This month’s topics:
- Do your homework, but not all at once
- Dancing through a topological insulator
- Soft cells, “a new class of shapes”
Do your homework, but not all at once.
For mathematics, the maximum benefit from homework comes with assignments due every day, but no improvement comes from extending the duration of each assignment beyond 15 minutes.
That’s the report from a new article in Learning and Instruction titled: “Little and often: Causal inference machine learning demonstrates the benefits of homework for improving achievement in mathematics and science.” The authors, all from Maynooth University (County Kildare, Ireland) begin by stating that while homework is a big part of student life, “its role in shaping academic outcomes is far from understood.” They report applying new analytical methods, including machine learning and Bayesian statistics, to data from 4,118 Irish eighth grade students. The data included information about class size, home environment, and duration and frequency of homework assignments, along with student performance on short mathematics and science tests. (The data were collected as part of the 2019 Trends in International Mathematics and Science Study.)
The study authors also addressed the concern that homework exacerbates inequality, since socioeconomically advantaged students enjoy better study conditions and more parental involvement. They found “regardless of the number of books a student has at home, or the level of education of their parents, all students are predicted to benefit by approximately the same amount from the homework they are assigned.”
The website Phys.org posted a press release on the article on September 19.
Dancing through a topological insulator.
Anna Demming’s “Dancing humans embody topological properties” was posted on Physics World on September 11. The particular notion addressed here is a topological insulator. This is a material that is an insulator (does not conduct electricity) in its interior but allows currents along its boundary. Topological insulators have lately been an intense focus of study. Nature alone has 120 articles on the topic in the last 5 years. But the connection with topology has been hard to grasp since they do not involve the usual phenomena associated with “topology,” such as surfaces, knots, or shapes in general. Demming’s article covers a dance that exhibits the key properties of a topological insulator. The dance, developed by Joel Yuen-Zhou of the University of California, San Diego, and his student Matthew Du, was motivated by the wish to “democratize the notions of topological phases of matter to a broader audience,” Yuen-Zhou told Demming.
The topological insulators in this story are 2-dimensional: non-conducting surfaces with edge currents. Yuen-Zhou and his team first devised a discrete model, published in “Chiral edge waves in a dance-based human topological insulator” on August 28 in Science Advances. In this model, an electron moves around a grid, influenced by a magnetic field. The electron begins on a starting square, then “hops” to one of its eight neighbors (the squares that share either a side or a corner with the starting square). Precisely which neighbor the electron hops to is random, according to probabilities that depend on the magnetic field.
Let us examine how the dance works and what it tells us about topological insulators. The dance floor is a checkerboard; each square represents a lattice site. The squares are decorated (see below) and are grouped in four-by-four square blocks (“unit cells”) which tile the entire lattice.
The dance proceeds by cycles. At the start of the first cycle, one dancer is chosen to be the “commander” and assigned a pose: arms raised (“up”) or lowered (“down”). In a first step, the excitation propagates from the commander to its neighbors, as follows. Crossing a red side strip in the commander’s square reverses the pose: arms up to arms down, and vice-versa; in crossing a green strip the pose is maintained. In a second step, the neighbors evaluate their matches. Neighbor a has a match (see below) with neighbor b if b‘s pose is related to a‘s by the color of the corresponding side strip in a‘s border: opposite if it’s red, the same if it’s green. If the commander was on the edge of the lattice, there will be exactly one neighbor, on the edge, with no matches. In the third and last step, the neighbor with no matches becomes the new commander, the other cells revert to their unexcited poses and a new cycle begins.
Characteristic of a topological insulator is that currents can move (always in the same direction) along the edge of the material but are impeded from propagating in the interior. The dance also manifests this second property.
One member of the team was Dylan Karzen from Orange Glen High School in Escondido. He recruited a group of second- and third-year students to dance through the algorithm on a checkerboard drawn in the school courtyard. The Science Advances article includes several photographs of the performance.
The authors underline the difference between this project and other physics-dance connections (for example this one where a line of students demonstrates different kinds of seismic waves, or the Rueda de Casino used to illustrate another aspect of topological insulators). Unlike the other projects, these dancers are not just interpreting or exemplifying the phenomenon, they are actually carrying out a “rigorous realization” of the underlying principles: they exhibit how in a topological insulator uniform local conditions can produce very different effects according to the position (interior or boundary) of the location of an excited element.
Soft cells, “a new class of shapes.”
“Mathematicians discover a new class of shape seen throughout nature” was Philip Ball’s news item in Nature for September 20. Soft cells do not have angular corners like ordinary polygons or polyhedra. Instead, as Ball explains it, “[They] have curved edges along with cusp-like corners where the edges meet tangentially at soft points.” The research in question, from a Hungarian-British team led by Gábor Domokos, appeared recently in PNAS Nexus. It finds that soft cells can tile space — that is, fit together in an infinite pattern that leaves no holes.
According to Ball, Domokos and his team were initally motivated by the shape of the 3D compartments in the shells of the chambered cephalopod Nautilus pompilius.
The authors define tilings as “space-filling patterns consisting of non-overlapping, finite domains.” Note that they do not require the pattern to be periodic. They examine the question: When can a convex polyhedral tiling be replaced by a soft tiling with a matching set of vertices, edges, faces, cells, and incidence relations? They give a sufficient condition for 3-dimensional tilings that satisfy some natural bounds on how many faces can meet along an edge, etc. To understand this condition, we need three definitions.
- Dual polyhedra: To every polyhedron is associated a dual polyhedron. The vertices of the dual (“dual vertices”) correspond to the faces of the original, and the dual faces correspond to the vertices of the original. The dual edges correspond to the edges of the original in this way: Two dual vertices are joined by a dual edge if the original faces have an edge in common; the duals of all the edges connected to a vertex form the boundary of its dual face. Some examples of dual pairs are the cube and octahedron, the dodecahedron and icosahedron, and the tetrahedron and itself (it is “self-dual”).
- A circuit on a polyhedron is a closed path made up of edges. The circuit is Hamiltonian if it visits each vertex exactly once. (Every regular polyhedron admits a Hamiltonian circuit.)
- Vertex polyhedron is defined for a vertex in a 3-dimensional tiling. A small sphere about the vertex will intersect each edge through that vertex at one point, and it will intersect each face touching the vertex in a curve between two of those points. These vertices and edges make the sphere into a spherical polyhedron. Flatten the faces and edges and you will get an ordinary rectilinear polyhedron; this is the vertex polyhedron. For example, in a cubical tiling, each vertex has an octahedron as its vertex polyhedron.
In these terms, the authors prove that if for every vertex of a polyhedral tiling M (satisfying the natural bounds mentioned above) the dual of its vertex polyhedron admits a Hamiltonian circuit, then there is a soft tiling with the same combinatorial structure as M, meaning that you can match its cells, faces, edges and vertices with those of M, and all the incidence relations (who is on the boundary of what) between the various elements will be preserved. The following sequence of images illustrates the proof, applied to the example of a cubical tiling of 3-space. These are stills from an animation in the Supplementary Material linked to the PNAS Nexus article. The images only show what happens at one vertex, but all the vertices in the cubic lattice look the same and are treated identically. The vertex polyhedron at that vertex is an octahedron (yellow in the images); its dual is a (purple) cube. The cube admits a Hamiltonian path (highlighted in the images). Since that path is a closed curve by definition, it splits the surface of the cube into two parts (red and blue rectangles, with three faces each). Each face gives its color to the corresponding vertex of the yellow polyhedron. That vertex corresponds to one of the edges incident to the lattice vertex being considered; that edge also gets the color. Since all three red colored edges pass through the red rectangle, a polygon that can be shrunk over itself to a point, there is nothing in the way of deforming them to all approach the vertex together, with a common tangent; same for the blue.
Note that the proof shows the vertex-polygon-dual-Hamiltonian-circuit hypothesis is sufficient for the tiling to have the same combinatorial structure as a smooth one. The authors conjecture that it is not necessary, and that their natural finiteness conditions would be enough.
The authors remark: “Beyond Nature and mathematics, we also find soft cells emerging in art.” They refer in particular to the work of the late Iraqi-British architect Zaha Hadid, who often used these shapes in her designs. And as Ball mentions, Domokos and one of his co-authors have collaborated with staff at the California College of Arts on an award-winning project that incorporates seashells. They called it: “Shell We Dance?”
—Tony Phillips, Stony Brook University