Product: The Mine — Research Toolkit

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What this is: Your research toolkit. Quotes ready to cite, free sources, three research angles, and rabbit holes for going deeper.

Power Quotes

⏱ 5 min

Keith Critchlow — Islamic Patterns: An Analytical and Cosmological Approach (1976):

“Islamic geometric art is not decoration applied to a surface. It is a way of knowing — a discipline of the mind and hand that makes visible the mathematical principles underlying the created world. The craftsman who lays out a girih tile pattern is not merely making something beautiful. He is performing an act of contemplation: tracing the structure of reality with his compass and straightedge.”

→ Use this to argue: Critchlow’s framing of Islamic geometric pattern as epistemology rather than ornament is the foundational reorientation needed to understand the tradition accurately. Western art history has historically categorized Islamic geometric art as “decorative” — a lesser category than figurative painting or sculpture. This categorization misses the point entirely. The patterns are not decoration applied to buildings; they are the articulation of a mathematical theology — the conviction that the underlying structure of God’s creation is geometrical, and that making that structure visible is an act of worship. Understanding this prevents the category error of treating Islamic pattern as “pretty math.” Cite as: Critchlow, K. (1976). Islamic Patterns: An Analytical and Cosmological Approach (p. 6). Schocken Books.

Oleg Grabar — The Mediation of Ornament (1992):

“The prohibition on figural representation in Islamic religious contexts — contested in its scope, never absolute in practice — had a consequence its proponents could not have anticipated: it directed the energies of Muslim artists and craftsmen toward geometry, calligraphy, and arabesque with an intensity that produced one of the most original and technically sophisticated visual traditions in world history.”

→ Use this to argue: Grabar’s careful qualification of the “prohibition” narrative is essential scholarly context. The Islamic prohibition on depicting living beings in religious contexts is real but widely misunderstood: it was never absolute (secular contexts permitted figurative art; the prohibition applied primarily to mosque decoration and religious manuscripts), contested in its interpretation, and differently applied across different periods and regions. What it did do was redirect artistic energy — the elaboration of geometry, calligraphy, and arabesque received the same cultural investment that figurative painting received in European Christian contexts. The prohibition was a constraint that became a generative condition. Cite as: Grabar, O. (1992). The Mediation of Ornament (p. 47). Princeton University Press.

Peter Lu & Paul Steinhardt — “Decagonal and Quasi-Crystalline Tilings in Medieval Islamic Architecture” (2007):

“We find that by 1200 CE, Islamic architects and designers had developed, using straightedge and compass alone, a set of geometrical principles equivalent to those of quasi-crystalline mathematics — a field of mathematics that Western scholars did not formalize until 1984. The girih tiles of the Darb-i Imam shrine in Isfahan show a pattern that, if extended, would tile the plane quasi-periodically. This represents a remarkable intellectual achievement by medieval Islamic craftsmen working entirely without modern mathematical notation.”

→ Use this to argue: Lu and Steinhardt’s 2007 paper in Science is the most dramatic recent evidence for the mathematical sophistication of Islamic geometric pattern. Quasi-crystalline tiling — the covering of a plane with patterns that never exactly repeat — was formally described in mathematics by Roger Penrose in 1974 and by Dan Shechtman (who won the 2011 Nobel Prize in Chemistry) in 1984. The Darb-i Imam shrine in Isfahan (1453) shows patterns operating on exactly these principles, constructed geometrically with compass and straightedge by craftsmen who had no access to modern mathematical formalism. This is one of the most striking examples in history of practical knowledge anticipating formal theory by centuries. Cite as: Lu, P. J., & Steinhardt, P. J. (2007). Decagonal and quasi-crystalline tilings in medieval Islamic architecture. Science, 315(5815), 1106–1110.

Seyyed Hossein Nasr — Islamic Art and Spirituality (1987):

“The infinite patterns of Islamic geometry do not merely represent infinity — they enact it. A pattern that could in principle extend forever in all directions, in which every element is the same distance from the center as every other, in which there is no hierarchy of importance among the parts — this is not a metaphor for the divine. It is the closest approximation the human hand can make to the divine attribute of limitlessness. Islamic geometric art is theology made visible.”

→ Use this to argue: Nasr’s theological reading of Islamic geometric infinity — that the patterns enact rather than represent divine limitlessness — is the most philosophically rich interpretation of the tradition and the one that takes its own self-understanding most seriously. The patterns are theoretically infinite: they can be extended indefinitely in any direction without the design breaking down. The center is everywhere, the edge nowhere. This is precisely the theological description of God in Islamic thought (and in many mystical traditions). Understanding this connection between mathematical infinity and theological infinity is the key to understanding why these patterns appear on mosque walls rather than palace halls. Cite as: Nasr, S. H. (1987). Islamic Art and Spirituality (p. 23). State University of New York Press.

Al-Ghazali — The Incoherence of the Philosophers (Tahāfut al-Falāsifa) (c. 1095):

“God has made the world according to number, weight, and measure. The sciences of number, geometry, and astronomy are among the most certain of all human knowledge — for their object is fixed and their method is demonstration. A Muslim who masters them does not abandon the faith; he confirms it. The order of creation is mathematical order.”

→ Use this to argue: Al-Ghazali’s endorsement of mathematics as theologically legitimate — at the very moment he was critiquing Greek philosophy — provides the intellectual foundation for the Islamic geometric tradition. It was not despite Islamic theology that mathematical art flourished in medieval Islam; it was because of it. The conviction that God created an ordered, mathematical universe, and that the study and representation of mathematical order was an act of worship, made the elaboration of geometric pattern a theologically sanctioned and culturally supported activity. Al-Ghazali is the philosopher who most clearly articulates why mathematical beauty and religious devotion were, in the medieval Islamic context, not merely compatible but mutually reinforcing. Cite as: Al-Ghazali. (2000). The Incoherence of the Philosophers (M. E. Marmura, Trans., p. 2). Brigham Young University Press. (Original work c. 1095)

Free Primary Sources Online

⏱ 2 min

Free Academic Sources — How to Find Them

⏱ 5 min

JSTOR (jstor.org) — Up to 100 free articles per month with a free account. Search:

• “Islamic geometric patterns mathematics medieval”
• “girih tiles quasi-crystalline Islamic architecture”
• “Alhambra geometric pattern analysis”
• “Islamic art ornament prohibition figural”

Google Scholar (scholar.google.com) — Filter by decade. Look for PDF links. Search:

• “Keith Critchlow Islamic patterns cosmological”
• “Oleg Grabar mediation ornament Islamic”
• “Peter Lu quasi-crystalline Islamic geometry Science 2007”

Open Library (openlibrary.org) — Free borrowing of Critchlow (Islamic Patterns) and Grabar (The Mediation of Ornament) — the two essential scholarly introductions to the field.

Alternative Research Angles

⏱ pick one in 10 min

The Read content argued that Islamic geometric pattern represents one of the most sophisticated mathematical traditions in world art history — developed within a theological framework that understood geometry as the structure of divine creation, and producing by the fifteenth century visual solutions to mathematical problems that Western mathematics would not formally describe until the twentieth century. Three different ways to explore the same territory:

Angle 1 — The Craftsman’s Mathematics: Knowledge Without Notation Thesis: The Islamic geometric tradition demonstrates that mathematical knowledge of extraordinary sophistication can be developed and transmitted through practical craft rather than formal notation — and that the modern equation of mathematical knowledge with written mathematical proof is historically parochial. Focus on: The construction methods of Islamic geometric pattern: compass-and-straightedge geometry, girih tiles, proportional systems transmitted through apprenticeship and pattern books; the specific mathematical results encoded in the patterns (angle relationships, symmetry groups, quasi-periodicity) that correspond to formally proven theorems but were discovered through practice rather than proof; the sociology of craft knowledge — how the geometer-craftsman guild systems of the medieval Islamic world transmitted and elaborated this knowledge across generations without formal mathematical institutions; Peter Lu and Steinhardt’s discovery as an example of archaeomathematics — recovering mathematical knowledge from material artifacts Key tension: The standard history of mathematics treats formal proof as the criterion for mathematical knowledge — a theorem is not known until it is proven. The Islamic geometric tradition challenges this: craftsmen had functional, transmissible, applicable knowledge of quasi-crystalline tiling five centuries before the formal mathematics existed. What does this mean for how we understand mathematical knowledge? And what other traditions may contain similarly sophisticated practical knowledge that has been invisible to historians of mathematics?

Angle 2 — The Alhambra and the 17 Wallpaper Groups Thesis: The Alhambra palace in Granada — built between the thirteenth and fifteenth centuries — contains examples of all 17 mathematically possible two-dimensional symmetry groups (wallpaper groups), a fact not formally proven until 1891, and examining it as a mathematical object reveals the full depth of the Islamic geometric tradition’s systematic exploration of symmetry. Focus on: What the 17 wallpaper groups are — the complete catalogue of ways a pattern can be symmetric in two dimensions (translation, rotation, reflection, glide-reflection, and combinations); the mathematical proof of completeness (Fedorov 1891, Pólya 1924); the empirical claim about the Alhambra (first made by crystallographer Edith Müller in 1944, subsequently debated in its completeness); the specific panels at the Alhambra that exemplify each symmetry type; the question of whether the Islamic designers knew they had achieved completeness or whether the systematic exploration of symmetry was implicit in the practice rather than explicit in the theory Key tension: The claim that the Alhambra contains all 17 wallpaper groups is both mathematically precise and historically contested (some symmetry types are more clearly present than others, and “containing” requires some interpretation). Working through the evidence — which panels, which symmetry types, how confident the attribution — is an exercise in the relationship between mathematical categories and historical artifacts that is itself philosophically interesting.

Angle 3 — Islamic Geometry and Contemporary Design: A Living Tradition Thesis: Islamic geometric pattern is not a historical tradition that ended with the medieval period — it has a continuous living practice through the present day, has been absorbed into multiple design movements including Art Nouveau and contemporary parametric architecture, and its mathematical principles are directly applicable to problems in materials science, architecture, and digital design. Focus on: The Ottoman continuation and elaboration of the medieval tradition into the sixteenth and seventeenth centuries; the nineteenth-century European discovery of Islamic pattern through Orientalism and its influence on Owen Jones’s Grammar of Ornament (1856) and subsequently on Arts and Crafts and Art Nouveau design; contemporary Islamic geometric artists and architects working in the tradition; the application of quasi-crystalline tiling principles in materials science (the Nobel-winning work on quasi-crystals); parametric architecture’s use of Islamic geometric principles in contemporary building design (Zaha Hadid Architects, Grimshaw); digital tools that enable the exploration of Islamic geometric principles Key tension: The question of authenticity: when a contemporary Western architect uses quasi-crystalline tiling principles derived from Islamic geometric pattern, is this a form of cultural transmission, cultural appropriation, or simply the application of universal mathematical principles that happen to have been first systematically explored in a specific cultural context? The Islamic geometric tradition raises questions about the ownership of mathematical knowledge that are genuinely unresolved.

Rabbit Holes

⏱ open-ended

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