One Thousand and One Conjectures

Fracture physics and Viking economics meet in the hack-silver hoard. When brittle materials are broken repeatedly and more or less at random, the resulting fragment masses follow a universal power-law distribution — a robust result from fragmentation physics that holds for shattered…

An astrolabe only works at the latitude its plate is engraved for, so every surviving instrument silently records where its maker expected it to be used. That turns the corpus of surviving astrolabes into medieval market-research data: the set of latitudes engraved…

The modern capital campaign — where a museum or university secures a headline anchor gift or star acquisition just before asking everyone else for money — is here read back into the medieval church. Relic translations, the ceremonial installation of a saint's…

The Polya urn — the classic mathematical model of path dependence, in which each ball drawn adds another of its color so that early luck compounds forever — is here applied to the medieval pilgrimage market. Pilgrims bought cheap metal badges at…

Currency unions are supposed to be a modern invention, and low variance in the unit of account is their hallmark: struck coinage keeps its weight uniform because mints are policed. The trans-Saharan trade, by contrast, ran for centuries on bars of rock…

Great Zimbabwe on its inland plateau and the Swahili port of Kilwa on the Indian Ocean coast are the two famous sites joined here, and the joint is metrological. Kilwa struck its own coins to a known weight standard and shipped the…

The backward-bending supply curve — the textbook anomaly in which workers with an income target respond to falling prices by working more, not less — is here joined to the fate of Norse Greenland. The colony's export staple was walrus ivory, and…

In the 13th century, English law adopted a rule for how far apart markets must be — the spacing derived in the classic legal reckoning from a day's return journey on foot, about 6.6 miles. The conjecture joins this statute to the…

Caravanserais — the fortified roadside inns of the Silk Road and the Islamic world — existed to break overland journeys into daily stages, so their spacing was governed by a physical constant: how far a loaded camel walks in a day. The…

The Cistercian order expanded by filiation — mother abbeys founding daughter houses — under statutes and economic pressures that discouraged crowding, since each abbey lived off its own granges and lands. That is the setup of competitive exclusion in ecology, where territorial…

The medieval Champagne fairs ran as a fixed annual cycle of six fairs in four towns, and the merchants of Flanders and Italy worked the whole circuit. Seen with modern eyes, scheduling that cycle is a traveling-merchant optimization: fair dates and durations…

The Shapley value, cooperative game theory's canonical answer to fair division, pays each member of a coalition its average marginal contribution to the coalition's worth. The Hanseatic League was such a coalition: member cities jointly controlled Baltic and North Sea trade routes,…

The Byzantine empire ran a fire-signal chain that could relay news of Arab raids from the Cilician frontier to Constantinople within hours — a communication channel in the exact information-theoretic sense, and one operating under noise, since fog and haze could blind…

As chess spread from India across Eurasia, both its objects and its rules mutated regionally: the piece carved as an elephant in the Islamic world became the bishop in Europe, and the moves themselves varied between documented regional rule sets. Linguistics maps…

Gothic churches were built in successive campaigns over decades, and masons could watch how earlier bays and earlier buildings cracked, leaned, or stood. Modern limit analysis can compute, for any buttress geometry, how far it sits from the thrust-line optimum — the…

This joins Gothic cathedral construction to the economics of competitive escalation — the same arms-race dynamics seen in armament races and advertising wars. A cathedral's height was a prestige signal aimed at specific rivals, and the relevant rivals were local: the sees…

This joins medieval craft practice to modern acoustics. An organ builder must decide how pipe diameter should vary with pipe length across a rank — too narrow and the trebles turn thin and stringy, too wide and the basses go dull —…

This joins historical linguistics to campanology. Dialectologists map isoglosses — the boundary lines where one regional speech feature gives way to another — and those lines famously follow the channels of contact: rivers, roads, trade routes. Bell founding, the conjecture claims, worked…

This joins Viking shipbuilding to modern fracture mechanics. A clinker hull is a shell of overlapping oak planks, and its worst enemy is a running split: a crack that starts at a fastening and propagates along the grain. Fracture mechanics says such…

This joins medieval pilgrimage history to tribology, the physics of wear. Every pilgrim who crossed a church threshold removed a microscopic layer of stone, and centuries of feet turned steps into shallow basins; the missing volume is a footfall counter that no…

This joins the Dresden Codex — the finest surviving Maya astronomical manuscript — to the mathematics of best rational approximation. The codex's Venus table tracks the planet with a canonical 584-day period, but the true synodic period is close to 583.92 days,…

This joins medieval church archaeology to geomagnetism. Churches were meant to face east, yet surveyed orientations scatter by several degrees, and the scatter is usually written off as sloppy surveying. But once the magnetic compass entered the building trades — plausibly around…

This joins two of the medieval world's best data series: the Nilometer readings of the Nile's annual flood, kept for centuries at Cairo, and the grain prices preserved in the Cairo Geniza's merchant letters. Egypt's harvest was the flood: a low Nile…

This joins the Classic Maya collapse to the physics of percolation — the mathematics of systems that fail not gradually but when a critical threshold is crossed. Lowland Maya cities rode out the dry season on constructed reservoirs, so each city's resilience…

This joins the vast water network of medieval Angkor to the theory of self-organized criticality — the sandpile physics in which a slowly loaded system tunes itself to a critical state where avalanches of every size occur, their sizes following a power…

Markowitz's portfolio theory says diversification pays exactly when the assets you spread across are weakly correlated — variance falls fastest where covariance is low. The medieval open-field system's scattered strips, a puzzle ever since they were read as peasant insurance, are a…

The Bass diffusion model, the workhorse of new-technology adoption studies, splits uptake into two forces: an innovation coefficient p (adopters persuaded by external influence) and an imitation coefficient q (adopters copying their neighbours). Medieval Europe's watermill boom is a textbook diffusion process,…

Airborne LiDAR has stripped the canopy from the Maya lowlands and revealed settlement in the tens of thousands of structures, making true regional settlement hierarchies measurable for the first time; settlement-scaling theory, meanwhile, holds that integrated urban systems produce Zipfian rank-size distributions…

The Swahili stone towns of the East African coast — Kilwa above all — were built of coral rag set in lime mortar, and construction there was governed by the monsoon: the rains halt building and lime-burning, so work proceeds in an…

RT60 — the time a sound takes to decay by sixty decibels — is the basic figure of architectural acoustics, and a great stone nave can hold a note for many seconds; a fast syllabic melody blurs into mud in such a…

The Norse thing — the open-air assembly at which law was recited, cases were pled and decisions proclaimed — was above all an exercise in unamplified speech before a crowd, and sites like Thingvellir have long invited the suspicion that their cliffs…

Muqarnas — the honeycomb vaulting of Islamic architecture, thousands of small stepped niches filling domes and squinches — is usually read as pure geometry made ornament. But a surface of many differently sized and angled cells is exactly what a modern acoustician…

Medieval masons cut personal marks into the blocks they dressed — piece-work signatures for the paymaster — and a cathedral wall therefore carries, frozen in stone, a record of who cut what. Work organisation leaves statistical fingerprints: if masons produced blocks in…

The Polynesian settlement of the Pacific is the greatest search problem ever solved by a preindustrial society, and optimal search theory says how a rational searcher orders targets: not by raw distance, but by how easy each target is to find. An…

The deep blue of medieval stained glass comes from cobalt, and medieval Europe mined very little of it: the colorant travelled along long-distance routes whose ultimate sources lay far to the east. Every ore body carries a trace-element fingerprint — the ratios…

Joins autocatalytic reaction kinetics to the sociolinguistics of conquest: in Egypt's dated documentary papyri, the replacement of Greek and Coptic by Arabic behaved like a chemical substitution running with a nearly universal rate constant. The mechanism is autocatalytic — every office and…

Joins the bunching estimators of modern public finance — the kink analysis that detects tax brackets as piles of taxpayers stacked exactly at the threshold — to the Chinese examination system. If jinshi degrees simply tracked the geography of talent and schooling,…

Joins index-number theory — the chained construction behind the modern consumer price index — to Tamil temple epigraphy. Thousands of dated Chola-era endowment inscriptions record in stone the rates of paddy, ghee, and oil required to keep a temple lamp burning in…

Joins stream-power hydrology to the great fiscal cadastre of 1086: the Domesday survey, in recording what each water-mill rendered to its lord, unwittingly logged a physics meter across England. A mill's earning power was set by the hydraulic power of its site…

Joins process-engineering labor accounting to Viking naval history. Tally the person-hours in a longship and the surprise is that the hull is the cheap part: replication labor budgets show a large woolen sail — the wool sorted, spun thread by thread, woven,…

Joins experience-curve economics to Islamic mathematical geography. In manufacturing, quality improves as a power law of cumulative output — the experience curve — because every unit produced teaches its producers. Every mosque is such a unit: it must face Mecca, its measurable…

Vertical-monopoly pricing from industrial organization shows that a chain of independent monopolists along one route charges more in total than a single owner would, because each ignores the damage its markup does to the others' traffic. This conjecture reads the toll castles…

Early-warning theory of critical transitions says complex systems approaching a tipping point betray themselves in advance — variance and autocorrelation rise as the system loses resilience. This conjecture applies those signals to Maya epigraphy. The Classic Maya katun-ending dedication network was a…

Priority queueing theory has a textbook pathology: give one class of customers near-absolute priority and their waiting times stay compact while low-priority waits blow up into a heavy tail. This conjecture finds that pathology in the Heian court's promotion ladder as recorded…

Common-value auction theory predicts the winner's curse — the highest bidder is the one who most overestimated the shared value — and that experienced bidders learn to shade their bids downward. This conjecture finds the earliest documented correction in Venetian state finance:…

Joins the two poetic registers of Song China to two enforcement institutions: regulated shi verse was examination technology, rhymed against the frozen categories of the official rhyme books long after speech had moved on, while ci lyrics were written to living melodies…

Maya polychrome cylinders pose a closed-loop composition problem, and workshop painters solved it the way a tailor does: by giving the garment a seam and placing it at the back. Because these vessels were presented and used along a principal viewing axis…

Medieval painters had almost no access to antique painting or free-standing portrait sculpture, but Roman coins passed through their world by the thousand. If coinage was the operative model medium for picturing antiquity, ancient rulers should be drawn in strict numismatic profile…

In Byzantium the coin was the fastest image medium the court controlled: dies were recut within months, while mosaics, ivories, and books ran on multi-year project cycles. Innovations in imperial iconography — new crown forms, loros arrangements, newly court-adopted Christ and Virgin…

Ornament in stained glass — border types, diaper grounds, grisaille foliage — is studied inside glass scholarship, but glaziers stocked their cartoon chests from painters' books. Ornament families should therefore appear in dated manuscripts before dated glass of the same region, with…