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# Tessellations

Created By Cisem, Catherine, and Myles
by

## Catherine Cho

on 17 October 2012

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#### Transcript of Tessellations

Tessellation Project
BY: Myles, Cisem, and Catherine Polygon
Tessellation Non-Polygonal
Tessellation

Step 1: Choose A Starting Shape
First start by choosing a shape. We chose a hexagon for our starting shape because it
is able to tessellate on its own (Part 1). Adding hexagons together allows for the sum of interior angles around each vertice to equal 360 degrees. Step 2: Begin by adding other regular polygons of your choice to your main figure to make your total angle measurement sum closer to the goal of 360 degrees. In this case, we added squares. Step 3: Next we continued by adding squares to each side.
We did this so all the sides of the hexagon would become equal and eventually be able to tessellate evenly in all directions. It looks like a bunny Step 6: Continue repetitiously placing the correct shapes on sides. By doing, eventually you will create a series of reoccurring figures continuing in all directions. This is called a tessellation. looks like carrots! (: Step 8: Finish up your last couple of shapes, and then decorate it. Be creative! We decorated ours to be cute and adorable (: Add squares to every available side on the polygon
so the angles around each shared vertice have
the same sum. You must do this in order
for the figure to tessellate. Step 4: To make up for the 60 degrees that was lacking from our 300 degrees, we added an equilateral triangle. This allowed the angles around each vertice to have a sum of 360 degrees, which they must have in order to tessellate. Step 5: Continue adding triangles so each set of angles around each vertice are the same. You want them to all be identical so that the combination of all the shapes create a regular polygon. By doing this you create something you can tessellate in all directions. What is a tessellation? CREATING A POLYGONAL TESSELLATION The interior angle of
a hexagon measures 120 degrees.
To form a tessellation, the total sum of the angles around a single point needs to be 360 degrees, so subtract 120 from 360 and 240 degrees is the difference, meaning that you still need to complete 240 degrees. The interior angles of square are 90 degrees. When you add this to the 120 degrees from the hexagon, you get a total of 210 degrees. You then subtract that from 360 and get a total of 150 degrees still needed to complete 360. Next you add another square with
an interior angle measurement of 90 degrees
and add that to the tessellation. The sum
of the interior angles around the chosen vertice
is now 300 degrees. This means we need
add a shape with an interior angle of 60
degrees to have the sum of interior angles
equal to 360 degrees. What is a
tessellation? What is a tessellation?
(continued)

Non-polygonal tessellations are tessellations that do not consist of polygons. This means that the shapes that are being tessellated do not have to have straight sides, however gaps and spaces remain unpresent. In order to make a non-polygonal tessellation, you can use several methods including:
A. Translation/Slide
B. Rotation/Turn
C. Rotation at midpoints

A) When you create a transitional / slide tessellation, you need to start with a polygon, who’s opposite sides are parallel and congruent, and create a line design on a side in which you wish to tessellate to. Then cut along that line and slide the detached part across the figure and tape it to the opposite side. This is now your stencil in which you need to trace onto another sheet of paper. After you finish tracing, slide and align the stencil, and trace again. Repeat these steps until your paper has been covered with your tessellation design.

B) To create a rotational/turn tessellation you must start with a polygon, whose adjacent sides are congruent to each other. On one side of the polygon, draw a designed line, in which your figure will tessellate to. Next, cut along the line that you have drawn and pick a vertice in which the figure can be rotated on. Take the detached part and rotate it 90 degrees until it is aligned with adjacent side and tape the part in place. This is now your stencil in which you need to trace onto another sheet of paper. When you are tracing with a rotational stencil, make sure you are rotating the stencil as your trace, to align your tessellation. Repeat these steps until your paper has been covered with your tessellation design.

C) Start with a polygon whose adjacent sides are congruent. Pick a side and find it’s midpoint, you can do this by using a ruler and finding the point exactly in the middle of the line segment. Draw a line design between one of line segment endpoints and the midpoint. Next, rotate that new line design 180 degrees around the midpoint to the other half of the same side. Repeat the same full line design on all sides of your polygon. Cut along the newly designed lines, creating your stencil in which you should trace onto another sheet of paper. This will create a tessellation using the midpoint rotation method. When creating a polygon tessellation with more than 2 polygons, you must think about every possible shared vertice that can be made when tessellating those 2 figures. 1. Define tessellation:
A tessellation is a pattern of shapes that cover a plane without gaps or overlaps.

2. Do all polygons tessellate? Why or why not?
Not all polygons can tessellate a plane because in order for a polygon to tessellate, all angles at a shared vertice must complete a full circle rotation of 360.

3. Define and explain polygon tessellations and give at least 2 examples.

Polygon tessellations are tessellations that contain polygons, closed shapes with at least 3 angles and 3 straight sides, in which their interior angles all add up to a 360 degree circle at the shared vertice, without gaps or spaces in between.

Some polygons can easily tessellate to create a tessellation, such as an equilateral triangle or a square. However, some polygons, such as pentagons, cannot tessellate. This is because in order to have a polygon tessellate a plane, you must have a 360°circle rotation. (#2)
You can see that each interior angle of a triangle measures 60°, and it takes 6 interior angles to make it equal 360°. The same thing applies to squares, 90 + 90 + 90+ 90 = 360
And hexagons. 120+ 120 + 120= 360.
In other words the measurement of the interior angle should be divisible by 360.

EX: hexagon
(180(n-2))/n
( 180(6-2))/6
(180(4))/6
(720)/6
120

360/120 = 3
3 hexagons complete the full circle rotation of 360°
http://www.mochimochiland.com/blogimages/dinotessellation.jpg

http://nrich.maths.org/4832/note

http://blog.learningtoday.com/Portals/60233/images//Tessellation.jpg

http://gwydir.demon.co.uk/jo/tess/bexample8.gif

http://paulscottinfo.ipage.com/polyhedra/tessellations/semiregulartess.html

http://library.thinkquest.org/16661/simple.of.regular.polygons/regular.1.html

http://www.angelfire.com/space/dianajhunter/index_files/Page298.html Sources: It is evident we used the rotation/turn method to create the non-polygonal tessellation. Our adjacent sides were congruent (equilateral triangle) and we also rotated the newly drawn side 90 degrees on a vertice, which defines this method. Method Used Thanks for Watching! (: CISEM+ CATHERINE+ MYLES
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