quadrilateral


last updated: 20060314 
The quadrilateral, quadrangle or tetragon
is a polygon with four sides.
There are many quadrilaterals to
distinguish. The tangent quadrilateral and the cyclic quadrilateral are special cases of
the tangent polygon and the cyclic polygon.
For the tangent quadrilateral the opposite sides are of equal length. A cyclic
quadrilateral for which the product of the opposite sides is the same, is called a harmonic quadrilateral.
A trapezium or trapezoid
is a quadrilateral with only two parallel sides. The trapezoid rule gives a way
to find the area under a give curve.
When the non parallel sides are equal in length, the figure is a isosceles trapezium. We see
it in the trapezium curve  a repetition of trapeziums  and in the
socalled trapezoid screwthread.
The parallelogram is a quadrilateral
for which the opposite sides are parallel. It can be seen as representing the vector sum
of two variables with length and direction (e.g. a force), resulting in the diagonal of
the parallelogram.
Anna Isabella Milbanke (who has been married to Lord Byron) had as nickname the
'princess of the parallelograms', for her interest in mathematics.
A crystal lattice in two dimensions is build of a repeating structure in two
directions, forming a parallelogram. This shows that a plane can be filled with
parallelograms.
The rectangle has only right ^{1)}
angles. It is a very handy form. How many objects have a rectangular form: the pages of a
book, this web page, the room I am sitting in, the piece of ground of our house, the
window I am looking through, the doors in my house: all rectangles. So while building a
house many right angles have to be constructed. We can make this construction easier when
we use the Pythagoras theorem, and make windows in a ratio of 3 to 4 for the height
related to the length. The right angle is easy to construct by pacing a diagonal of 5
units. In Holland in old farms such windows can be seen.
Also many other object have this rectangular form: writing paper, postcards, football
fields, paintings, tables, shelves. Package material as boxes consists often of
rectangles, what makes piling up more easy.
In perspective the rectangle seems deformed, for the eye. In such a way that parallel
lines  in the direction of the observer  meet each other in the socalled
vanishingpoint. Our brains can compensate for this deformation, and we will often be able
to recognize a rectangle.
In a perspective view the scale is not constant, so that is difficult to measure and
and count. An oblique projection, where a rectangle is shown as a parallelogram, is
easier for this purpose.
The golden section is a ratio between the sides of a rectangle, that looks quite natural.
We define this figure so that when you cut a square from the rectangle, the resulting
rectangle is similar to the original rectangle ^{2)} . We call
it the golden rectangle.
The rectangular numbers 2, 6, 12, 20 and so on can be defined analogous to the triangular
numbers:
and so on.
The
rectangle is repeated in the rectangular curve.
A rhombus is an oblique square: the four sides are equal
in length, but there are no right angles.
In Dutch a rhombus is called a 'ruit', a word which originated from the medieval
'ru(y)te', which also denoted the similar formed plant. A strange phenomenon is that some
rectangular objects have the rhombus in their Dutch name: this is the case for the words
for: pane, squared paper and tartan. The checker 'ruit' is a real rhombus, as is the
'wyber' (liquorice) and the Renault logo.
When the sides are in pairs equal in length, we see a kite.
The square is the regular quadrilateral. Of all rectangles
with equal circumference the square has the maximum area (what has been proved already by Euclid,
about 300 BC).
Lifting a square to three dimensions gives a cube.
The square numbers 1, 4, 9, 16 and so on, were for the Pythagoras group the
symbol of justice. The quadratic equation and the quadratic root are also based on the
power two.
A magic square is a matrix of numbers, where the sum of rows, columns and diagonals has
the same result.
A curve of constant width is a curve
that makes contact with all four sides, when rotated in a square. Such a curve
is also known as a roller.
Examples are:
A pentamino
is a curve formed by joining squares together.
notes
1) rectus (Lat.) = right.
The Dutch word 'loodrecht' (lead right, from: leadline) means: right (perpendicular).
2) The ratio has a value of about 0.62.
