Math and the gerrymander

On March 10, 2017 the Chronicle of Higher Education ran "Meet the Math Professor Who’s Fighting Gerrymandering With Geometry," by Shannon Najmadabi; her article begins: "A Tufts University professor has a proposal to combat gerrymandering: give more geometry experts a day in court. Moon Duchin is an associate professor of math and director of the Science, Technology and Society program at Tufts. She realized last year that some of her research about metric geometry could be applied to gerrymandering—the practice of manipulating the shape of electoral districts to benefit a specific party, which is widely seen as a major contributor to government dysfunction." Now, in partnership with the Lawyers' Committee for Civil Rights Under Law, she is running a five-day summer program at Tufts, this August, advertised as having "the principal purpose of training mathematicians to be expert witnesses for court cases on redistricting and gerrymandering." Besides legal history and courtroom demeanor, the program includes "a course in metric geometry and mathematical ideas for perimeter-free compactness." [Compactness in this context is a measure of the roundness of a planar region. Usually, the ratio of the area of the region to the area of a circle with the same perimeter.]

On March 14, 2017 PNAS published "Assessing significance in a Markov chain without mixing" (Abstract) by Maria Chikina (U. of Pittsburgh), Alan Frieze and Wesley Pegden (Carnegie Mellon), who study how Markov chains can be used in statistics. "We remind the reader that a Markov chain is a discrete time random process; at each step, the chain jumps to a new state, which only depends on the previous state." They apply their conclusions to the problem of detecting bias in Congressional districting, and explain how this would apply to Pennsylvania: "The Markov chain that we use has as its state space the space of all valid districtings (with 18 districts) of Pennsylvania. Note that there is no simple way to enumerate these, and there is an enormous number of them. A simple way to define a Markov chain on this state space is to transition as follows.

1. From the current state, determine the set $S$ of all pairs $(p, D)$, where $p$ is a precinct in some district $D_p$, and $D\neq D_p$ is a district that is adjacent to $p$.
2. From $S$, choose one pair $(p_0, D_0)$ uniformly at random.
3. Change the district membership of $p_0$ from $D_{p_0}$ to $D_0$ if the resulting district is still valid."

(The actual chain they use is a "regularized" version of this one). The assumption is that under normal circumstances, the movement of a precinct from one district to another should occur to equalize population, and should otherwise be at random. Their theoretical work allows them to measure, with a given reliability, how plausible a districting scheme is; in particular they can identify "outliers" (unlikely schemes) and quantify how improbable they are. They conclude that "the current Congressional districting of Pennsylvania is an outlier at significance thresholds ranging from $p \approx 2.5\cdot 10^{−4}$ to $p \approx 8.1\cdot 10^{−7}$."

Top: a plausible districting scheme for Pennsylvania, dividing the state into 18 Congressional districts of equal population. Bottom: Pennsylvania's actual Congressional districts. Images from PNAS 114 2860-2864.

Flexible structures from 3D tessellations

Snapology is a paper-folding technique invented by Heinz Strobl for constructing regular and semi-regular polyhedra (a nice demo here). If a polyhedron has edge-length 1, its snapologized version has each face represented by a hollow prism of height one based on the corresponding polygon. This idea has been extended to uniform tesselations of 3-space by Johannes T. B. Overvelde, James C. Weaver, Chuck Hoberman and Katia Bertoldi, who report on their work in "Rational design of reconfigurable prismatic architected materials" (Nature, January 19, 2017).

For example, the uniform tesselation of the plane by hexagons and triangles (detail shown at left; side length, say, one) can be thickened by one unit and stacked to give a uniform tesselation of 3-space by hexagonal and triangular prisms. When each of the tessellating polyhedra is snapologized, those hollow prisms fit together to give a 3-dimensional mesh-like material.

The "3D prismatic architected material" constructed from the hexagonal, triangular prism tessellation of 3-space. Image from Nature 541 347-352, used with permission.

Depending on the tessellation chosen, the resulting architected material may be rigid or may have one, two or three degrees of freedom. For the example illustrated here, the number is two:

The two deformation modes of the architected material illustrated above. Images from Nature 541 347-352, used with permission.

As the authors report, when this construction is applied to the 28 uniform tessellations of 3-space, 15 are rigid, 2 have one degree of deformational freedom, 10 have two degrees and 1 (coming from the tessellation by cubes) has three. They also describe a method for creating many more 3D architected materials by keeping rigid some of the faces in the unit cell, while "extruding" the others. "Given that the underlying principles are scale-independent, our strategy can be applied to the design of the next generation of reconfigurable structures and materials, ranging from metre-scale transformable architectures to nanometre-scale tunable photonic systems."

Fractal dimension of Rorschach blots

Alison Abbott contributed a News Item, "Fractal secrets of Rorschach's famed ink blots revealed," to the February 14, 2017 issue of Nature. The occasion was the appearance of an article on the topic published that day in PLOS ONE by R. P. Taylor (Physics, U. of Oregon) and eight collaborators.

A Rorschach blot: a copy of Plate I from Hermann Rorschach's 1921 series of ten.

The blots are used in the Rorschach Test: They are shown one by one to a subject, who is asked (e.g. "What might this be?") to free-associate about what they could represent. As Taylor et al. explain, "Originally adopted as a projective psychological tool to probe mental health, psychologists and artists have more recently interpreted the variety of induced images simply as a signature of the observers’ creativity."

Recognizing namable images in a Rorschach blot is an example of a common phenomenon ("That cloud looks like a battleship") with a fancy name: pareidolia. Earlier work in the field "was motivated by the observation that humans readily perceive and identify meaningful images in many naturally occurring but largely unstructured configurations ranging from clouds, rocks, and cracks in the ground to the surface of the Moon. All of these structures are fractal, featuring patterns that repeat at increasingly fine magnifications. The study explored the relationship between the scaling characteristics of fractal patterns, as quantified by their fractal dimension $D$, and the patterns’ ability to evoke the perception of namable objects. Using computer-generated stimuli, fractals with low $D$ values were found to elicit a higher ability to evoke the perception of namable objects." Taylor and collaborators were able to quantify this relationship using Rorschach blots. They measured the fractal dimensions of the five black blots, nos. 1, 4, 5, 6, 7 (the different inks in the colored ones had diffused differently, making the fractal dimension hard to determine). On the other hand, they had numbers for the "ability to evoke the perception of namable objects;" these date back to 1936, when the psychologist Marguerite Hertz tabulated the responses of 1050 subjects to each of the ten blots. The team counted the number $n$ of of different "percept types" associated to each blot (Hertz had recorded what the percepts were: person, bat, etc.). Here is the plot of $n$ vs. $D$, for the five black blots:

Plot of number of elicited percept types against fractal dimension, for each of the five black blots in the Rorschach test. (Larger image). Images from "Seeing shapes in seemingly random spatial patterns: Fractal analysis of Rorschach inkblots" by R. P. Taylor et al..

In this image Taylor et al. also exhibit a second set of percept numbers, compiled by Wirt and McReynolds in 1953. That study recorded the average number of reported percepts per subject. "Despite these differences in recording $n$, the standardized $z$-scores for the average number of total responses reported by Wirt and McReynolds, plotted as blue symbols [the figure], closely follow the same inverse relationship between the $n$ and $D$ values found in the Hertz data."

Art note: "The Abstract Expressionist Jackson Pollock's poured paintings are composed of fractals with $D$ values that increased from 1.1 to 1.7 over the decade 1943–1952. Intriguingly, Pollock seems to have been aware that his drive towards higher complexity paintings would reduce the number of induced precepts: 'I try to stay away from any recognizable image; if it creeps in I try to do away with it ... I don't let the image carry the painting ... It's extra cargo—and unnecessary.'"

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