Islamic Geometric Ornament: The 12 Point Islamic Star. VI: 8 Plus 12 Point star.

7/28/2019 Islamic Geometric Ornament: The 12 Point Islamic Star. VI: 8 Plus 12 Point star.

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Part VI: Twelve Point plus Eight Point Star Tiling

Islamic Geometric Ornament:

Construction of the Twelve Point Islamic Star

The tilings of the twelve pointed Islamic star studied so far have been simple. The entire pattern was developed

by extension of the parent 12 point star. The common and appealing historic pattern shown here is different.

Still, it is not terribly complex. How are two perfect Islamic star patterns constructed to blend seamlessly?

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



This construction depends on an eight point Islamic star. Only a short introduction is needed. The basic

structure and layout procedure of the eight point Islamic star are absolutely identical to the 12 point star. Only

the polygon changes. A circle divided into 16 parts is required. The 16 divisions are drawn with two polygons,

exactly as for the 12 point star. For the eight fold star, The tiling polygon is usually a square. Two squares are

inscribed in the major layout circle by simply connecting intersections. Vertices and intersections are connected

by radii and inter-radii.

The two divided circles above are identical for our purposes. The polygons can be drawn inside or outside of

the basic layout circle. Drawing them outside is a bit less crowded.

The star will tile in the square tiling polygon circumscribed around the basic layout circle. Four arms of the star

will meet the polygon at points (a). As for the 12 point star, a minor layout circle is drawn from point (o). Point

(o) lies on the tiling edge at the next inter-radius from (a). The radius of this minor layout circle is (o a). The

bisector is constructed as for all previous examples; more construction details are found in appendix II.[Link ]

http://www.scribd.com/doc/146370381/Islamic-Geometric-Ornament-Appendix-II-The-Minor-Layout-Circle-and-Bisecting-Angles





This will be a parallel arm star, so the ends of the arms follow the layout octagon. As before, points (g), (b) and

(c) are defined by the minor layout circle. Each decisions here is taken to maximize symmetry in the tiling as

for the 12 point star. Layout is transferred around the divided circle with circles though points (b) and (c).

The arms are drawn in as before; connecting the circles through (b) and (c) yield the star polygon. Extending

these lines inward, to intersect each other, and outward, to intersect the layout polygon, gives an exact parallel

arm star which will tile with the best possible symmetry. If the ends of the arms are extended to intersect the

tiling polygon, as usual, one of the most common infinite tilings of Islamic art results. This is the best

symmetry eight point star in the most commonly encountered proportions



The eight point star deserves, and will have, its own chapter later. For now, the question is how does one

integrate the two layouts below, the eight and 12 point stars? The layouts are extremely closely related. The

same decision, to maximize symmetry, has defined the proportions of both eight and twelve point layouts.

They do differ in an important parameter. The external angle of the eight point star is 135°, of the 12 point star

150°. They cannot be connected arm end to arm end as they are drawn here. One star must be chosen to set the



angles at the ends of the arms. In almost all historic patterns, the smaller star is a parallel arm star; the parallel

arm star defines the end angles.

An external angle of 135° for the 12 point star, to match the 8 point star arms, will yield a tapered arm 12 point

star. That taper in the 12 point star will be set by the minor layout circle and its bisector. This decision is taken,

again, to preserve the best possible symmetry in the minor five point star formed by the tiling.

The key to the layout isrecalling this: recalling that the

five point star is formed by

tiling.

Half of the star will belong to

the 12 point layout and half

will be defined by the eight

point layout. If all arms of the

minor five point star are to be

of the same length, the two

minor layout circles are

identical.

They map onto each other

exactly, as shown here. The

figure will tile in a square

centered on the 12 point star.

A quarter eight point star will

appear in each corner.

The diagonal red line will be

the common side of the layout

octagon of the eight point star

and the layout dodecagon of

the 12 point star.

The questions to address are; how is this layout drawn to a specific size? What is the separation of the centers

of the eight and twelve point stars?

The basic layout circle of the 12 point star does not have an obvious relationship to the tiling polygon, the red

square.

For both the basic layout circles and the layout polygons, dodecagon and octagon, no obvious size relationship

is found. They are in fact in a ratio of approximately 1.54586 : 1.

The length of the sides of the layout polygons must be equal, but that is not helpful for layout

The starting point must be what is known to be fixed. That fixed definition is the structure just constructed

here. The minor layout circle. It seems odd, but it is the key to constructing the figure.



The triangle defined by points (o12), (e) and (o8) contains an enormous

amount of information. This triangle is useful since distance (o12-e) is one

half of the repeat dimension of the pattern. This would be used to scale the

layout for a defined space.

Since the layout centered at (o8) defines an eight point star, we know thatangle (a o8 e) is 45° and that the blue layout inter-radius is its bisector. A

layout defined by bisecting these angles is simple.

Drawing a divided square and it diagonal is easy. The repeat spacing is

determined and a layout square is drawn. [See App. I] Points (o8), (o12)

and (e) are obvious. The angle at (o8) is easily bisected as shown. [See App. II]

The next step is not quite as obvious but is equally simple.

The three radii and inter-radii from (o12) trisect an angle.

Trisecting an angle exactly is not generally possible, but this

case, trisecting a 45 angle is exact and easy. This is one

eighth of a 24 fold divided circle, which has been

constructed here many times. It is not clear what size to

make the layout for this 24 fold division step, so size is

ignored for the moment. Any convenient size layout circle

is used. The usual two staggered hexagons are used to

divide the circle and the radii and inter-radii are drawn.

An interesting result is produced. Point (o’) is defined

without further effort. The point is defined uniquely by the

definition of the eight fold and twelve fold divisions of the

circle. No other information is needed

Several things are known now. Point (o’) lies on the tiling

edge of the dodecagon and octagon. It can define the

common side of the layout polygons.

The common side defines both the layout circle and octagon

for the eight fold star and both the layout circle and the

dodecagon for the twelve fold star.

Point (o’) on that shared side also defines the minor layout

circle. That circle has an equal radius, (o’ a) on both the

eight and twelve fold star.

The decision to make the eight fold star a parallel arm star

completely defines the pattern and the layout can be

completed easily from here.

http://www.scribd.com/doc/146369736/Islamic-Geometric-Ornament-Appendix-I-Polygon-Construction







Two complete Islamic star layouts are being defined simultaneously, so there will be numerous steps. They are

not difficult. The red circle from (o12) through point (o’) is used to transfer the layout for the eight point star

to the remaining radii in the left figure. The two layout lines from point (o8) to that layout circle define the

shared side, (s s). The basic layout circle for the eight point star can now be drawn; it is defined by the shared

side as shown. The minor layout circle is then drawn at point (o’).

Two bisectors are now needed. The arms of the eight and twelve point stars have different relationships to the

shared side. A bisector for the 12 point star and a bisector for the eight point star are drawn in the right side

figure to points (k) and (k’) respectively. They are defined by the common side (s s) and the inter-radii of the

eight and twelve point layouts which define point (o’). The required sides for the octagon defining the ends of

the eight point star can be drawn in and the definition of the arms can now be laid out below.

Note that only the quarter circle is needed to construct the eight point stars with this layout method.



The layout of all four eight point stars can be filled in. The parallel arm eight point star defines the tapered arm

12 point star by extending the arm end to cross to the 12 point star bisector. This defines the arms of the 12

point star completely.

The intersections of the 12 point arm layout are transferred around the radii with the usual layout circles and the

star polygon is completed by connecting the inside layout circles on the radii and inter-radii. The point of

intersection of the stars is transferred around with a new layout circle; this circle is the same as the base layout

circle of the 12 point star which would usually be drawn as step one. It was determined after the common side

is constructed in this case.

The star polygon sides are extended inward to intersect and outward to the layout circle. The ends of the arms

are completed to the base layout circle for the 12 point star. The figure is almost completely defined. Only the

extensions of the ends for eight of the arms need to be determined.



Two of the extensions on each side intersect the tiling polygon exactly; this results from the strict geometric

definition. Two of the extensions need a further layout circle.

These two remaining extensions should intersect the extended arms of the eight point star. Due to the partial

layout, these need one more arc to define them. Circles from point (o8) as shown define the intersections of the

extended arm of the eight point star and the arm of the 12 point star. Adding this final layout line completes the

figure.

The layout of this composite tiling is complex, but no step is

difficult or introduces any new complex ideas. The constructionorder seems surprising, driven by the small corner stars. This is

by far the easiest and most accurate layout. The geometry of the

interaction of the two stars is allowed to define the layout.

This is a general method for constructing several cases of figures

with two stars of different symmetry. The intersections of the

divided circles define the layout. Tiling rules only allow some

cases, but 7 plus 28, 8 plus 16, 9 plus 12, 12 plus 8, 14 plus 7, 18

plus 6 and 30 plus 5 symmetry stars are all possible using this

method. The peculiar case of the composite 9 point star plus 12

point star tiling will be dealt with in the next chapter. [Link] Both

12 plus 18 and 12 plus 16 can then be derived.

Most cases will not be addressed in this series of chapters. See

A.J. Lee’s notebooks for details on other cases.

(See reference 1b, p12 ff. in the introduction.)

Islamic Geometric Ornament: The 12 Point Islamic Star. VI: 8 Plus 12 Point star.

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