# data[].attributes.body
Geometry for Elementary School/Transformation
Wikibooks
CC by SA 3.0 License

# data[].attributes.attachment_link.readable_text
Transformation is when we change the size, orientation, and/or position of a shape. Note that transformation is usually done on graph paper to avoid excessive meaurements and ensure accuracy.

Reflection is when a shape is reflected along an axis to produce a reflectionally symmetrical figure. The axis of reflection is also the axis of symmetry of the new figure.

Look at the diagram on the right. Imagine you are given the left part. How can you reflect the figure? First, Find out the distance between A and the axis. That's four. Then find point A′ (pronounced 'A prime'), which should be the same distance from the axis but on the different side. Look at the figure that is on the right of the figure. If you found that point, you're right. Put a little cross there - if your teacher doesn't let you do it, just erase it later! Now do the same for the other two points. Join them up. Now you have formed the reflected figure!

A common mistake while reflecting a figure is forgetting to mark the points. That costs you a lot in marks, so don't make this mistake! Also, never mark the point wrongly. Remember to add the ' ′ ' symbol and check if the points are corresponding!

Rotation is the most difficult kind of transformation. It requires rotating a figure with reference to a single point. Therefore, you will only be asked to rotate 90°, 180° and 270° at this stage. (360° is meaningless, but if you do manage to come across it you're very lucky!) There are three things that we need to note before rotating a figure.

The number of degrees to rotate

Whether you should rotate something clockwise or anticlockwise

Where the centre of rotation is

Let us look at the example on the left. Imagine we are only given the top, left one. We need to rotate it in a number of ways.

1. 90° anticlockwise through the point in the middle

2. 180° anticlockwise through the point in the middle

3. 270° anticlockwise through the point in the middle

4. 90° clockwise through the point in the middle

5. 180° clockwise through the point in the middle

6. 270° clockwise through the point in the middle

Look at #1 and #6. If you look at them carefully, you should be able to see that #1 and #6 are actually the same! This can be explained through the example. To translate triangle ABC 90° anticlockwise through the point in the middle, we can use point A first. Point a is three squares to the left and one square to above the centre of rotation, so we can rotate that 90 degrees. This cannot be explained in words, so try looking at the figure on the right. As you can see, A and A′ are the same distance from the centre of rotation. We do the same to the other two points, producing a triangle that looks like the one in the figure. If we do the same for number 6, you will produce an identical triangle! The same goes for points number 3 and 4.

What about 2 and 5? As we can see, they both involve translating 180°; however, one is clockwise and the other is anticlockwise. Let's try rotating it clockwise first. You should get the triangle A′′B′′C′′ in the figure. Then do the same for the anticlockwise. Do they look the same?

Now that we have tried rotating 6 possibilities from one centre of rotation, and figuring out there is only three, can you try rotating triangle ABC through point A? If you are reading this as an e-book, please copy out triangle ABC on a piece of graph paper. If you are using the print version, you can draw the new triangle directly on your book. Name your new triangle DEF.

Translation is a simple transformation. Put simple, translation is the change of the position of a shape. For example, if we want to transform a figure five units to the left, then we just move it five units to the left.

The process of translating is very easy. Imagine we have a right-angled triangle ABC on a piece of graph paper. Our task is to translate it four squares upwards and two squares to the left. We take the vertex of the right angle, named A, as our point, move it four squares upwwards, then two squares to the left. Draw a little cross there. and mark it A′. Now we re-create the shape by referring to the original shape. Remember to name the points correctly.

We are often asked to trace back the translation gone through in a given translation. We are given the original figure and the new figure. When dealing with this type of questions, it may be helpful to use a point like we did above. Take the same right-angled triangle. We see that A has been translated four squares upwards. So we write: Translate A upwards four squares. Then we see that A has been translated two squares to the left, so we write, then translate A two squares to the left. Then we have finished the question!

The following table shows what things are changed when a transformation is gone through.

Note: For all of these changes the shape never changes. A triangle will always be a triangle, a rectangle will always be a rectangle.

# data[].attributes.attachment_link.description
Transformation is when we change the size, orientation, and/or position of a shape. Note that transformation is usually done on graph paper to avoid excessive meaurements and ensure accuracy. Reflection is when a shape is reflected along an axis to produce a reflectionally symmetrical figure.

# data[].attributes.body
Maisonet Math
© 2020 Maisonet Math

# data[].attributes.attachment_link.description
Watch and learn how to work with different types of transformations.

# data[].attributes.body
Common Types of Transformation
Math Planet
CC by NC-ND License

# data[].attributes.attachment_link.readable_text
Any image in a plane could be altered by using different operations, or transformations. Here are the most common types:

Translation is when we slide a figure in any direction.

Reflection is when we flip a figure over a line.

Rotation is when we rotate a figure a certain degree around a point.

Dilation is when we enlarge or reduce a figure.

Video lesson

Determine which of the triangles that is a reflection, a rotation and a dilation

# data[].attributes.attachment_link.title
Common types of transformation (Geometry, Transformations) - Mathplanet

# data[].attributes.attachment_link.description
Any image in a plane could be altered by using different operations, or transformations. Here are the most common types: Translation is when we slide a figure in any direction. Reflection is when we flip a figure over a line. Rotation is when we rotate a figure a certain degree around a point.

# data[].attributes.body
Chris Hansford
CLTC maths: transformations
CC by Attribution
https://www.geogebra.org/m/yEkNc3VF

# data[].attributes.body
How Tessellations Work
HowStuffWorks
Copyright © 2020

# data[].attributes.attachment_link.readable_text
We study mathematics for its beauty, its elegance and its capacity to codify the patterns woven into the fabric of the universe. Within its figures and formulas, the secular perceive order and the religious catch distant echoes of the language of creation. Mathematics achieves the sublime; sometimes, as with tessellations, it rises to art.

Tessellations -- gapless mosaics of defined shapes -- belong to a breed of ratios, constants and patterns that recur throughout architecture, reveal themselves under microscopes and radiate from every honeycomb and sunflower. Pick apart any number of equations in geometry, physics, probability and statistics, even geomorphology and chaos theory, and you'll find pi (π) situated like a cornerstone. Euler's number (e) rears its head repeatedly in calculus, radioactive decay calculations, compound interest formulas and certain odd cases of probability. The golden ratio (φ) formed the basis of art, design, architecture and music long before people discovered it also defined natural arrangements of leaves and stems, bones, arteries and sunflowers, or matched the clock cycle of brain waves [sources: Padovan, Weiss, Roopun]. It even bears a relationship to another perennial pattern favorite, the Fibonacci sequence, which produces its own unique tiling progression.

Science, nature and art also bubble over with tessellations. Like π, e and φ, examples of these repeating patterns surround us every day, from mundane sidewalks, wallpapers, jigsaw puzzles and tiled floors to the grand art of Dutch graphic artist M.C. Escher, or the breathtaking tile work of the 14th century Moorish fortification, the Alhambra, in Granada, Spain. In fact, the word "tessellation" derives from tessella, the diminutive form of the Latin word tessera, an individual, typically square, tile in a mosaic. Tessera in turn may arise from the Greek word tessares, meaning four.

Mathematics, science and nature depend upon useful patterns like these, whatever their meaning. Beyond the transcendent beauty of a mosaic or engraving, tessellations find applications throughout mathematics, astronomy, biology, botany, ecology, computer graphics, materials science and a variety of simulations, including road systems.

In this article, we'll show you what these mathematical mosaics are, what kinds of symmetry they can possess and which special tessellations mathematicians and scientists keep in their toolbox of problem-solving tricks.

First, let's look at how to build a tessellation.

# data[].attributes.attachment_link.description
We study mathematics for its beauty, its elegance and its capacity to codify the patterns woven into the fabric of the universe. Within its figures and formulas, the secular perceive order and the religious catch distant echoes of the language of creation. Mathematics achieves the sublime; sometimes, as with tessellations, it rises to art.

# data[].attributes.headline
Basic Translations, Reflections and Rotations

# data[].attributes.subject
Basic Translations, Reflections and Rotations

# data[].attributes.body
Types of Transformations
Kyle Webb
CC by Attribution

# data[].attributes.body
Transformations
Khan Academy
© 2020 Khan Academy

# data[].attributes.attachment_link.title
Transformations | Geometry (all content) | Math | Khan Academy

# data[].attributes.attachment_link.description
In this topic you will learn about the most useful math concept for creating video game graphics: geometric transformations, specifically translations, rotations, reflections, and dilations. You will learn how to perform the transformations, and how to map one figure into another using these transf…

# data[].attributes.body
Tessellations and Symmetries
Malin Christersson
CC by NC-SA License

# data[].attributes.attachment_link.description
An interactive tessallation, a comparison between rotations and reflections, and interactive reflections in circle.

# data[].attributes.headline
Rotations around the origin or a given point

# data[].attributes.subject
Rotations around the origin or a given point

# data[].attributes.body
Transformations and Rotations
FuseSchool- Global Education
CC by Attribution

# data[].attributes.attachment_link.title
Transformations & Rotations | Geometry | Maths | FuseSchool

# data[].attributes.headline
Introduction Video of Congruent Transformations

# data[].attributes.subject
Introduction Video of Congruent Transformations

# data[].attributes.body
What are transformations and the different types
Brian McLogan
CC by NC

# data[].attributes.attachment_link.title
What are transformations and the different types

# data[].attributes.body
Mathematics for Elementary Teachers
Michelle Manes
CC by SA 4.0 License

# data[].attributes.attachment_link.readable_text
A tessellation is a design using one ore more geometric shapes with no overlaps and no gaps. The idea is that the design could be continued infinitely far to cover the whole plane (though of course we can only draw a small portion of it).

Many tessellations have translational symmetry, but it's not strictly necessary. The Penrose tiling shown below does not have any translational symmetry.

It's actually much harder to come up with these "aperiodic" tessellations than to come up with ones that have translational symmetry. So we'll focus on how to make symmetric tessellations.

The first two tessellations above were made with a single geometric shape (called a tile) designed so that they can fit together without gaps or overlaps. The third design uses two basic tiles. Tessellations are often called tilings, and that's what you should think about: If I had tiles made in this shape, could I use them to tile my kitchen floor? Or would it be impossible?

On Your Own

Work on these exercises on your own or with a partner. You will need lots of copies (maybe 10-15 each) of each shape below. In each problem, focus on just a single tile for making your tessellation.

1. Start with the square tile. Can you fit the squares together in a pattern that could be continued forever, with no gaps and no overlaps? Can you do it in more than one way?

2. Now try one of the triangular tiles. Can you use many copies of a single triangle to tessellate the plane?

3. Repeat this process with each of the other tiles. Keep track of your findings.

Think / Pair / Share

Which of the tiles given above tessellate, and which do not?

Do you have any conjectures based on this experience, about which shapes will tessellate and which will not?

The artist M.C. Escher created many works of art inspired by mathematics, including some very beautiful tessellations. Below you will see some images inspired by his work. You can view the real thing at http://www.mcescher.com/ in the "Symmetry" gallery.

You can make your own Escher-like drawings using some facts that you learned while studying tessellations.

Theorem: Tessellations

Any triangle will tessellate. So will any quadrilateral.

The explanation for this comes down to what you know about the sums of angles. The sum of the angles in a triangle is 180°.

So if you make six copies of a single triangle and put them together at a point so that each angle appears twice, there will be a total of 360° around the point, meaning the triangles fit together perfectly with no gaps and no overlaps.

You can then repeat this at every vertex, using more and more copies of the same triangles.

Think / Pair / Share

Use the fact that the sum of the angles in any quadrilateral is 360° to explain why every quadrilateral will tessellate.

Use angles to explain why regular hexagons will tessellate.

Explain why regular pentagons will not tessellate.

On Your Own

Work on the following exercises on your own or with a partner. Here's how you can create your own Escher-like drawings.

1. Select your basic tile. The first time you do this, it's easiest to start with a simple shape that you know will tessellate, like an equilateral triangle, a square, or a regular hexagon.

2. Draw a "squiggle" on one side of your basic tile.

3. Cut out the squiggle, and move it to another side of your shape. You can either translate it straight across or rotate it.

or

4. It's important that the cut-out lines up along the new edge in the same place that it appeared on its original edge.

5. Tape the squiggle into its new location. This is your basic tile. On a large piece of paper, trace around your tile. Then move it the same way you moved the squiggle (translate or rotate) so that the squiggle fits in exactly where you cut it out.

6. The shape will still tessellate, so go ahead and fill up your paper.

7. Now get creative. Color in your basic shape to look like something - an animal? a flower? a colorful blob? Add color and design throughout the tessellation to transform it into your own Escher-like drawing.

8. If you want to try a more complicated version, cut two different squiggles out of two different sides, and move them both.

For this activity, you will need some construction materials:

You'll need lots of toothpicks.

You'll also need something to connect the toothpicks together. The best material for this is mini marshmallows; you can stick the ends of the toothpicks into the marshmallows to connect them. You can also use pieces of clay, bits of gummy candies, or other similar (sticky) material.

Problem 16

Try this as a warm-up activity. Grab exactly six toothpicks. Your job is to make four triangles using all six toothpicks. You cannot break any of the toothpicks or add any other materials besides the marshmallow connectors.

Problem 17

Now comes the main challenge. You have ten minutes to build the tallest free-standing structure that you can make. "Free-standing" means that it will stand up on its own. You can't hold it up or lean it against something. When the ten minutes are up, back away from your tower and measure its height.

Think / Pair / Share

Look at your own tower and at other students' towers. Talk about these questions:

What design choices led to taller free-stranding structures? Why do you think that is?

If you had another ten minutes to try this activity again, what would you do differently and why?

# data[].attributes.attachment_link.description
A tessellation is a design using one ore more geometric shapes with no overlaps and no gaps. The idea is that the design could be continued infinitely far to cover the whole plane (though of course we can only draw a small portion of it). Many tessellations have translational symmetry, but it's not strictly necessary.