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مقرنس

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"باییی سازد و ذهن فرد را هدایت مییییی وارد استدلال هندسی شود زیرا به خوبی آرایش یافته و سازمان پیین ذهنییی گیرد" .

— Ibn Khaldun, 1332-1406 CE, polymath, 732-808 Hijri, أبو زيد عبد الرحمن بن محمد بن خلدون الحضرمي‎

Kitab al 'Ibar, Book One, Chapter VI. The various kinds of sciences, 20. The geometrical sciences

مقرنس چیست؟

ییستم ساده، در عین حال قویاً پیچیده، ساختاری سه بعدی و در بر گیرنده فضا.

  • مقرنس دارای جایگاهییز در معماری اسلامی می باشد
  • پروفسور شیی تاریخچه معمارییر پلان هندسی منظم می باشد، ثبت نموده است
  • مقرنس می تواند بر اساس تنوع گسترده ای از تقارن ها باشد
  • مقرنس ها می توانند تعداد زیادی از فرمهای قدیم و جدید را تولی
  • توسعه مقرنس با نرم افزار سه بعدیییی ثبت نشده، فراهم نموده است

یادداشتی بر این ترجمه: در سرتاسر متن من از تعبیر عربی، مقرنس، به جای تعبیر فارسی مقرنس استفاده نموده ام

یادداشتی بر تاریخچه مقرنس

ی اسلامی در شمال آفریقا و ایران در قرن چهارم هجری دهم میلادی ظاهر شد.

  • به نظر می از رییییی زمیی باشد.
  • ی، شواهدی از وجود دو بالغ بر دو هزار مقرنس در معماریی اسلامی پی
  • الحمراء در اصفهان ایران و مسجد امام در میان ایی معروفی نظیر؛ در گرانادای اسپانیا وجود دارد.
  • مقرنس در درجه اول از اگر چه گچ و مثالهای چوب و آینه ساخته شده است (شیرو ی).

منشاء هندسی

هندسه پایه مقرنس های سه بعدی از چند ضلعی های منتظم به وجود آمده است.

  • ی دارای
  • اساساً آنها حداقل نیاز به پنج عنصر(واحد) دارند. سیستمی من در اینجا آنرا گسترش داده ام از 16 عنصر(واحد) مقرنس منحصربفرد استفاده می.
  • احتمالاً نزدییی عملی می باشد
  • هر چه تعداد انواع عناصر(واحدها) منحصربفرد بیشتر باشد، سیی لاغرتر خواهد بود.
  • این شیوه چالشهای مهندسی و مصالحگونه را مطرح می

یری مقرنس

ییید (n ضلعی). من ییN ضلعی را n-1ی و دوران دهیی را 360/n درجه حول ییر، دوران دهید.

    • زمانی اییان رسید، این ییر پلان از گنبد مقرنس سنتی می باشد.
    • هر عنصر(واحد) مقرنس یی در تصویر پلان می باشد.
    • تمام لبه های واحد های مقرنس طول یی در تصویر پلان و در سه بعدی (اندازه واقعی) دارند، این ییژگییدی از عناصر(واحدهای) مقرنس می باشد.
    • هر لبه یی دهد تمام لبه های همه عناصر مقرنس بدون هیی با ییگر به هر لبه هر آجر مقرنس منحصربفرد دیگری
    • عناصر مقرنس 32 ضلعی 16 لوزی منحصربفرد می باشد

گىبد مقروس سىتی

Traditional Muqarnas Dome

رابطه بین تصویر پلان و گنبد مقرنس سنتی مربوط به آن.

تص یْش

عناصر(واحد های) مقرنس

Tile 1

Tile 1, 11.25º = 1 x 11.25

Tile 5

Tile 5, 56.25º = 5 x 11.25

Tile 9

Tile 9, 101.25º = 9 x 11.25

Tile 13

Tile 13, 146.25º = 13 x 11.25

Tile 2

Tile 2, 22.5º = 2 x 11.25

Tile 6

Tile 6, 67.5º = 6 x 11.25

Tile 10

Tile 10, 112.5º = 10 x 11.25

Tile 14

Tile 14, 157.5º = 14 x 11.25

Tile 3

Tile 3, 33.75º = 3 x 11.25

Tile 7

Tile 7, 78.75º = 7 x 11.25

Tile 11

Tile 11, 123.75º = 11 x 11.25

Tile 15

Tile 15, 168.75º = 15 x 11.25

Tile 4

Tile 4, 45º = 4 x 11.25

Tile 8

Tile 8, 90º = 8 x 11.25

Tile 12

Tile 12, 135º = 12 x 11.25

Tile 16

Tile 16, 180º = 16 x 11.25

رنگ بندی اجزاء مقرنس، ی ارزشی برای شناسایی الگوهاییید را می سازد.

لبهها و عناصر مقرنس

  • تعداد عناصر منحصربفرد (غیی) مقرنس بوسیله n ضلعی اولیه مشخص می شود.
  • تعداد عناصر منحصربفرد (غیی) مقرنس مساوی است با n/2، وقتیn زوج باشد و n-1/2 وقتی n فرد است.
  • لبه های عناصر مقرنس هم اندازه هستند و بایستی در تصویر پلان خطوط مستقیمییم ، منحنی ، زاویه دار ییبی از ییر جانبی باشند.
  • لبه هایی های من دارای قوس 60 درجه هستند در حالیی قوس ها مماس با صفحه موازی با تصویر پلان می باشند.
  • لبه بین نیمه پائینی و بالایی هر عنصر مقرنس ییییه راس افزایش پیدا مییاد می شود.
  • ی مقرنس از دو سطح منحنییی لبه های بالایی و لبه های پائینی را بهم وصل می.
  • دیواره های عمودی در مقرنس استفاده می شوند تا ارتفاع را زییی از عمودی بودن بوجود آورند.ییبییبات) تاثیر نمی گذارند در عوض آنها میین رو زیبایی طراحییی دهند همچنین در تصویر پلان آنها هیچ نوع سطحی ندارند.

انواع مقرنس


نوع مقرنس
در برابر
روش گسترش

گسترش گنبد مقرنسی

ی اساسی آنها به شمار می آیند، بنابراین آنها جایگاه منطقی براییچیده تر می باشند.

  • یییی منحصر به فرد مقرنس را اییم
  • این گروه به سمت پائین و به سمت خارج گنبد برای ایجاد قطار اولیه ی شود
  • این N-1 بار در زاویه N/360 در تصوییی می شود
  • عناصر مقرنس مورد نیاز برایی بینابینی تعیین می شوند و به روشیی می شوند
  • قطار دومیی بینابینی استفاده می
  • ین روش قطارهای های سوم و چهارم و ... می تواند افزوده شود

روش شعاعی گسترش گنب

Ray Method of Traditional Dome Expansion

گنبدهای سنتی دارای 2 شعاع منحصربفرد هستند. جای دهیی.

برای نمایش انیمیشن روی تصویییید

روش هلالی گسترش گنبد

Crescent Method of Traditional Dome Expansion

گنبدهای سنتی دارای یی باشند.یه های جایگزین حالت دلپذیری بوجود می آید

برای نمایش انیمیشن روی تصویییید

روش شعلهای گسترش گنبد مسطح

Crescent Method of Traditional Dome Expansion

ییار زیاد ناشناخته از روش های گسترش گنبد وجود دارد. من این را شعله ای می نامم. این چیزی به جز ییست .

برای نمایش انیمیشن روی تصویییید

VARIOUS METHODS OF DOME EXPANSION

Ray method*

Crescent method

Flame method

Angle method

Zigzag method

Lightning method

Zipper method

Jagged method

Snake method


*The ray method expands each group of similar-colored tiles separately. The other methods shown all expand as single groups. These are just a few more of what are probably hundreds of dome expansion methods. I'm more interested in exploring new families of muqarnas forms, rather than methodically and exhaustively listing members of a single family. Almost all methods (except ray) display chirality. That is, they come in right- and left-handed pairs, which is useful in design. (See crescent dome.) It may very well be that any configuration can be expanded in either a one or two color group manner. Each configuration has one of each of the fifteen different muqarnas tiles.

Onion Dome Technique

Muqarnas onion domes use the flat, vertical 16th muqarnas tile to create a geometric condition that allows the dome to narrow below it.

  • Adding the 16th tile to the traditional dome creates an 'equator', a widest radius.
  • The next tile added is the 15th tile, but it and all successive tiles are rotated 180º from their normal placement. That is, all muqarnas tiles below the 'equator' are 'inside out'.
  • Then the 14th tile, the 13th tile, etc. are added until one chooses to stop narrowing the 'waist'.
  • If all 15 normal tiles are added below the 'equator', then the shape is closed at a point at both top and bottom. This would be a joined dome/anti-dome.
  • I chose to stop at the 10th tile and then add the primary and secondary tiers. This was an arbitrary choice.
  • The ray and crescent methods of dome expansion for creating primary and secondary tiers are used in the following examples.

ONION DOME RAY EXPANSION

ONION DOME RAY EXPANSION

The ray approach seems much more 3D than with the traditional dome.

Click image for animation.

ONION DOME CRESCENT EXPANSION

ONION DOME CRESCENT EXPANSION

The crescent expansion appears more dramatic with the onion approach.

Click image for animation.

Flattened Domes

Flattened domes have a different arrangement of muqarnas tiles in the central dome.

  • Several types of flattened domes have the same plan view, the same number and kind of each muqarnas tile as the traditional muqarnas dome.
  • However, the tiles are in different positions than the traditional domes.
  • This is possible because there are seven pairs of unique muqarnas tiles 1/15, 2/14, 3/13, 4/12, 5/11, 6/10, 7/9 that are identical in the plan view.
  • Other flattened domes differ in all respects from traditional domes.
  • Nonetheless, the same methods of dome expansion can be used on flattened domes.
  • With less symmetry flattened domes have more types of rays and crescents, making their expansion more complex and painstaking.

CHRYSANTHEMUM FLATTENED DOME

Chrysamthemum Flattened Dome

This 'dome' contains the same number & type of muqarnas tiles as the traditional dome, but arranged differently. It is identical to the traditional dome in the plan view.

Click image for animation.

RAY METHOD OF FLATTENED DOME EXPANSION

Ray Method of Flattened Dome Expansion

This flattened dome has eight unique rays. It takes a very careful inspection to see the differences.

Click image for animation.

WHIRLPOOL FLATTENED DOME

Whirlpool Flattened Dome

This flattened dome has 64 tiles in the center, 32 of which are the flat, vertical 16th tile. As a result of this it is twice the diameter of other domes.

Click image for animation.

DRAGON TEETH FLATTENED DOME

Dragon Teeth Flattened Dome

This flattened dome also has 64 tiles in the center and is twice the diameter of other domes. Interestingly, unlike most forms the primary tier switches back to that of a non-flattened dome style.

Click image for animation.

RISING STAR DOME

Rising Star Dome

This is not a true flattened dome since it has a 1:3 pitch vertical to horizontal from the lowest (outside) up to the highest (center). However, since it is similar to them rather than forms based on traditional domes, I include it with the flattened domes.

Click image for animation.

Muqarnas Seed

Another method for generating muqarnas compositions uses what I call the 'seed' method. The total of the tile numbers is always equal to 32. The traditional dome has 32 #1 tiles in the center (32 X 1 = 32).

  • For the top of the composition one chooses a pre-arranged or arbitrary arrangement of muqarnas tiles sharing their apex.
  • The only restrictions are they do not overlap and there are no gaps.
  • Then the next layer is added etc., etc. until:
    • Either the limit is reached and the next tier is begun, OR
    • The vertical muqarnas tile (#16) is added and the next tier is begun, OR
    • In reverse order the muqarnas tiles (rotated 180º) are added pulling the 'waist' in to create an 'onion' dome,
    • Which in turn allows additional tiers to be created.

SEED DOME

Seed Dome

This example of the 'seed' approach to muqarnas creation uses a repetition of the #1 and #3 tiles eight times for the seed. Thus, as seen at its apex the seed for this one is (1-3)8, shorthand for 1-3-1-3-1-3-1-3-1-3-1-3-1-3-1-3. In the animation it is expanded to the primary, secondary and tertiary tiers with the ray method.

Click image for animation.

PEACOCK DOME

Peacock Dome

This example of the 'seed' approach uses a repetition of the #2 and #4 tiles four times for the seed. So at its apex the seed for it is (2-4-2)4, shorthand for 2-4-2-2-4-2-2-4-2-2-4-2. In the animation it is expanded to the primary, secondary, tertiary and quaternary tiers with the peacock motif which appears to be unique to this form.

Click image for animation.

LESS SYMMETRICAL DOME

Less Symmetrical Dome

This example of the 'seed' approach uses a seed of 2-1-3-1-5-2-3-3-2-5-1-3-1 (no shorthand, sorry). In the animation it is expanded to the primary and secondary tiers with the ray method. It displays mirror symmetry about a single vertical axis, making it the least symmetrical form.

Click image for larger image.

MUQARNAS ARCH

Muqarnas Arch

Another example of the 'seed' approach. The seed for this one is 8-8-8-8, (8)4. Instead of continuing the pattern out I mirrored the secondary tier like a portal entrance. An additional mirrored copy created an internal arched space which shared an edge with the opposite side.

Click image for animation.

Composite Muqarnas Forms

Some muqarnas forms can be combined to create composite, hybrid or mixed compositions.

Scarab Hybrid MUQARNAS

Spiral Staircase

Below the tradtional dome the primary tier is of the ray type. The secondary tier is a complex novel form made up of two pairs of interlocking patterns.

Click image for animation.

Spiral Staircase Muqarnas

A variety of muqarnas forms can be developed to serve as the ceilings and/or roofs of spiral staircases.

  • The inner radii of the spiral staircases is determined by the muqarnas expansion from which the repeated form is derived.
  • Each spiral muqarnas element needs to be shaped like a wedge or truncated wedge from the plan view.
  • Of course, the leading and trailing edges of the spiral elements must join with a fixed change in elevation and angle around the center.
  • The wedge angle and truncation determines the radius of the empty cylindrical center. This is similar to the expansion method for domes - but with the inner elements removed.

Crescent Motif MUQARNAS SPIRAL STAIRCASE

Spiral Staircase

Crescent motif muqarnas ceiling/roof for a spiral staircase. Derived from the central dome - tightest radius.

Click image for animation.

RAY MOTIF MUQARNAS SPIRAL STAIRCASE

Ray Motif Spiral Staircase

This ray motif spiral staircase is derived from the primary tier and thus has an intermediate radius.

Click image for animation.

CRESCENT MOTIF SPIRAL STAIRCASE

Crescent Motif Spiral Staircase

The crescent element used is from the secondary tier - giving it a wider radius. Stairs omitted.

Enclosing Muqarnas Forms

If muqarnas forms are placed adjacent to each other sharing a vertex or edge sometimes an enclosing form can be developed.

FORM ENCLOSING DOME & FLATTENED DOME

Enclosing Form

A dome and flattened dome share a lower, outer vertex in the same vertical plane with their central, top vertices. This form has a very complex type of symmetry. It is mirrored on the other side of the domes and is related to the whirlpool flattened dome.

Click image for animation.

FORM ENCLOSING COMPLEX DOME ARRANGEMENT

Enclosing Form

This incomplete enclosing form shows how large enclosing forms can be. The 'center' of this enclosing shape ix 2/5 from one side and 3/5 from the other. This pattern is so complex that the type and placement of each tile must be individually determined. It might be completed in six months, while working on other less demanding forms.

Click image for larger image.

Unique Muqarnas Forms

When combining disparate muqarnas elements, novel forms may result. Some of these other types of muqarnas forms are difficult to categorize.

PAPRYRUS COLUMN HEAD

Papyrus Column Head

I tried unsuccessfully to create a column, but the result is interesting.

Click image for animation.

TWIN PEAKS DOME

Twin Peaks Dome

I tired of the 'super' symmetricality of muqarnas and tried unsuccessfully to create a totally irregular form. It still has two axes of mirror symmetry at right angles.

Click image for animation.

What Muqarnas Aren't

Muqarnas are not a structural support system. They require the support of an enclosing and/or supporting architectural structure.

  • As such they must be hung or otherwise attached to form a ceiling of an interior space, or
  • They must be placed upon a supporting roof structure.
  • Spiral staircases are a special case that imposes a variety of challenges in designing suitable muqarnas tiles, since each tile could potentially be both ceiling and roof simultaneously.

Postscript

I have intentionally ignored the specific design of traditional muqarnas stone blocks for a variety of reasons.

  • Historically, to 'force' muqarnas to fit onto square or rectangular walls numerous additional tile shapes were added, such as kite-shaped, half-blocks and others.
  • This proliferation of tile types may have solved specific immediate problems, but it did so at the expense of realizing the elegant potential of a smaller set of 'pure' muqarnas tiles.
  • My interest in muqarnas lay, not in accommodating them to known architectural forms, but in allowing them to develop inspiring new forms and methods of composing them that may be used in current architectural settings.
  • The minimum muqarnas tile set, the most elegant, is also the most cost effective for manufacturing and building.
  • Modern material science, structural engineering and architectural techniques should be applied to muqarnas to bring them into the present. This is a compliment, not a threat, to traditional muqarnas techniques and designs.
  • For the above reasons analyses of the historically extant muqarnas are of limited interest to me and are not mentioned here.


Muqarnas

Acknowledgements

To Dr. Mamoun Sakkal, Arabic calligrapher, font designer, artist and designer, whose architectural thesis, Geometry of Muqarnas in Islamic Architecture, inspired me. I use his technique for generating muqarnas blocks to create a traditional dome. Dr. Sakkal also allowed me to use his Sakkal Majalla light font for the Arabic word muqarnas.
To Professor Emeritus Branko Grünbaum who introduced me to Dr. Sakkal's work and encouraged my interest in geometry.
Thank you Ameneh Karimian for your helpful suggestions to provide clarity and indicate limitations.
Thank you also to Hakim El Hattab, amazing web developer, for making your reveal.js HTML5/CSS3 presentation framework available.

THE END

Daniel P. Owen / ensightful.com

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