Islamic Geometric Ornament: Appendix I: Polygon Construction

7/16/2019 Islamic Geometric Ornament: Appendix I: Polygon Construction

http://slidepdf.com/reader/full/islamic-geometric-ornament-appendix-i-polygon-construction 1/19

Islamic Geometric Ornament: Appendix I

Compass and Straight Edge Constructions of the polygons.

This appendix collects many of the common operations of polygon construction in one place. The illustrations

here and throughout the chapters were created with a drafting program. They comply, however, with the

traditional rules of compass and straight edge geometric construction rules. Where an exception is made, it will

be mentioned.

Constructions are presented for the common constructible polygons; 4,5,6,7,8,10,12, 14, 15 and 16 sides . The

emphasis is on the case where a specific size and orientation is needed, as for constructing a figure. Only the

exact constructions and one approximate construction are presented. There are many approximate constructions

not mentioned here. Coincidentally these are likely to be the same methods used by historic artist; geometry

has not changed in the last 1000 years.

i) 6 & 12 Hexagon or Dodecagon of a specific size, vertical or horizontal orientation

ii) 4, 8 &16 Square or Octagon drawn around a line or in a circleiii) 5 &10, 15 & 30 Pentagon, Decagon and Pentadecagon drawn in a circle

iv) 7 & 14 Heptagons drawn in a circle, an approximate construction used by the Arab geometers



The Basic layout Structure.

Almost all Islamic Star figures are radial symmetry figures; their symmetry is based on the circle. A divided

circle is needed to construct the star. It does not appear in the final figure, but it is the foundation. All of the

polygon constructions will start with a circle.

For an Islamic star figure we need to construct a layout circle with a set size. If the required layout

measurement is length (A B), a radius, the construction usually starts with the figure below. The layout line isextended, to points (a) and (b), and the perpendicular bisector is constructed between (A) and (B).

Smaller circles could be used to erect the bisector, but smaller is not better in compass and straight edge work.

Longer layout lines are usually more accurate.

The next step varies greatly depending on whether the layout line (A B) is a side of a polygon, the centerline, or

a radius of a layout circle.

The case most relevant to this discussion of the 12 point Islamic star is the division of the circle into 24 parts by

two hexagons. The hexagons are constructed first.



i) The Hexagon and the Dodecagon. Basis of the 12 Point Islamic star.

For the basic 12 point star, the relationship of the layout circle and the tiling figure is very simple. The layout

circle is the inscribed circle in the tiling hexagon. Both are constructed together in this case.

The exact order of operations depends on the orientation of the tiling hexagon. For a “horizontal” tilinghexagon, R2 is used as the repeat spacing. R2 is the circumscribed circle defining the tiling hexagon. For a

“vertical” layout hexagon, R1 is used as the repeat spacing equal to the distance across the points of the star

arms. R1 is both the inscribed circle of the tiling hexagon and the circumscribed circle defining the layout

hexagon. The size of the area to be covered by the repeating pattern is divided into the desired number of

repeats and the radius R2 is determined.

There is a strong preference in historic examples for patterns which have half or quarter patterns at corners and

sides of the field. As a result, the field is usually an even multiple of R2 in one, R1 in the other dimension.

This is usually the only “Measurement” you will use in a pattern construction.

Case one is for a “horizontal” hexagon with vertices to the left and right. Case two is for the “vertical” oriented

hexagon, with vertices up and down.



Case 1: The Horizontal Layout Hexagon, The Divided

Layout Circle.

For the “horizontal” hexagon with vertices to the left and right

the figure is constructed from the outside in. The hexagon has a

remarkable relationship to the circumscribed circle, drawn

through its vertices. The length of the sides of the hexagon is

exactly equal to the radius of the circle.

We simply draw a circle of the desired Radius. For a tiling

hexagon, this radius R2 is the repeat spacing for the pattern.

From the indicated point (a), we draw arcs of radius R2 to

define the sides of the hexagon.

Repeating the arcs from point (a’) yields a hexagon of exact

known width, with sides of length R2.

If the blue hexagon is the tiling polygon, it will be the limits of the tile which repeats to form the pattern. For

the twelve point Islamic star which is discussed here, the layout of the star lies inside that tiling hexagon

touching the center of each side.

For this definition, the red circle of radius R1, from the origin to point (c), is the base layout circle for the star.

This circle will need to be divided into 24 parts to define a 12 point star. The new radius R1 is used and theexact hexagon construction used above is repeated to inscribe a layout hexagon. The first side is (d - e2). The

polygon is completed exactly as for the tiling hexagon to divide the circle into six parts.

A second hexagon is needed for 24 divisions, so the process is repeated.



The second layout hexagon is constructed rotated 90° from the first, beginning at points (c) and (c’). These are

called staggered polygons.

Completing the figure results in three hexagons; a tiling hexagon defining the repeat and two layout hexagons

defining the division of the circle into twelve parts, where each vertex touches the layout circle.

In general the lines connecting the vertices of the layout hexagons are called “radii,” in dark blue, which divide

the circle into 12 parts. The division of the circle into 24 parts is finished with the second set of light blue line,

“inter-radii,” connecting the intersections of the layout polygons.

This layout uses a total of two circles and six arcs. All points are geometrically defined and nothing is

measured with a ruler.

The repeated pattern field.

At this point, the layout will define one star. To repeat the star, a repeated series of layouts is needed. The

division of the larger space to be covered by the layouts was already determined when R2 was chosen.

Fortunately, defining the hexagon repeat is very easy. The repeat is easily laid out with each new center defined

by R1, not the layout radius for the tiling hexagon, but the layout radius for the star.



The repeat centers are identified and the repeated

figures are laid out at each center.

The centers are located by one of two equivalent

methods. The tiling is edge to edge, so the new

hexagon is located from the face of the first tilinghexagon. The sides of the tiling hexagon can be

extended to intersect, locating point (o’) or the radii

which cross the center of the faces, from the vertices of

the layout hexagon, can be extended and point (o’’)

located by striking an arc.

Extending the tiling hexagon sides is somewhat neater

since the lines will be used in the layout of the divided

circle in any case. The layout can be extended as far as

desired.

Case 2: A Vertically Oriented Layout

Hexagon.

The layout of the vertically oriented tiling hexagon proceeds exactly as for the horizontal orientation with the

exception of order of operations. The radius, R1, of the base layout circle which will define the right to left

repeat spacing is used directly for defining of the layout hexagon in this case.

This layout proceeds from the inside out; the layout hexagon is constructed before the tiling hexagon. The

process of drawing the two layout hexagons can start at either point (c) or point (d), as drawn here.

The layout hexagon is then exactly as in the first case. A hexagon is drawn starting from point (d) using radius

R1.



The process is repeated starting at point (c), 90° from point (d)to give the staggered polygons.

To draw a circumscribed hexagon, a side is needed. This side is easily defined by constructing a vertical

tangent at point (c). The required point (h) is defined as shown.

Two radii define the ends of the side drawn through points (c) and (h). With one side defined, a layout circle for

the circumscribed hexagon can be drawn and it is completed as for any hexagon layout based on R2, the radius

of the circumscribed circle through points (k1) and (k2).



Radii and inter-radii are drawn in to yield the 24 fold division of the circle. Exactly the same result is

produced, starting from a different known dimension to allow precise design.

Connecting the vertices of the interior hexagons, at the radii, defines a dodecagon, of 12 sides. Repeating the

process by connecting the inter-radii at the layout circle to yield a staggered dodecagon, produces the 24 sided

polygon. This is technically the icosikaitetragon but it is usually called the “24 gon”. The resulting 48 fold

division produces a layout for a 24 point star.



ii) The Square the Octagon and the Hexadecagon (16 sides)

It is perhaps surprising that all constructible figures, even the square, start with a circle layout. A square can be

defined without a full circle layout, but most convenient layouts will start with a circle layout. A defined size is

usually required, and the line (AB) is the defined width of the square here.

Perpendiculars are drawn on the basic figure at (A) and (B). These are defined by arcs from points (a) and (B)

for (AF); points (A) and (b) for (BG).

The line (AB) may be defined as the centerline or as one side of the square as required by the layout. Squares

for both cases can now be drawn.



All points are defined for the square with line (AB) as a side. Points (A), (B), (G) and (F) define a perfectsquare. Only on arc is needed, from point (A) with radius (AD) to define the missing horizontal bisector line

(HJ).

For the case where line (AB) is the center bisector of the square, two arcs, (AD) and (BD) are drawn from

points (A) and (B) to define the corners, (K), (K’), (M) and (M’). Both horizontal and vertical bisectors are

already present. Connecting the corners completes the eight fold divided square.

Defining the next related polygon with 2x4, the octagon, is done by adding a 90° staggered square exactly as for

the hexagon cases.

All points already exist for the layout. Using the case where line (AB) is the bisector, a circle is drawn from (D)

through a corner to generate a base layout circle for an eight point figure. The intersections at the vertical and

horizontal bisectors are connected to create the staggered square. All of the vertices of the two squares are

connected to generate a regular octagon.

Vertices of the squares are connected as radii to divide the circle into eight parts; the intersections of the two

squares are connected as inter-radii to divide the circle into 16 parts.



It should be obvious how that 16 sided figure, a hexadecagon or hexakaidecagon, is drawn.

A 16 sided figure requires division into 32 parts;

division into 32 parts can be done with two eight

sided figures. One is already drawn in above,

defined by the radii of the two squares. The

second is drawn in by using the inter-radii to

define the sides.

The tiling of the square is obvious. In the case

of the higher figures, they are not usually tiled

independently. Neither octagons nor 16 sided

polygons give perfect tilings, so the size of these

figures is almost always determined by the base

layout circle diameter, circle (DK). They are

part of a larger tile. When the octagon is tilededge to edge, it tiles exactly as for the hexagon,

based on the radius of an inscribed circle.

This division process can go on indefinitely, and

figures up to 96 sides are reasonably common in

Moroccan art.



iii) The Pentagon the Decagon and the Pentadecagon (15 sides) and the Triacontagon (30 sides)

The pentagon and its relatives are the least obvious of the compass and straight edge constructions. An exact

construction has been known since antiquity, and is still the common construction. It does require a few steps.

The critical step in the construction of a polygon can be described as determining the length of a side.

The side of the pentagon is constructed from exactly the same base layout used for the square. In this case, the

length of the side is determined by the arc drawn from point (D) to (F). The point (G) at the intersection with

the layout line (A) (B) defines the length of a side, (GF). These are exact. The sides are transferred to the

layout circle with arcs starting from point (F) and continuing around the circle.



The arc from point (F) defines points (s1) and (s2). Arcs from (s1) and (s2) define points (s3) and (s4). The

construction is fast and exact, but it is not easy to construct as an exact size relative to a side or the distance

across the vertices. Fortunately, the pentagon does not give a perfect tiling. Tilings containing pentagons and

decagons are almost always defined by another tile constructed around them. The problem almost always

reduces to a definition based on the layout circle.

The only remaining problem is to define the orientation of the vertices of the figure. This is simply solved by

constructing the perpendicular (CDE) parallel to whatever radius defines the first vertex of the pentagon.

The pentagon already divides the circle into 10 parts. Each vertex is connected to the center of the base layout

circle and extended across to intersect that circle. This is true for any polygon with an odd number of sides. A

five point star could be constructed with just one polygon as a result. Five point Islamic stars are very

uncommon, probably due to difficulties tiling them. Ten point stars are extremely common, particularly in the

central and eastern part of the Islamic world. The probable reasons are discussed in the chapters.

The division of the circle into 20 parts for layout of the decagon, 10 sided polygon, is achieved by constructing

one more pentagon, staggered by 180°. The new pentagon is drawn at the same radius as the first, at the

opposite side of the circle. The radii and inter-radii will overlap with the first pentagon. By connecting the

intersections of the sides of the pentagons, the 20 fold division is completed.

Tiling the pentagon or decagon is normally done by defining a new tile with appropriate symmetry by extension

of the sides of the layout polygon. There are several possibilities and they are usually defined by other

considerations of the tiling of the figure. The important point is that the size of the layout pentagons is almost

always defined by the layout circle. This is good since constructing a pentagon based on a side is quite difficult

The last relatively common polygon derived from the pentagon is the triacontagon, the 30 sided polygon, to

define the 15 point star.



Each pentagon allows a 10 fold division of the circle, so three staggered polygons are required 120° apart. This

layout requires an equilateral triangle, which has not been constructed yet. It is normally constructed as for the

hexagon layout, but an isolated triangle in this circle is required.

The triangle is simply a half hexagon

layout. An arc from point (H) to point

(A) defines two points (K) and (M).

Points (F), (K) and (M) define an

equilateral triangle. If pentagons are

constructed on lines (AF), (AK) and

(AM) a thirty fold division of the circle is

possible.

A modern compass is used to transfer the

lengths of the sides for the remaining two

pentagons. The strictest classic compass

and straight edge layout requires

reproducing the layout for all three pentagons on (AF), (AK) and (AM).

The layout to the right is used for the

strict construction. Completing a triangle

from (CDE) provides the analogous

radius bisectors, (b’) and (c’) for (b) and

(c). Radii drawn from (C) and (E)

provide the perpendiculars (b’’) and (c’’).



The thirty fold division is completed as usual.

Divisions of the circle beyond 30 and 32 fold are not

common for the construction of the Islamic star

discussed in these chapters. Forty fold division for

the 20 arm star is sometimes found.

The comparison of the 15 and 30 arm stars below

demonstrates why. This design method begins to

produce artistically unsatisfactory stars beyond about

20 arms.

The Islamic star with very high arm counts,

commonly found in Morocco and the Maghreb,

almost always uses an altered layout for the outer

cells of the arms. This is a discussion for another

day.



iv) The Heptagon. An approximate construction.

There is no exact compass and straight edge construction of the heptagon, a seven sided polygon. It is unusual

to make a statement that a negative statement is proven, but a method is commonly used to prove that a

particular angle or figure cannot be constructed.

Despite this, seven and 14 arm stars and seven sided figures are uncommon but not rare in Islamic geometric

art.

As remarked above, constructing a polygon in a circle is equivalent to defining the length of a side. An

approximation exists for the side of a polygon inscribed in a circle which has been known since the ancients. It

is known to have been used by the Islamic geometers as an approximation, i.e. the method was known and it

was known to be an approximation.

It was noticed long ago, probably in India, that the segment (CD) in the standard beginning layout is very close

to the length of a side for a heptagon.

The line (CD) is exactly Rsin60°,

or the radius times one half of the

chord of 120°, as it would have

been known to the Greeks or

Arabs. We do not need to know

that to observe that it is very

close to the correct length for a

side of the heptagon.

If it is drawn out over seven sides

we find that the sides are about

0.2% too short, or two

millimeters per meter. One

problem with approximate

constructions is that errors add

up. If the sides are drawn one

after another around the circle,

the errors add up to 1.5% and the

ends do not meet. An error of

15mm per meter is a problem for

just about any purpose.

There is also a minor layout problem. The heptagon is lying at an unusual angle. There will be no vertex at the

top, bottom or either side. A construction with the vertex at the top is usually most useful, but a general method

to put a vertex where it is needed is required.

Both issues are easily solved. There is also an easy method to reduce the maximum error by almost tenfold, to a

maximum of 0.09%, less than one millimeter per meter.



Surprisingly, the layout for a heptagon starts with a hexagon. It is very easy to put the vertex of a hexagon at

any point on a circle. The two sides drawn in blue are used to set the line (AB) which is used to construct the

standard starting figure.

The sides are drawn by drawing the circle (CD) from (C). The point (C) sits at the top of the circle as desired.

The circle (CD) defines points (s1) and (s2). Circles (s1-C) and (s2-C) define two more sides. We stop here to

analyze errors.



The figure above defines both the length of the sides, drawn in red, and their bisectors. If the remaining two

sides are drawn based on the same length, (CD), the length error in line (s5-s6) will be determined by 6 times

the error in (CD).

This can be reduced, or redistributed, by extending the bisectors as shown to define side (s5-s6). The five sides

in red are now exactly equal. The length errors in sides (s3-s5) and s4-s6) will be one half of the error which

would have added up in line (s5-s6) from sequentially drawn sides. This is as accurate as this approximation

can be in on layout. For figures drawn under 100 mm, this is an acceptable approximation.

The approximation error is obvious on the lower left where a bisector and a radius are extended. They should be identical and they are not. This offers an interesting way to correct the error in length. It is reasonably

obvious from the figure that this error, measured at the layout circle, is about 3.5 times the length error in the

sides. If eight sides are drawn in sequence, the error, or mismatch, at the last side will be 7.5 times, about 8

times the error. If the error interval is bisected 3 times, divided by 8, a corrected length can be drawn. The

maximum error using this single step correction is reduced to about 0.09%. That is a good approximation.

The figure for the 14 fold division and 28 fold division are completed normally.



The two sides (s3-s5) and (s4-s6) are bisected separately and the figure is complete for a seven point star. If the

radii and bisectors are connected as shown to the right, the intersections of the two heptagons complete the 28

fold division of the circle.

The 14 arm star to the right was drawn with the least accurate

approximation, all of the error accumulated in one side. Even

with the maximum error, the star looks good and the errors

are not huge. The error in the extended arms, far from the

layout center of the star, are almost exactly one percent. This

is not negligible, and not as small as errors in a well drawn

patten can be.

The errors are not periodic, they are not evenly distributed

around the star. As a result, some care is required to be sure

that the errors do not magnify in a layout, but the more

accurate “second order” approximation shown above is

acceptable for most purposes.

There are many other approximate polygon constructions. In

principle, any polygon can be approximated. The other

common approximate constructions, particularly nine and

eleven point stars, occur in specific layouts and will be

discussed there.

Alan D Adams, Holland, New York, 6 June 2013. License: Creative Commons -Attribution 3.0 Unported (CC BY 3.0)Text, photos and drawings.

Documents

Islamic Geometric Ornament: Appendix I: Polygon Construction