters- Border Patterns and Wallpaper Patterns.

Wallpaper Pattern:

A motif translated in 2 nonparallel directions.

Example of Wallpaper Pattern

https://s-media-cache-ak0.pinimg.com/236x/ff/b9/83/ffb983ae1d7dd59607c7196bd6a2407b.jpg

Border Patterns:

A repeated motif that has been translated in just one direction. There are two symbol names assigned to each pattern by the "International Crystallographic Union": First Symbol: m, if there is a vertical line of symmetry. 1, otherwise. Second Symbol: m, if there is a horizontal line of symmetry. g, if there is a glide-reflection symmetry (but no horizontal line of symmetry) 2, if there is a half-turn symmetry (but no horizontal or glide-reflection symmetry). 1, otherwise.

Example of Border Pattern

"Border Design." Artlandia. N.p., n.d. Web. 09 July 2015.

Point Symmetry:

Has symmetry when rotated 180 degrees around the center of symmetry.

The BLOCK letters H, I, N, O, S, X, & Z are examples of Point Symmetry.

H I N O S X Z

Rotational Symmetry (Also called Turn Symmetry):

The figure is superimposed (or spun around a center point) on itself when it is rotated through a certain degree (0 degrees-360 degrees). Rotation occurs around the Center of Symmetry.

Example of Rotational Symmetry

"Rotational Symmetry." Hotmath. N.p., n.d. Web. 09 July 2015.

Reflection Symmetry (Also called Bilateral Symmetry or Line Symmetry:

The reflection across some line, the line that is also called Line of Symmetry or a Mirror Line.

Example of Reflection Symmetry

"Reflection Symmetry." CK-12 Foundation. N.p., n.d. Web. 09 July 2015.

Definition: Any rigid motion of the plane that moves all the points of the figure back to points of the figure. There can be Horizontal, Vertical, and many more!

Games! Here I have put a couple of links to sites that have games about gemetric transformations, symmetries, and tiling. I am only allowed one hyperlink attachment, so the web address is in the notes section.

1. www.sheppardsoftware.com/mathgames/geometry/shapeshoot/TranslateShapesShoot.htm

^^This game is fairly basic, but a great way to keep reminders about what each transformation is. Parents could work with their child on this game.

2. http://www.mathplayground.com/index_geometry.html

^^This site has LOTS of different games geared around geometry and has options for tiling, transformations, and symmetry!