December’s topics:

College Algebra in Kansas

The Ohio Star ran “University System Weighs Gutting Math Standards After Students Keep Failing Algebra” on December 13, 2022. They were relaying information from a report by Suzanne Perez on the site of the NPR affiliate station KCUR datelined Wichita the day before. (The item generated significant local coverage and even made it to Fox News).

The story is that Daniel Archer, the vice-president for academic affairs for the Kansas Board of Regents, recently recommended to the Board that they consider not requiring College Algebra for so many of their undergraduates. As Perez explains, one reason for the change is that one in three students fail the course the first time they take it. Here is some background:

  • College Algebra in fact covers high-school-level material; it is identical to Algebra 2, except taught much faster.
  • The University of Kansas organizes its general graduation requirements into 6 goals, each with 2 outcomes. Goal 1, outcome 2 is Quantitative Literacy. The courses satisfying this outcome are listed on the site. Almost all have MATH 101 (College Algebra) as a prerequisite.

Archer is proposing a switch to Math Pathways, a program developed at Teacher’s College, Columbia, with the credo “algebra is designed to prepare students for calculus rather than for the type of math many students need in their majors, jobs, and lives.” Perez tells us that the program has helped Georgia State improve their graduation rate by 5% over the last 7 years.

While the population of the United States has approximately doubled since 1950, college enrollment has grown almost by a factor of 10. We cannot expect today’s students to have either the training or the goals of those who came to college back then. A pedagogically and scientifically informed reworking of the mathematics curriculum for the current audience is just a matter of common sense.

Extra dimensions?

Mind Matters News, an online outlet of the Walter Bradley Center for Natural and Artificial Intelligence at Discovery Institute, posted “Are extra dimensions of the universe real or imaginary?” on December 25, 2022. The existence of extra spatial dimensions beyond the standard three sounds like science fiction, but it is a fundamental ingredient of much current research in physics. People are usually willing to include time as a fourth dimension (even though it seems quite different from left-right, up-down and forward-back), but they naturally balk at more dimensions for space. For one thing, where are they? The Mind Matters News posting is a look back at Margaret Wertheim’s essay “Radical Dimensions” (Aeon, 2018), which investigates this problem. They quote several paragraphs from Wertheim (in one she shows us how the very notion of an ambient, Euclidean “space” only dates back to the Renaissance) and link to several explanatory videos, which are unfortunately of unequal scientific quality. (The clip they link to of a 3-dimensional projection of a rotating 4-dimensional cube has nothing wrong with it, but their text: “There is also the concept of a fourth spatial dimension, often pictured as a tesseract, a four-dimensional cube” is meaningless). Another link, to The 11 dimensions EXPLAINED, has that same projection floating over a grassy field as example of what a 4-dimensional cube would look like as it passed through our space. This is wrong. The video shows a series of projections from 4-space to 3-space; but if a tesseract (a 4-dimensional cube) happened to pass through our space a spectator would see a series of three-dimensional sections.

Image showing the cross-sections of a tesseract that one would see in if it were passing through 3D space.
If a tesseract passes vertex-first through 3-space, moving along one of its diagonals, the 3D sections one would see start with a point, then a tetrahedron swelling until its vertices become snubbed off. The new areas grow until they equal the old, forming a regular octahedron. As the process continues, the new areas take over, leaving a new tetrahedron which then shrinks to a point and vanishes.

Another possibility, as drawn by Tom Banchoff, is shown in “Understanding the hidden dimensions of modern physics through the arts”. Mind Matters News does have some useful links: Flatland (the movie) and an interview with Brian Greene (“How to visualize the 10 dimensions of String Theory”). I also recommend his TED talk on the topic.

How honeybees deal with geometric frustration

An article in PNAS (November 23, 2022; picked up in Physics Today) analyzes the dynamics of honeycomb construction by setting a computer to do the same job. A regular hexagonal grid is the most efficient way of partitioning the plane into equal areas. (Presumably known to bees tens of millions of years ago, this was only proved mathematically in 1999). But what happens when that grid does not exactly fit in the physical space it occupies? The PNAS authors (Golnar Gharooni Fard, Francisco López Jiménez, and Orit Peleg of University of Colorado, Boulder and Daisy Zhang of Princeton University) set up experiments in which bees were presented with incomplete combs made up of misaligned structures that could not be part of the same hexagonal grid.

Image showing three misaligned honeycombs, consisting of two half-honeycombs with a gap between them. In each example, the cells of the honeycomb on the right are shifted so as not to fit together with the one on the left.
Three kinds of misalignment: angular, horizontal and vertical. In each case, for most parameter choices, the two sides cannot be completed to a regular hexagonal grid. Image from PNAS, 119 (48), e2205043119.
The bees’ solution to a misalignment problem. Left, the problem: the two partial combs are separated by twice the distance between
the centers of adjacent cells, and the grids have been rotated $30^{\circ}$ with respect to each other. Center, the bees’ solution. Right, analysis of the solution. Gray: original cells; white: new six-sided cells; blue: new 5-sided cells; red: new 7-sided cells. Images from PNAS, 119 (48), e2205043119.

How would a computer solve the same problem? The team started by taking a clue from the bees: how many new cells they introduced to complete the comb. They began by distributing that number of points randomly into the space between the two partial combs, and then constructing a Voronoi diagram using those points and the centers of the cells along the problem edges.

Given a collection $A$ of points in the plane, the Voronoi cell about a point $p$ in $A$ is the set of points which are closer to $p$ than to any other point in $A$. Here the Voronoi diagram formed by the cells is drawn in blue. A uniform hexagonal grid is in fact the Voronoi diagram if $A$ is the set of centers of the hexagons.

This gives them a starting configuration, and now the computer goes to work to find out how to place the points so as to use the least amount of material (wax, when bees are doing the work). To make this into a more standard math problem, they use a special potential function $f(r)$, “a variation of the Lennard-Jones potential, known to produce hexagonal lattices in the absence of constraints”, depending on a distance variable $r$. The potential function is adjusted so as to have a minimum at $r=5.4$mm (“the distance between the center of the cells built by bees under no geometric frustration,” measured experimentally). The computer is tasked to find the minimum of the function $\sum f(d(p,q))$ where $d$ is distance and the sum is taken over all pairs $(p,q)$ of centers of adjacent Voronoi cells in the part of the grid being constructed. For a problem like the one considered above, there are some 30 new cells, so this is a function of some 60 variables. A minimum configuration is found using a simulated annealing process, an iterative method borrowed from theoretical physics.

Results of filling in a misaligned honeycomb with original cells in gray and new cells in red, white, and blue. The results of the experiment and the simulation are similar though not exactly the same.
A comparison between experiment and simulation for the frustration problem described above. In the rendering of the simulation, the new 6-sided cells are marked with a large gray dot; the other new cells are colored as described above. The authors report that achieving this minimal configuration took about 700,000 iterations of the simulated annealing process. Image from PNAS119 (48), e2205043119.

Read other recent Tony’s Take columns.