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

遍及清真寺、學校、宮殿及私人住宅

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

此傳統始於西元八世紀，伊斯蘭教早期

when craftsman took preexisting motifs from Roman and Persian cultures

這時期的工匠已從羅馬、波斯帶來當時已有的藝術題材

and developed them into new forms of visual expression.

並從中發展出新的視覺表達形式

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

這個時期是伊斯蘭文化的黃金時代

during which many achievements of previous civilizations

許多過去的文明成就在此時

were preserved and further developed,

被保存下來，並有更進一步地發展

resulting in fundamental advancements in scientific study and mathematics.

造就了科學研究及數學方面的重大進步

Accompanying this was an increasingly sophisticated use

隨之而來的，是他們越來越嫻熟地

of abstraction and complex geometry in Islamic art,

運用抽象概念及複雜幾何以表現伊斯蘭藝術

from intricate floral motifs adorning carpets and textiles,

從裝飾在地毯及織品上的精細花卉主題

to patterns of tile work that seemed to repeat infinitely,

到看似無限重複的排序圖案

inspiring wonder and contemplation of eternal order.

啟發了讚嘆，以及對於不朽秩序的凝思

Despite the remarkable complexity of these designs,

僅管這些圖像設計複雜地驚人

they can be created with just a compass to draw circles and a ruler to make lines within them,

卻只需圓規繪製圓，再用尺拉出分割線，就能創造出來

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

這些簡單的工具，能變化出萬花筒般多樣的圖形

So how does that work?

怎麼做到的呢?

Well, everything starts with a circle.

一切都從圓形開始

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

首先要決定的是，你如何分割這個圓?

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

大多數的圖樣是把這個圓分成四、五或六等分

And each division gives rise to distinctive patterns.

每種分割都會構成獨特圖樣

There's an easy way to determine whether any pattern is based on fourfold,

用簡單的方法就能判別任一圖樣是四重

fivefold,

五重

or sixfold symmetry.

或六重旋轉對稱 (N重旋轉對稱 :圖形旋轉360度，能與原圖形重合N次的幾何性質)

Most contain stars surrounded by petal shapes.

大多是星狀周圍環繞著花瓣的形狀

Counting the number of rays on a starburst,

計算星芒的數量

or the number of petals around it,

或周圍花瓣的數量

tells us what category the pattern falls into.

我們就能知道這個圖樣屬於哪個一類

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

被六個星芒，或六個花瓣環繞的

belongs in the sixfold category.

是屬於六重旋轉對稱這一類的

One with eight petals is part of the fourfold category, and so on.

而八片花瓣環繞的，則屬於四重旋轉對稱的類型，依此類推

There's another secret ingredient in these designs:

另外，這些設計還有另外一種隱藏結構

an underlying grid.

基礎網格

Invisible, but essential to every pattern,

是無形的，但是對於每種圖樣而言都十分重要

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

網格能幫助你在工作開始之前，決定圖像構成的規模

keeps the pattern accurate,

維持圖樣的精準度

and facilitates the invention of incredible new patterns.

更有利於創造令人驚艷的新圖樣

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

我們來看個例子，說明如何組合所有元素，

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

一開始，先在方形裡畫個圓，並劃分為八等分

We can then draw a pair of criss-crossing lines

然後畫一對縱橫交錯的線

and overlay them with another two.

重複疊加兩次

These lines are called construction lines,

這些線稱為建構線

and by choosing a set of their segments,

從中選出一組線段

we'll form the basis of our repeating pattern.

我們就能組成重複排列的基本圖樣

Many different designs are possible from the same construction lines just by picking different segments.

同一種建構線，光是挑選線段組合，就有很多設計的可能性

And the full pattern finally emerges

而完整的花紋終於

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

在我們依著網格，重複鋪排這一片片圖樣後呈現

in a process called tessellation.

這個過程稱為" 密鋪 " (亦稱為: 密鋪鑲嵌、細分曲面、平面鑲嵌)

By choosing a different set of construction lines,

透過挑選出建構線裡的不同線段組合，

we might have created this pattern,

我們有可能創造出這種圖樣

or this one.

或這種

The possibilities are virtually endless.

有幾乎無限的可能性

We can follow the same steps to create sixfold patterns

我們能以相同的步驟創造出一個六邊形的圖樣

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

繪製建構線把一個圓分成六等份

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

然後"密鋪"這個圖樣，做出來的成品就像這樣子

Here's another sixfold pattern that has appeared across the centuries and all over the Islamic world,

這是另一組流傳好幾世紀的六邊形圖樣，在伊斯蘭世界廣為流傳

including Marrakesh, Agra, Konya and the Alhambra.

包括馬拉喀什(摩洛哥南部)、阿格拉(印度)、科尼亞(土耳其)、阿蘭布拉宮(西班牙的清真寺)

Fourfold patterns fit in a square grid, and sixfold patterns in a hexagonal grid.

四邊形圖樣搭配方網格，六邊型圖樣搭配六角網格

Fivefold patterns, however, are more challenging to tessellate

而五邊型，要以密鋪的方式呈現比較有挑戰性

because pentagons don't neatly fill a surface,

因為五角形無法剛剛好填滿整面

so instead of just creating a pattern in a pentagon,

所以，不能僅以五邊型來設計

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

要加入其他圖形，讓圖樣可重複鋪排

resulting in patterns that may seem confoundingly complex,

這樣一來，花紋可能會複雜到令人眼花撩亂

but are still relatively simple to create.

但其創作仍然是相對容易的

Also, tessellation is not constrained to simple geometric shapes,as M.C. Escher's work demonstrates.

還有，密鋪鑲嵌並不限於簡單的幾何形，例如 M.C. Escher(艾雪，荷蘭藝術家) 的作品

And while the Islamic geometric design tradition doesn't tend to employ elements like fish and faces,

雖然在伊斯蘭傳統圖樣設計裡，不太用魚或臉這類的元素，

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

有時也仍會使用多重圖形來打造複雜的圖樣

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

這種超過千年的傳統藝術，揮灑自如地應用基礎幾何學

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

來創造引人入勝、裝飾精美，賞心悅目的藝術作品

And these craftsman prove just how much is possible with some artistic intuition, creativity, dedication and a great compass and ruler.

這些工匠也證明了藝術的直覺、創作力、奉獻精神、圓規和尺，有無限可能