The complex geometry of Islamic design - Eric Broug

In Islamic culture,
geometry is everywhere.

You can find it in mosques,
madrasas, palaces and private homes.

This tradition began in the 8th Century CE
during the early history of Islam,

when craftsman took preexisting motifs
from Roman and Persian cultures

and developed them into new forms
of visual expression.

This period of history was a golden age
of Islamic culture,

during which many achievements
of previous civilizations

were preserved and further developed,

resulting in fundamental advancements
in scientific study and mathematics.

Accompanying this was an increasingly
sophisticated use of abstraction

and complex geometry in Islamic art,

from intricate floral motifs
adorning carpets and textiles,

to patterns of tilework that seemed
to repeat infinitely,

inspiring wonder
and contemplation of eternal order.

Despite the remarkable complexity
of these designs,

they can be created
with just a compass to draw circles

and a ruler to make lines within them,

and from these simple tools emerges
a kaleidoscope multiplicity of patterns.

So how does that work?

Well, everything starts with a circle.

The first major decision
is how will you divide it up?

Most patterns split the circle
into four, five or six equal sections.

And each division gives rise
to distinctive patterns.

There's an easy way to determine
whether any pattern is based on four-fold,

five-fold,

or six-fold symmetry.

Most contain stars surrounded
by petel shapes.

Counting the number
of rays on a starburst,

or the number of petals around it,

tells us what category
the pattern falls into.

A star with six rays,
or surrounded by six petals,

belongs in the six-fold category.

One with eight petals is part
of the four-fold category, and so on.

There's another secret ingredient
in these designs:

an underlying grid.

Invisible, but essential to every pattern,

the grid helps determine the scale
of the composition before work begins,

keeps the pattern accurate,

and facilitates the invention
of incredible new patterns.

Let's look at an example of how these
elements come together.

We'll start with a circle within a square,
and divide it into eight equal parts.

We can then draw a pair
of criss-crossing lines

and overlay them with another two.

These lines are called construction lines,

and by choosing a set of their segments,

we'll form the basis
of our repeating pattern.

Many different designs are possible
from the same construction lines

just by picking different segments.

And the full pattern finally emerges

when we create a grid with many
repetitions of this one tile

in a process called tessellation.

By choosing a different set
of construction lines,

we might have created this pattern,

or this one.

The possibilities are virtually endless.

We can follow the same steps
to create six-fold patterns

by drawing construction lines
over a circle divided into six parts,

and then tessellating it,
we can make something like this.

Here's another six-fold pattern
that has appeared across the centuries

and all over the Islamic world,

including Marrakech, Agra, Konya
and the Alhambra.

Four-fold patterns fit in a square grid,
and six-fold patterns in a hexagonal grid.

Five-fold patterns, however,
are more challenging to tessellate

because pentagons
don't neatly fill a surface,

so instead of just creating
a pattern in a pentagon,

other shapes have to be added
to make something that is repeatable,

resulting in patterns that may seem
confoundingly complex,

but are still relatively simple to create.

Also, tessellation is not constrained
to simple geometric shapes,

as M.C. Escher's work demonstrates.

And while the Islamic
geometric design tradition

doesn't tend to employ elements
like fish and faces,

it does sometimes make use of multiple
shapes to craft complex patterns.

This more than 1,000-year-old tradition
has wielded basic geometry

to produce works that are intricate,
decorative and pleasing to the eye.

And these craftsman prove just how
much is possible

with some artistic intuition, creativity,
dedication and a great compass and ruler.