### Tony Phillips' Take Blog on Math Blogs

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This month's topics:

Aztec area algorithms

The Aztec numbering system is pretty well understood: it is a base-20 place-value system with a symbol for zero. But some extra numerical symbols ("arrow," "hand," "heart," "bone") appear on land surveys, where they seem to represent quantities smaller than 1. An explanation for these mysterious symbols was recently (April 4, 2008) published in Science. Barbara Williams (Wisconsin-Rock County) and Maria del Carmen Jorge y Jorge (UNAM) exploited the data from the Codex Vergara (in the Bibliothèque Nationale) and the Codice de Santa Maria Asuncion (Fondo Reservado de la Biblioteca Nacional de Mexico, UNAM), where a number of plots are recorded with their side measurements and their areas. The areas are invariably whole numbers of the squared unit; by a process of trial and error they were able to reconstruct some of the algorithms the Aztecs had used to calculate the areas, and work backwards to figure out values for the unknown symbols.

Part of the holdings of a sixteenth-century Mexican landowner, shown with linear side measurements and with areas. Details of two pages from the Codice de Santa Maria Asuncion (Fondo Reservado de la Biblioteca Nacional de Mexico, UNAM). The sides of the plots are labeled with their lengths in Aztec numeration: Solid dot = 20, vertical line = 1, grouped by 5s. The areas are written in Aztec place-notation with the 20s place in the center and the 1s place in a tab on the upper right. Some of the notation for "fractional" length measurements appears in this chart: the arrow and the hand. The dimensions of the right-most plot are, clockwise from the top, 35, 34+hand, 29+arrow, 39; its area is given as 59X20+12 or 1192 square units. In the left-most plot the 17 is presumably a copying error for 37; otherwise area 767 is impossible. Images courtesy of Prof. Maria del Carmen Jorge.

• Starting with the simplest example: the Codex Vergara contains many examples of plots with length 20, width 10 and area 200. It also has an example of a plot with length = 20, width = 10 + ->, and area = 210. The authors infer that in all these cases the Aztecs used "area = length times width," and that consequently -> is worth 1/2 the unit.
• In a more complex example, the plot is a near-trapezoid with sides 26, 32, 30, 10 and area 588. They infer that the Aztecs are using the surveyor's rule: "area = the product of the averages of the opposite sides," because this algorithm is simple and gives that exact answer: (26 + 30)/2 X (32 + 10)/2 = 588. Then they consider the quadrilateral with sides 36, 12, 37, 12 and area 438. The only plausible explanation they find is that the surveyor's rule (here identical with the trapezoid rule) was used, with (36+37)/2 = 36 + -> and 12 X (36 + ->) = 432 + 12 -> = 438. This implies that the Aztecs used -> in calculation, and not only in measurement.
• Other algorithms are similarly inferred. For example the quadrilateral with sides 24, 16, 25, 24 and area 492 was most plausibly calculated with the "triangle rule" (taking it as two right triangles joined at the hypothenuse): (24 X 16)/2 + (25 X 24)/2 = 492.

As the authors remark, the explanations for the hand (3/5 unit), the heart (2/5 unit) and the bone (1/5 unit) are mathematically less compelling. [In fact, in every case that I checked the author's "calculated value" involves arbitrary approximations and/or roundings before matching with the "recorded area" from the Codex. For example their plot 03-030 has sides 42, 20, 47, 23 and area 1005, for which they use the rule "average of one pair of opposite sides times an adjacent side" as follows, using the arrow (-> = 1/2) and the heart ((=) = 2/5): (23+20)/2 = 21+-> ~ 21 + (=), 47(21+(=)) = 987 + 47(=) = 987 + 9(5(=)) + 2(=) = 987 + 9X2 + 2(=) = 1005 + 2(=) rounds down to 1005. The problem is that there are too many unknowns. We have no clue as to the exact shape of many of the plots. We do not even know whether the surveyors always used a formula going from side lengths to areas or whether other estimates or measurements were involved. But the examples spelled out above do give very strong evidence that sixteenth-century Aztecs had good algorithms for computing areas of rectangles, trapezoids and triangles, and that they could and did calculate with quantities less than the unit. -TP] This article was featured in ScienceNOW   ("How Aztecs Did the Math"), and picked up in the Los Angeles Times   ("Aztec math finally adds up"), the Scientific American  ("Aztec Math Used Hearts and Arrows") and National Geographic News  ("Aztec Math Decoded, Reveals Woes of Ancient Tax Time").

Polyhedral cages in nerve synapses

Conduction of signals along the membrane of a single neuron is an electro-chemical phenomenon, but conduction from one neuron to another is mediated by molecules diffusing at a synapse (where the axon of the sender cell is in close contact with dendrites of the receiver cell) through the small extracellular space between the cells. These molecules ("neurotransmitters") are released from the "active zone" of the sender's membrane upon receipt of an electrical impulse; when they reach the membrane of the receiver, they start a chain of events leading to generation of a new electrical signal propagating in the receiver cell. The whole electrical --> molecular --> electrical scenario plays out every time a signal is transmitted from one nerve to another.

The exact mechanism by which the arrival of an electrical impulse generates the release of neurotransmitters is not completely understood. It is known that the neurotransmitters are stored in (roughly spherical, diameter approximately 50 nanometers) vesicles, and that they are released when a vesicle fuses to the inside of the neuron membrane and by topological "surgery" becomes part of the membrane, releasing its contents to the outside. The April 16 2008 Journal of Neuroscience ran an article by a UCLA-Milan team (leader: Guido Zampighi) with the title: "Conical Electron Tomography of a Chemical Synapse: Polyhedral Cages Dock Vesicles to the Active Zone." The authors' working hypothesis was that "the structure of the active zone of chemical synapses has remained uncertain because of limitations of conventional electron microscopy." To get a better and more three-dimensional picture they developed the technique of conical electron tomography: a sample --of rat cortex in this case-- is tilted (hence "conical") and rotated 360o in the field of an electron microscope, with images taken every 5o. "Tomography" refers to a reconstruction of the 3D structure from the resulting series of electron micrographs.

Details of three from a series of electron micrographs of a sample of rat cortex. Image B is rotated 30o towards the viewer with respect to image A; image C an additional 60o. Membrane white, syndesomes green, vesicles yellow; in image C the vesicles have been cut away to give a better view (arrow) of the polyhedral structure of the central syndesome. Scale bar 90 nm. Images courtesy of Guido Zampighi.

The paper describes the considerable amount of geometry that went into the experimental algorithm. Just making sure that the same spot was at field center after each rotation required elaborate analysis (gold particles had been sprinkled on the sample; individual particles were used as markers); also, a tissue sample shrinks under bombardment by electrons; to keep successive images in registration the change in distance between specific gold particles was used to quantify the shrinkage and to subsequently recalibrate images; etc. For better distinction between subunits, the authors used "a specialized micro-graphics software package" to color-code each voxel (3D-pixel) in the images "on the basis of density thresholds and/or topological considerations." The segmentation is is visible in the images above : it distinguishes the cell membrane (shown in white) from the synaptic vesicles (yellow) and the docking cages ("syndesomes," green). These last structures were a discovery: polyhedral cages which seem to play a shepherding role in "docking" vesicles at the membrane. Disappointingly, "The cages were irregular and did not correspond to any of the perfect polyhedra described in the literature."

Teaching math with concrete examples?

"Abstract knowledge, such as mathematical knowledge, is often difficult to acquire and even more difficult to apply to novel situations. It is widely believed that a successful aproach to this challenge is to present the learner with multiple concrete and highly familiar examples of the to-be-learned concept." This is the start of "The Advantage of Abstract Examples in Learning Math" by Jennifer Kaminsky, Vladimir Sloutsky and Andrew Heckler (Center for Cognitive Science, OSU), an Education Forum report in the April 25 2008 Science. As can be surmised from the title, the authors present evidence that this widely shared belief is wrong.

 Generic Instantiation Concrete Instantiation

Two ways of presenting the concept commutative group with three elements. In the generic example, the concept is presented as a set of rules (--->) linking pairs of objects to a third object. In the concrete example, "participants were told that they needed to determine a remaining amount when different measuring cups of liquid are combined."

In one of their experiments, "undergraduate college students learned one or more instantiations of a simple mathematical concept. They were than presented with a transfer task that was a novel instantiation of the learned concept." The instantiations in question were a "generic" instantiation (top figure above) and three different "concrete" instantiations, one of which is illustrated above (the others involved slices of pizza or tennis balls in a container). They authors report that "all participants successfully learned the material" but that when transfer was tested participants who had been taught the generic condition "performed markedly higher than participants in each of the three concrete conditions." The authors also investigated the advantage of teaching a concrete instantiation and then a generic one, and found that "participants who learned only the generic instantiation outperformed those who learned both concrete and generic instantiations." They conclude that "grounding mathematics deeply in concrete contexts can potentially limit its applicability." [It is curious that the authors did not investigate the standard paradigm: abstract definition followed by concrete example, which has the advantage of showing students an example of how to transfer. -TP]

Tony Phillips
Stony Brook University
tony at math.sunysb.edu