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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.

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

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    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

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    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.

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    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-Constructionhttp://www.scribd.com/doc/146370381/Islamic-Geometric-Ornament-Appendix-II-The-Minor-Layout-Circle-and-Bisecting-Angleshttp://www.scribd.com/doc/146370381/Islamic-Geometric-Ornament-Appendix-II-The-Minor-Layout-Circle-and-Bisecting-Angleshttp://www.scribd.com/doc/146369736/Islamic-Geometric-Ornament-Appendix-I-Polygon-Constructionhttp://www.scribd.com/doc/146369736/Islamic-Geometric-Ornament-Appendix-I-Polygon-Construction
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    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 poin