QUADRILATERALS

A quadrilateral is a polygon with four sides. It has:

• Four sides (edges)
• Four vertices (corners)
• The interior angles add up to 360 degrees
• Two diagonals: a diagonal of a quadrilateral is a line segment whose end-points are opposite vertices of the quadrilateral. In picture below, ABCD is a quadrilateral, AC, BD are the two diagonals.

We name a quadrilateral by naming the four vertices in consecutive order. So we can name the quadrilateral as ABCD.

TYPES OF QUADRILATERALS

1. Parallelograms (paralelogramos)

A parallelogram is a quadrilateral where both pairs of opposite sides are parallel. Also opposite angles are equal.

Types of parallelograms

 Square
 All sides equal, all angles 90°, the diagonals are equal and perpendicular.

 Rectangle
Opposite sides equal, all angles 90°, the diagonals are congruent but they are not perpendicular. 

Rhombus
Opposite sides parallel and equal, the diagonals are not equal but they are perpendicular.

Rhomboid
A rhomboid has opposite sides that are parallel and equal.

2. Trapezium (Trapecios- trapezoid US)

A trapezium is a quadrilateral with a pair of parallel sides.

The pair of parallel sides (SR||PQ) are called the bases of the trapezium, and the non-parallel sides (PS,QR) form the legs of the trapezium. The distance between the bases or the trapezium is called the height or the altitude. 

Types of trapeziums

If we have two right angles and two parallel sides, then we have an

rectangle trapezium

 

If the two legs of the trapezium are congruent to each other, then we have an isoceles trapezium

If the four sides or the trapezium have different lengths, then we have a scalene trapezium

 

3. Trapezoid (trapezoide)

A trapezoid (trapezium in US) is a quadrilateral with NO parallel sides

You can find a overview abour quadrilaterals here: 

TESSELLATIONS

TESSELLATIONS
The tessellations are patterns made of identical shapes:
• the shapes must fit together without any gaps
• the shapes should not overlap


REGULAR TESSELLATIONS

A regular tessellation is a pattern made by repeating a regular polygon. It is a regular division of the plane.
There are only 3 regular tessellations:

                Squares                    Triangles                   Hexagons

SEMI-REGULAR TESSELLATIONS
A semi-regular tessellation is made of two or more regular polygons. The pattern at each vertex must be the same!
There are only 8 semi-regular tessellations:

MODULE
A module is a regular or irregular form that makes a uniform surface by repeating itself a specific number of times and in a certain order.

In this image the module is the octopus

  

The Nazari tessellations are created by the transformation of a basic geometric shape. The final figure has the same surface as the original but the shape is different.

 

TRANSFORMATIONS

To modify the modules we can use three concepts:

-Translation
- Reflection
- Rotation 

1. Translation

To translate means to move a figure to a new location with no other changes.

2. Reflection

The reflection is a transformation where each point in a shape appears at an equal distance on the opposite side of a given line - the line of reflection.

 3. Rotation

A transformation where a figure is turned about a given point.

HOW TO CONSTRUCT TESSELLATIONS

To draw your own tessellation you can start with a square (or other basic geometrical shape) and make the transformations. 

1. Translation

Click on the picture to practise this method.

2. Reflection

  3. Rotation

 M.C. Escher

Maurits Cornelis Escher (1898-1972) was a great artist who studied deeply this technique.

He became fascinated by the regular Division of the Plane, when he first visited the Alhambra in 1922. He made 137 Regular Division Drawings in his lifetime.

You can find more artworks here

http://www.mcescher.com/gallery/symmetry/ 

And now you can visit an exhibition in Madrid: 

http://www.rtve.es/noticias/20170201/mundos-imposibles-escher-llegan-madrid/1482166.shtml

 LET'S MAKE TESSELLATIONS!

1. Start with a square
2. Draw a design on of side of the square
3. Cut  the design piece out and make one of the three types of transformations (or the three)
4. Tape it tho the square
5. Trace the design on your white paper until it is covered completely

Marking criteria:
- Pattern tessellates the plane
- Module or template with modifications (at least 1 type)
- Neatness
- Creatitivity
- Effort