This study investigates the structural and epistemological role of numerical techniques (ḥiyal ʿadadīyah) in shaping architectural form during the Timurid period, with a particular focus on the design, measurement, and proportional logic of domed areas (gonbadkhānehs). Within the post-Mongol tradition of Islamic sciences, especially in works influenced by al-Fārābī’s classification of the sciences, ḥiyal (ingenuity, contrivance, or device) is defined as a branch of practical science (ʿilm al-ḥiyal) that connects theoretical mathematics with applied arts and crafts. This science consists of two principal components: geometrical ḥiyal, which operates through constructive diagrams and spatial transformations, and numerical ḥiyal, which relies on calculation, measurement, and the manipulation of units and ratios. Whereas the geometrical dimension—often treated under headings such as handasah ʿamaliyyah (practical geometry) and associated with instruments like compasses, rulers, and templates—has attracted considerable attention in modern scholarship on Islamic art and architecture, its numerical counterpart remains largely understudied. However, it is precisely numerical ḥiyal, with its systematic use of units, modules, and multipliers, that can clarify how abstract number and concrete space were intertwined in the design of Islamic buildings.
The article seeks to recover the importance of numerical ḥiyal in architectural practice during the Timurid dynasty. This period has often been characterised in historiography by its “obsessive geometry” and sophisticated decorative programmes, but less frequently analysed in terms of its underlying systems of measure and number. Rather than approaching geometry and arithmetic as separate analytical categories, the study proposes that they should be understood as two interdependent dimensions of a unified design methodology. In this methodology, abstract mathematical principles—such as proportional series based on 3 and 6, or modular grids expressed in units of gaz—are translated into repeatable rules for laying out plans, organising structural spans, and calibrating the relationship between main domed spaces and their subsidiary chambers. The central research question is therefore not only how Timurid architects drew complex geometries, but how they counted, measured, and standardised space through numerical operations.
The methodological framework combines descriptive, analytical, and comparative approaches. First, the study undertakes a close reading of Persian and Arabic scientific and historical texts produced between the thirteenth and fifteenth centuries. Among the mathematical works, special attention is given to Abū al-Wafāʾ al-Būzjānī’s Manāzil al-sabʿ and Ghiyāth al-Dīn al-Kāshānī’s Miftāḥ al-ḥisāb. These treatises discuss practical procedures for measuring areas and volumes, estimating quantities of materials, and converting between different units. They record the use of instruments such as the dhirāʿ (cubit), qabḍah (hand-span), ashl (yardstick), and grid-based drafting boards. Such sources testify to an advanced culture of precision in the workshop, where numerical rules were not abstract formulae but operative tools for surveyors, architects, and master builders. In parallel, narrative chronicles such as Sharaf al-Dīn Yazdī’s Ẓafarnāmah and Ghiyāth al-Dīn Khwāndamīr’s Ḥabīb al-siyar are analysed for their references to architectural commissions, building campaigns, and royal supervision of construction, which illuminate the institutional context in which these numerical practices were embedded.
Second, the study incorporates architectural documentation and plan-based measurement. Existing surveys and measured drawings are collected for a selection of major Timurid monuments. The criteria for selecting case studies are threefold: (1) clear attribution to the Timurid period and, where possible, to specific patrons within the Timurid dynasty; (2) the availability of sufficiently accurate plans or the possibility of reconstructing the geometry of the building from archaeological and archival sources; and (3) the presence of a domed area functioning as a central spatial and symbolic unit, whether in a funerary, congregational, or commemorative context. Based on these criteria, the corpus includes the mausoleum of Khwāja Aḥmad Yasavī in present-day Kazakhstan, the Gur-e Amīr Mausoleum and the Mosque of Bībī Khānum in Samarqand, the Goharshād Mosque in Mashhad, the shrine complex of Mīr ʿAlī Shīr Navāʾī in Mazar-e Sharīf, and several other domed structures for which reliable plans exist.
The mausoleum of Khwāja Aḥmad Yasavī occupies a pivotal place in this analysis and serves as the reference point for deriving a modular framework. Commissioned directly by Timur and extensively documented in both textual and architectural sources, the monument consists of a large square domed hall (30 gaz × 30 gaz) and a smaller rear domed chamber measuring 12 gaz per side. By reconstructing these dimensions at scale and superimposing a square grid divided into gaz-based modules, the study identifies a hierarchical system of measures governing the distribution of walls, piers, openings, and secondary spaces. This grid not only structures the main domed volume but also regulates the alignment and width of corridors, side rooms, and transitional zones. The resulting “measurement diagram” of the Yasavī mausoleum is treated as an exemplary instance of numerical ḥiyal applied to architecture, and it provides a comparative template for examining other Timurid buildings.
Applying this modular diagram to the remaining case studies reveals strong patterns of consistency. In monuments such as the Gur-e Amīr and the Goharshād Mosque, the principal spans and bay widths can be expressed as simple multiples of the same basic unit. At the same time, secondary spaces frequently adhere to subdivisions of one-half or one-third of the main module. The analysis shows that large-scale compositional elements—such as the diameter of domes, the length of principal axes, and the dimensions of major iwans—tend to be organised according to a series of multiples of six (12, 18, 24, 30, and 48 gaz). In contrast, smaller cells, staircases, and ancillary rooms are more often based on multiples of three. In other words, the overall system emerges as a dual structure combining a macro-scale order rooted in 6 and a micro-scale order rooted in 3. This duality resonates with the emphasis in mathematical texts on proportional progressions and on the practical advantages of working with highly divisible numbers for layout and estimation.
The findings further suggest that these numerical schemes were not accidental or merely convenient approximations. Historical evidence points to the existence of state-regulated standards of length—such as the dhirāʿ shāhī, dhirāʿ hāshimī, and regional variants associated with Samarqand and Herat—used in administrative, mercantile, and architectural contexts. When the measured data from the architectural plans are converted back into historical units using a Timurid gaz of approximately 60.6 cm, the resulting values align closely with the discrete series of integers reconstructed from the Yasavī mausoleum. This alignment implies a conscious attempt to implement consistent royal standards across important state-sponsored projects, especially those that functioned as dynastic mausolea, Friday mosques, or shrines associated with charismatic saints.
In light of these observations, the article argues that numerical operations in Timurid architecture should be understood as a central component of design rather than a secondary or purely symbolic layer. Numerical ḥiyal, as reconstructed here, constitutes an applied intellectual system that integrates measuring instruments, modular grids, and proportional series into a coherent framework for planning, construction, and visual order. It underpins the repetitive yet carefully calibrated rhythms of bays and piers, the precise registration between structural spans and decorative programmes, and the ability to extend or reproduce standardized domed units across different sites. At the same time, by linking these practices to broader intellectual currents—such as the post-Mongol revival of mathematics, the circulation of astronomical and occult sciences, and the Timurid court’s investment in what might be called a “Pythagorean politics of architecture”—the study situates architectural measurement within a larger landscape of knowledge and power.
Ultimately, the article contributes to the historiography of Islamic architecture in three main ways. First, it introduces the notion of numerical ḥiyal as a productive analytical lens for investigating the relationship between number, measure, and architectural form, complementing existing work on geometrical design methods. Second, it proposes a concrete modular model, grounded in the Yasavī mausoleum and verified across a set of major Timurid monuments, that clarifies how historical units such as the gaz were operationalised in design and construction. Third, it demonstrates that the domed architecture of the Timurid period cannot be fully understood without recognising the role of standardized numerical systems in stabilising spatial organisation, encoding symbolic values, and facilitating the transmission of design knowledge across regions and generations.