POLYGONS

Polygons

Polygons play a fundamental role in Islamic geometrical design. When combined with identical or other-shaped polygons they form grids. These grids in turn form the structure behind geometrical designs. Two examples of polygons are the snowflake on the left of this page and the tiles in the main picture. Both of these polygons are sixe sided and are called hexagons.



A six-sided polygon is also known as a hexagon
A five-sided polygon is known as a pentagon
A seven-sided polygon is known as a heptagon
A eight-sided polygon is known as a octagon
A nine-sided polygon is known as a enneagon (or nonagon)
A ten-sided polygon is known as a decagon
An eleven-sided polygon is known as a hendecagon
A twelve-sided polygon is known as a dodecagon

Traditional craftsmen were able to draw polygons with just a compass and a ruler. They did not need to calculate angles. Here are some examples of how they did this:

How to draw a square with just a compass and a ruler:
1. Draw a horizontal line and place a circle on the line.
2. Draw 2 arcs of equal size, the diameter is not important. Place the compasspoints in the intersections indicated with red circles
3. Draw a line that connects the intersecting arcs.
4. Draw 4 circles of equal size. Place the compass points in the indicated intersections. The circles meet each other at the intersection of the horizontal & vertical lines. 
5. Draw a square by connecting the 4 indicated intersections.

How to draw a pentagon with just a compass and a ruler:
1. Draw a circle and 2 lines as seen above in steps 1-3 of "how to draw a square".
2. Draw 2 arcs of equal size, the diameter is not important. Place the compasspoints in the intersections indicated with red circles. 
3. Draw a line that connects the intersecting arcs.
4. Place the compass- point in the indicated intersection & draw an arc that intersects the vertical line & the circle
5. Place the compass- point in the upper indicated intersection & draw an arc that intersects the lower indicated intersection.
6. Draw 2 lines that connect the 3 indicatted intersections. These 2 lines from the 'roof' of the pentagon.
7. Place the compass- point in the indicated intersections & draw 2 arcs that intersect the top of the 'roof'.
 8. Draw 3 lines that connect the four indicated intersections
9. The pentagon is complete. 

How to draw a heptagon with just a compass and a ruler

the twinned stars on the right in the minbar panel in the mosque of Qiqmas al-Ishaqi in Cairo.
For many centuries, mathematicians and geometrists have tried to find a method to divide a circle into seven equal parts. The consensus now is that it is impossible. However, we can see occurrences of geometrical designs in Islamic architecture that use 7-pointed stars, such as the twinned stars on the right in the minbar panel in the mosque of Qiqmas al-Ishaqi in Cairo. 

This means that there were very skilled Islamic craftsmen who were able to divide a circle into 7 parts with a compass and a ruler. Mathematicians cannot accept anything less than 100% accuracy but craftsmen can accept a more or less equal division of a circle into more or less equal parts. The method shown here is devised by me and shows a possible approach to the division of a circle into 7 equal parts. The margin of error is less than 2%.
   
1. Draw circle on a horizontal line and divide it into six equal parts.
2. Draw a pentagon as indicated.
3. Find the halfway point on the circle between the indicated points of the pentagon and hexagon. 
4. Draw a line that connects the point from the previous diagram with the intersection of the circle with the horizontal line. Draw a circle that touches this line as indicated.
5. Draw a vertical line that touchces the circle as indicated.
6. Find the halfway point on the circle between the two intersections indicated by red circles.
7. Draw two lines that connect the indicated intersections with the halfway point from the previous diagram, these are the first two sides of the heptagon. 
 8. Draw 3 arcs with the compasspoint on the indicated intersections. The radius is the length of one of the sides of the heptagon. 
9. Draw the five remaining sides of the heptagon
10. A heptagon drawn with a compass and a ruler. The margin of error is less than 2%
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