Pure mathematics is, in its way, the poetry of logical ideas. --- Albert Einstein ...Economics is nothing but Mathematics -Dr.Ahsan Abbass...Symmetry is Ornament of Mathematics-Zulfiqar Ali Mir...Law of Nature are But Mathematical Thoughts of God - Euclid (Father of Geometry)...Mathematics is about Number and pattern among these nos.-Sir Zulfiqar A Mir,... Number Theory is Foundation of Mathematics-Sir Zulfiqar Ali Mir

Sunday 16 October 2011

Transformations

Transformation involves moving an object from its original position to a new position. 

The object in the new position is called the image. Each point in the object is mapped to another point in the image. 


The three main Transformations are:

Rotation Turn!
Reflection Flip!
Translation Slide!

After any of those transformations (turn, flip or slide), the shape still has the same size, area, angles and line lengths.
If one shape can become another using Turns, Flips and/or Slides, then the two shapes are called Congruent.

Resizing

The other important Transformation is Resizing (Dilation) (also called dilation, contraction, compression, enlargement or even expansion). The shape becomes bigger or smaller:
Resizing

If you have to Resize to make one shape become another then the shapes are not Congruent, but they are Similar.

Congruent or Similar

So, if one shape can become another using transformation, the two shapes might be Congruent or just Similar
If you ... Then the shapes are ...
... only Rotate, Reflect and/or Translate 

Congruent

... need to Resize

Similar



Rotation

"Rotation" means turning around a center:

The distance from the center to any point on the shape stays the same.
Every point makes a circle around the center.


Here a triangle is rotated around
the point marked with a "+"


Reflection


the reflected image is always the same size, it just faces the other way:
Left-Right Reflection 75 Degrees Reflection

A reflection is a flip over a line

How Do I Do It Myself?

Just approach it step-by-step. For each corner of the shape:
1. Measure from the point to the mirror line (must hit the mirror line at a right angle) 2. Measure the same distance again on the other side and place a dot. 3. Then connect the new dots up!
Step 1 Step 2 Final Reflect

Labels

It is common to label each corner with letters, and to use a little dash (called a Prime) to mark each corner of the reflected image.
Here the original is ABC and the reflected image is A'B'C'
Mirror Image Prime

Some Tricks

Reflect About X Axis

X-Axis

If the mirror line is the x-axis, just change each (x,y) into (x,-y)

Y-Axis

If the mirror line is the y-axis, just change each (x,y) into (-x,y)
Reflect About Y Axis

Fold the Paper

And if all else fails, just fold your sheet of paper along the mirror line and then hold it up to the light !

Reflection Symmetry

Reflection Symmetry (sometimes called Line Symmetry or Mirror Symmetry) is easy to recognise, because one half is the reflection of the other half.





Rotational Symmetry

With Rotational Symmetry, the shape or image can be rotated and it still looks the same.
How many matches there are as you go once around is called the Order.

If you think of propeller blades (like below) it makes it easier.

Examples of Different Rotational Symmetry Order

Order Example Shape Artwork
... and there is also Order 5, 6, 7, and ...
... and then there is Order 9, 10, and so on ...

Is there Rotational Symmetry of Order 1 ?

Not really! If a shape only matches itself once as you go around (ie it matches itself after one full rotation) there is really no symmetry at all, because the word "Symmetry" comes from syn- together and metron measure, and there can't be "together" if there is just one thing.


 

W 

 

 

 

 

Real World Examples

A Dartboard has Rotational Symmetry of Order 10 The US Bronze Star Medal has Order 5



The London Eye has Order ... oops, I lost count!






























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