Tony’s Take April 2022

This month’s topics:

Quipus and quantum computing

Frank Wilczek’s column in the the Wall Street Journal (April 14, 2022) had the title “A Quantum Leap, With Strings Attached; The Inca system of quipu—tying a series of knots to record information—is providing a surprising model to modern physics and quantum computing.”

Left: photo of a quipu. Right: a drawing of a quipu with four pendants and one main cord, representing the number 805.
Left, a quipu from the American Museum of Natural History, image taken from L. Leland Locke’s The Ancient Quipu (AMNH, 1923). Here the main cord supports 24 pendants, tied off in groups of four by the top cords. Right, Locke’s analysis of the third 4-cord group, image adapted from his book. Overhand knots, represented by circles and crosses, represent 10 or 100 depending on their position on the cord. The other symbols represent the special knots used to record the numbers from 1 to 9. In this quipu each top cord records the sum of the numbers encoded on the four corresponding pendants.

This textile document is a typical quipu in that its data is numerical and recorded in a decimal system. (For another example and more details, see Nicole Rode’s YouTube video from the British Museum). While they were used by earlier pre-Columbian Andean cultures, most of the surviving specimens date from the period of Inca domination, c. 1400–1532 CE. (We can only guess what the numbers recorded on quipus were actually counting —these civilizations left no written records).

Wilczek compares quipus with the information storage and transmission systems we encounter today: “written human language, the binary code of computers and the DNA and RNA sequences of genetics,” and remarks that the Andean system involves “something unique: topology, the science of stable shapes and structures.” In fact the difference between one knot and another, which is contrastive in quipus, to borrow a term from linguistics, is one of the most basic examples of a purely topological concept.

The connection between quipus and “modern physics and computing theory” comes precisely through topology. The equivalents of Andean cords are the world-lines of particles. Wilczek asks us to suppose our particles are only moving in two dimensions, and that we add a third dimension to represent time. Then as time progresses the successive positions of a particle trace out a curve: this is its world-line. And if several particles are observed at once, their world-lines can tangle (“these are not our ancestor’s strings”) and form what mathematicians call braids.

Two examples of how the world-lines of three particles form braids.
The paths of $n$ planar particles, tracked through time, form an $n$-strand braid. Given a finite piece of the braid, joining tops to bottoms, proceeding left to right, can give a knot. Here (top) $n=3$ and the knot is in fact the figure-eight knot. But for some braids (bottom) this process produces a link involving two or more disjoint curves.

“There are certain particles, called anyons, whose quantum behavior keeps track of the braid that their world-lines form [and therefore could be used to store information]. The anyon world-lines form a quantum quipu.” Wilczek’s terminology has to be taken with a grain of salt. Knots and braids are very different mathematical objects, although fundamentally related (see the drawing above and Alexander’s Theorem on Wikipedia); quipus use one and not the other. Nevertheless it is striking that the topology of curves in space turns up both in an antique recording system and in the latest quantum science.

The science really is very new. Anyons were only experimentally detected two years ago (Wilczek himself had conjectured their existence, and named them, some 40 years back). He tells us that “the simple quantum quipus that were produced in those pioneering experiments can’t store much information” but that only last month “Microsoft researchers announced that they have engineered much more capable anyons.” This is presumably the research described in the Microsoft Research Blog on March 14.

Geometry, a human language?

Siobhan Roberts contributed Is Geometry a Language That Only Humans Know? to the March 22, 2022 New York Times. The subtitle is more specific: “Neuroscientists are exploring whether shapes like squares and rectangles — and our ability to recognize them — are part of what makes our species special.” The neuroscientists in question are Stanislas Dehaene (Université Paris-Saclay and Collège de France) and his collaborators.

The first part of Roberts’s article concerns the research that Dehaene and his team published last year in PNAS: “Sensitivity to geometric shape regularity in humans and baboons: A putative signature of human singularity.” In a typical experiment they report, subjects were presented with a display of polygons. Five of the six were similar, differing only in size and orientation; the sixth was like the others except that its shape had been changed by moving one vertex. Subjects were asked to pick out the oddball.

The “normal” polygons were chosen from a family of eleven quadrilaterals that can be ranked, starting with a square, by how unsymmetrical they are.

Eleven quadrilaterals ranging from a square to a completely irregular shape.
The collection of quadrilaterals used in this experiment. The colors of the labels will identify points in the display of results. “Irregular” has no symmetries, no parallel sides and no right angles. The last diagram shows various ways in which an oddball can be generated by moving one vertex. Images for this item used under PNAS License.
typical test slide: 6 quadrilaterals arranged in a circle. Five of the quadrilaterals are rectanges, and the last is a right hinge (a rectangle with one corner jutting out).
A typical display during the experiment. Here the “normal” quadrilateral is the rectangle. Image redrawn from original for better resolution.

The first experiment, involving 605 French adults, showed that the number of errors they made “varied massively” with the lack of symmetry/orthogonality/parallelism of the “normal” exemplar.

graph with error rate on the y axis and the shapes on the x axis. More irregular shapes had higher error rates.
The geometric regularity effect in an experiment with French adults. “The error rate varied massively with shape,” the researchers wrote. Shapes are ordered by performance.

The team repeated the experiment with French kindergarteners and with Himba adults (“a pastoral people of northern Namibia whose language contains no words for geometric shapes, who receive little or no formal education, and who, unlike French subjects, do not live in a carpentered world.”) The results correlated strongly with those of French adults. “Both findings converge with previous work to suggest that the geometric regularity effect reflects a universal intuition of geometry that is present in all humans and is largely independent of formal knowledge, language, schooling, and environment.”

The experimenters had access to a colony of baboons (Papio papio) in the south of France; they managed to train the baboons to where they had a “clear understanding of the task”—they could recognize the oddball apple in a group of watermelon slices, and even a regular hexagon in a group of non-convex polygons, but “although error rates differed across the 11 shapes, with a consistent ordering across baboons, […] they correlated weakly and nonsignificantly with the geometric regularity effect found in human populations.”

After speaking with Moira Dillon (a psychologist at New York University) Roberts puts this research in a historical context: “Plato believed that humans were uniquely attuned to geometry; the linguist Noam Chomsky has argued that language is a biologically rooted human capacity. Dr. Dehaene aims to do for geometry what Dr. Chomsky did for language.” But Frans de Waal (a primatologist at Emory University) cautioned her: “Whether this difference in perception amounts to human ‘singularity’ would have to await research on our closest primate relatives, the apes.”

Roberts reviews connections between this research and work in artificial intelligence, and then moves on to Dehaene et al.‘s latest project, essentially figuring out what in the human mind makes geometric regularity so significant. Here’s a clue, quoting from Dehaene: “We postulate that when you look at a geometric shape, you immediately have a mental program for it. You understand it, inasmuch as you have a program to reproduce it.” The team explored an algorithm, DreamCoder (the authors overlap with Dehaene’s collaborators) that “finds, or learns, the shortest possible program for [drawing] any given shape or pattern.” Then they tested human subjects on the same shapes. “The researchers found that the more complex a shape and the longer the program, the more difficulty a subject had remembering it or discriminating it from others.”

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