Land surveying in ancient states is documented not only for Eurasia but also for the Americas, amply attested by two Acolhua-Aztec pictorial manuscripts from the Valley of Mexico. The Codex Vergara and the Códice de Santa María Asunción consist of hundreds of drawings of agricultural fields that uniquely record surface areas as well as perimeter measurements. A previous study of the Codex Vergara examines how Acolhua-Aztecs determined field area by reconstructing their calculation procedures. Here we evaluate the accuracy of their area values using modern mathematics. The findings verify the overall mathematical validity of the codex records. Three-quarters of the areas are within 5% of the maximum possible value, and 85% are within 10%, which compares well with reported errors by Western surveyors that postdate Aztec-Acolhua work by several centuries.and surveying played an integral role in the development of ancient states as growing economic demands and political complexity required increasingly precise records of distribution, amount, and quality of agricultural resources. In Eurasia such records begin several millennia ago. In the Americas, time depth of land surveying is unknown, but two extant native-style pictorial records attest that a sophisticated system was used by AcolhuaAztec people prior to European contact.Painted circa A.D. 1543-1544, the Codex Vergara* and the Códice de Santa María Asunción* provide unparalleled data to reconstruct Acolhua-Aztec (hereafter Acolhua) metrology and arithmetic and to evaluate the credibility of the records. The first comprehensive study of these codices (1) demonstrated that Acolhua land surveyors/scribes recorded side lengths of hundreds of agricultural fields using a standard linear measure, the tlalcuahuitl ("land rod"; T, equal to 2.5 m) and shorter-than-standard distances depicted by hearts, arrows, and hands (metrological monads, which are simple, indivisible units). Unexpectedly, it was also demonstrated that surveyors reported field areas in square tlalcuahuitl (T 2 ), which were depicted pictographically by a spatially distinctive form of numerical notation (Fig. 1).A follow-up study of 367 quadrilateral fields in the Codex Vergara (hereafter the Vergara) reconstructed both Acolhua survey metrology and area algorithms from an emic perspective (that of cultural insiders) (2, 3). The study validated the previously known Acolhua standard linear measure and amassed quantitative evidence establishing both the metric values of Acolhua monads and their use as fractions in area computation. Using Acolhua congruence arithmetic, five recurrent algorithms were detected that exactly reproduced 78% of the recorded areas. These results indicated that areas were indeed computed rather than measured by some physical means.Although Acolhua arithmetic was functionally accurate within their cultural context, in this work we analyze the accuracy of their recorded areas from the etic (cultural outsider) perspective of Western mathematics (3). Such assessment would be fairly st...

We study a singularly perturbed semilinear elliptic partial differential equation with a bistable potential on an oval surface. We show that the transition region of minimizers of the associated functional with a suitable constraint converges in the sense of varifolds to a minimal closed geodesic on the surface.

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