Al-Kashi's modular muqarnas mensuration

How can a ceiling assembled from hundreds of little faces become an arithmetic problem? In the Key to Arithmetic,al-Kashi organized measurement of muqarnas,the tiered cellular construction articulating and decorating ceilings. He selects a module,counts elements by shape,and adds the surfaces of facets,roofs and connecting pieces. This is a calculation method within an older architectural practice,not an attribution of muqarnas itself to one inventor ².

Yvonne Dold-Samplonius explains the procedure and its average coefficient,but also identifies a projection inconsistency and values whose derivation she does not know. Many printed digits therefore do not promise absolute accuracy for every curved surface. The reported1427/830AHdates completion of the book,not the first building measured with it or the invention of every muqarnas vault ^{3 4}.

In the definition discussed by the study,a muqarnas is a staircase-like ceiling:a cell consists of two facets and a roof,and cells with bases in one horizontal plane form a tier. Surfaces may be flat or curved. Some visible pieces belong to cells;others fill gaps between them. Counting hollows in a photograph is not enough to determine their exposed surface area ⁵.

The contribution is mensuration in BookIVof an arithmetic text,not a plaster object,tile or observing instrument. The study distinguishes simple,clay-plastered,curved and Shirazi muqarnas. Element inventories and measurement procedures change with type. A convenient method does not make every real ceiling identical in geometry or curvature coefficient ⁶.

Builders already knew how to assemble cells. The historian explains that al-Kashi did not need to teach his practical readers the whole craft from its beginning;her modern explanation uses reconstructions of excavated Takht-i Suleiman elements. Older components contradict the image of a solitary mathematician inventing the architecture on paper. Modern archaeological comparison is not evidence that al-Kashi designed those buildings ⁷.

Area problems belonged to practical mathematics. The book places arches,vaults and buildings among matters a calculator needs to understand. MacTutor describes its intended mathematics for astronomy,surveying,architecture,accounting and commerce. This is an account of audiences and problems,not proof that its author practised every listed occupation or that every school taught his text ^{4 8}.

Al-Kashi completed the Key within Samarkand's scholarly setting,where he worked with Ulugh Beg. The study dates it1427/830and locates muqarnas in the last section of chapter9,BookIV,on measuring structures and buildings. That organization connects the subject to earlier plane-area problems rather than presenting ornament as an isolated activity unrelated to elementary calculation ^{9 4}.

The study separates textual statements from historical hypotheses. Paying artisans according to completed area in the Arab world is suggested by comparison with an Italian practice,not established through a discovered al-Kashi wage contract. Likewise,the historian's explanation for relatively careless treatment of simple muqarnas is an interpretation,not direct testimony about the author's intentions. Probability must survive the summary ¹⁰.

MacTutor explicitly identifies al-Kashi as an Islamic mathematician. That provides the individual Muslim basis here;the museum's declared Sunni frame remains distinct from assigning a personal school from birthplace or patronage. His earlier biography,sources and attribution are preserved. No belief is inferred from a ceiling,and working in Samarkand does not by itself establish a jurisprudential affiliation ^{11 12}.

This method is distinct from his circumference computation,sine-one-degree computation and conjunction plate. Within mensuration itself,the addendum attributes panel construction to masons:the author calculates from existing craft knowledge. Dold-Samplonius is a modern interpreter and calculator,not a medieval coauthor,and her corrections cannot be silently put into al-Kashi's own voice ¹³.

Calculation starts with a module,a standard length associated with a facet base and assigned the value one. Lengths and areas are first expressed through it,then converted to cubits or another unit. If one cubit equalsbmodules,numerical area in module-squared units is divided byb²to obtain square cubits. Dividing bybonly would confuse length conversion with area conversion ¹⁴.

For plane surfaces,facet bases are summed and multiplied by height;roof-element areas are added,and then tiers are combined. Figure8gives approximately0.707107for a45-degree rhombus,0.353553for its half,0.414214for an almond and0.292893for its biped complement. These are areas in squared-module units,not centimetres or volumes of cavities ¹⁵.

An independent modern explanation gives a unit-sided45-degree rhombus area√2/2and half-rhombus√2/4. The illustrated almond and complement have areas√2−1and1−√2/2,summing to√2/2. Checking these plane identities corroborates the rounded planar values,not the exact area of every curved element. Doubling every length quadruples area:scale and units are part of the mathematical answer ¹⁶.

For curved muqarnas,cell bases are summed and multiplied by a coefficient;intermediate-element areas are then added. The historian explains the coefficient through a straight segment plus half a curved intersection line in a construction panel,using an average that unifies calculation. It is not a magical constant valid for any chosen curvature. She explicitly does not know how some intermediate-element values were obtained,and her reconstructions yield different numbers ¹⁷.

Unat reports advanced teaching of the Key in Ottoman madrasas and Ibrahim Kami's Turkish translation of BookIV. That concerns the book and its mensuration section generally,not every teacher's use of the muqarnas passage or every ceiling measured from it. The translation was not inspected here;bibliographical notice does not become a complete examined manuscript genealogy ¹⁸.

The study uses architectural forms and earlier panels to make the arrangement intelligible. Similar craft can persist without documented dependence on the Key. The modern Qom photograph is therefore not the endpoint of an arrow claiming all its calculations came from Samarkand. It illustrates the problem's subject. Textual reception and the visible presence of ornament are different kinds of evidence ^{19 20}.

The procedure combines familiar plane areas,basic shapes and the module that allows pieces to fit. Many roof shapes had already been discussed in BookIV's chapter2. A numerical entry is not automatically a newly invented mathematical object. The organization reuses plane knowledge to understand a complicated spatial arrangement ²¹.

Masons' knowledge enters through curved panels. A mean value avoids following every different cell diagonal,and the historian explains why this makes practical calculation manageable. Writing the method preserves a connection between craft and mathematics. It does not turn an approximation into an exact equality or eliminate thickness,material variability and construction errors from real architecture ¹³.

The written account permits a modern historian to reconstruct calculations and compare values. Some comparisons support planar numbers;others reveal differences for curved elements. These are outcomes of studying a text,not proof that every modern design program directly inherits al-Kashi's algorithm. The readings do not establish such an extensive route of transmission ²².

The Key contains much beyond muqarnas. MacTutor discusses decimal fractions and root extraction while warning against erasing predecessors. Each heading does not become another invention without its own investigation. This article does not recount circumference or sine achievements under a new architectural name. A bounded record makes the contribution understandable and testable rather than a slogan about all mathematics ²³.

Dold-Samplonius connects area with possible estimates of needed material,such as paint or gilding,and proposes area-based artisan payment as a possible application rather than a verified local event. The importance of the question can be understood without inventing a budget,saving percentage or historical wage. Surface area alone is not final material quantity either;thickness and further conditions would be needed ²⁴.

The educational point is that complicated ornament need not be an unanalyzable mass. The author separates it into classes and tiers and reconstructs a whole from parts. This describes arithmetic,not structural certification or permission to restore a monument from our example. Modern engineering needs measured material conditions and structural knowledge beyond this historical presentation ⁶.

The text witnesses a meeting of textbook mathematics and architectural craft. One module converts size differences into numerical transformations,and counting elements organizes a compound form. MacTutor values the Key as a substantial teaching work,but that judgement is not a count of buildings altered by it. Such a number is not supplied by the consulted sources ^{4 25}.

Criticism contributes to its historical value. The historian notes that equating roof area with the area on which muqarnas is built conflicts with inclined roofs in the definition and examples. Reading is neither celebration that suppresses inconsistency nor wholesale dismissal because an inconsistency exists. It preserves the method's contribution and limits while distinguishing modern interpretation from the author's statement ²⁶.

The attribution is the studied written mensuration procedure,not the whole architectural form. Muqarnas predates this book,masons possess panel-making knowledge,and the modern researcher explains how evidence helps reconstruct it. Our reading does not establish absolute priority for every area rule or coefficient in world mathematics. A book title should not become exclusive ownership of shared craft ²⁷.

**Sunni frame:**the maker's Muslim identity is documented,and personal affiliation remains separate from ceiling geometry. **Academic account:**Dold-Samplonius describes a practical average and discloses recomputation limits. The museum does not convert numerical differences into a new judgement of the person. Precisely printed digits alone cannot validate a physical model's accuracy ^{11 28}.

An independent paper example distinguishes projection from surface:a tilted rectangle of width2and surface length3has area6;if its projected length is2,the projection has area4. These hypothetical numbers are neither measurements of the photograph nor al-Kashi's values. Their difference does not change the unit;it reminds us to define the desired quantity before multiplying ²⁹.

An upward view also omits slopes and depths. Counting visible stars in a panorama cannot yield a reliable historical coefficient. This explanation shows why the mathematician cares about modules and elements,without offering construction instructions or structural advice. The comparison remains on paper,not a climbing exercise or attempt to measure a high ceiling ^{6 20}.

The examined witness is the1992study's selected pages and embedded translated statements,not al-Kashi's manuscript or the article's complete Arabic text and translation. Figure8on printedp213was visually inspected to confirm planar forms and numbers;it is not republished. Cairo1967and Damascus1977editions are bibliographical information through Unat,not editions claimed read here ^{30 18}.

Amir Pashaei's photograph shows an iwan ceiling in Atabki Sahn at Fatima Masumeh Shrine,Qom,underCC BY-SA4.0. Commons describes an HDRpanorama combining25images. Blue/gold ornament,written tile borders and an opening to the sky are visible. Neither the building nor verified use of the method is attributed to al-Kashi. It is modern visual context,not an original application witness ³¹.

Names and mathematical shapes meet here:almond,biped and barley-kernel designate distinct elements in the analysis,not a licence to interpret every decoration as one religious symbol. The text distinguishes facets,roofs and intermediate surfaces;the historian similarly distinguishes older craft from a beautiful modern picture of it ³².

Calculation does not abolish beauty;it asks another question. A visitor can attend to light and colour,then ask about surface area,without reducing architectural value to a number or giving every arch a single origin. The gap between real construction and mathematical assumptions is precisely what makes critical study richer than drawing a ceiling and calling it perfectly accurate ³³.

On paper,a square with side1has area1,and one with side2has area4. Two matching half-squares make one square's area. This demonstrates parts and units,not a method for building muqarnas overhead. The studied diagram contains different classes of elements;equal counts do not imply equal areas ¹⁶.

Ask whether we calculated a surface or its shadow on paper,and whether the answer approximates a real object or belongs to an ideal shape. If a historian's calculation differs from an old number,record the difference rather than hiding it. These questions turn arithmetic history into inquiry rather than a contest to assign everything in a ceiling to one hero ³⁴.

• Dold-Samplonius,selected actual pages,especially module/coefficient and qualifications ³⁵.
• MacTutor,maker identity,teaching work and qualified priority accounts ³⁶.
• Unat,work7,reception and editions not read here ¹⁸.
• Commons,photograph/context/rights,not al-Kashi building attribution ³⁷.