April’s topics:
- Non-Euclidean geometry in everyday life
- Three bodies in deep space
- Expected and unexpected findings in Math Education research
Non-Euclidean geometry in everyday life.
Visual space has a geometry of its own. When objects are close, our brain integrates information from our two eyes into a three-dimensional impression. But if an object is very distant, our eyes are too close together for this effect to be useful, and everything appears projected on a sphere. What happens at intermediate distances? What is the relation between our perception of depth and the actual Euclidean reality of where things are?
The problem of this “visual geometry” has been studied since the 19th century, but until now, never with a large number of subjects and under outdoor, daylight conditions. That’s according to eight researchers from Western Kentucky University, who published “Visual outdoor space perception” in Nature Scientific Reports on April 4, 2025. In one of the new experiments, observers with normal binocular vision were placed in a large, grassy field, along with two poles. This created a triangle with the observer at one vertex. One of the poles was kept fixed by the experimenters, while the observer was asked where the other pole should be placed to make the triangle equilateral.
This experiment was repeated with the fixed pole set at varying distances: 2m, 6m, 11m, and 15.5m from the observer. Once the observer had chosen a placement for the second pole, the experimenters measured the angle at the observer’s vertex. If visual space is Euclidean, that angle should be 60$^{\circ}$ (to make an equilateral triangle). If the space is negatively curved, the angle should be smaller; if the space is positively curved, like the surface of a sphere, it should be larger.
The authors report that for the smallest triangle ($d=2$m) the measured angle was significantly larger than 60$^{\circ}$, and that it decreased for larger values of $d$, clustering near 50$^{\circ}$ for $d=15.5$m.
The curve and error bars do not tell the whole story. The authors report that while 22 of the 30 subjects followed the general pattern, the remaining 8 gave idiosyncratically diverging responses.
The results of this experiment, and another completely analogous one conducted with right isosceles triangles and 45$^{\circ}$ viewing angles, support the conclusion that in general (but not always), nearby space is perceived as positively curved, while the observers’ judgments are “consistent with hyperbolic geometry for the largest stimulus size.”
Three bodies in deep space.
Astronomers may have discovered a new way for three masses to orbit one another. “Evidence for a polar circumbinary exoplanet orbiting a pair of eclipsing brown dwarfs” by Thomas Baycroft (University of Birmingham) and collaborators ran in Science Advances on April 16, 2025. “Brown dwarfs” are cosmic objects intermediate in size and luminosity between planets and stars. (Here each has about 1/30 the mass of the Sun.)
Interpreted mathematically, the postulated equations of motion of the exoplanet and the dwarfs constitute a new solution to a problem that dates back to Newton and was studied by Euler, Lagrange and Poincaré. The “three-body problem” asks about the motion in space of three masses subject only to the gravitational attraction between them. For Newton, those three masses were the Earth, the Moon and the Sun. Newton’s law of motion, force = mass $\times$ acceleration, completely determines the trajectories of the three bodies given their initial positions and velocities. The problem is to solve for these trajectories.
The three-body problem is enormously difficult. Even the simplest set-up, in which the bodies begin at rest and the evolution takes place within the plane containing their initial positions, leads to very complicated trajectories.
Mathematicians have looked for more intelligible solutions, especially those which are periodic, or almost periodic, like the Earth-Sun-Moon system. They have mostly concentrated on the planar case, since it involves many fewer equations, and over the years have found several solutions. Details are outlined in a Feature Column on this site.
Around 1960, Kirill Alexandrovitch Sitnikov found the first solution with potentially unbounded oscillations. His solution was non-planar, but it also involved a major simplification: Two of the masses are comparable and the third is negligible with respect to the others. This is the restricted 3-body problem. In Sitnikov’s solutions, the two larger masses are equal and move in the $xy$-plane, where they traverse congruent elliptical orbits about their joint center of mass. Meanwhile, the smallest body oscillates up and down along a line perpendicular to the $xy$-plane, passing through the center of mass.
In the special case where the two ellipses merge into one perfect circle (in this case, the two large masses occupy diametrically opposite positions on the circle), the system is well-behaved. The smaller mass cannot move too far from the other bodies. Otherwise, in a configuration like the one illustrated here, the oscillations of the smaller mass may grow arbitrarily large. Bill Casselman analyzed this behavior in another Feature Column on this site.
The Science Advances paper describes a configuration in which two brown dwarfs, like Sitnikov’s large masses, traverse elliptical orbits about their common center of mass. The orbital plane of this binary system almost exactly contains our line of sight, which is why the authors refer to the brown dwarfs as “eclipsing.” The third body in this system is a polar exoplanet, “polar” meaning that its orbit is perpendicular to the plane of the system. It is much smaller in mass than the dwarfs, but not negligible. In fact the only way we know it exists is by its effect on the binary. The binary system precesses in its plane, and in a “retrograde” direction, i.e. opposite to the direction in which the dwarfs traverse their own orbits. This precession is a small effect, only about 340 seconds of arc per year, but the authors assure us that the only way to explain it (even though such a system is “exotic and seemingly unlikely”) is to posit the existence of a planet in polar orbit.
Some details. The binary system in question is located about 150 light-years from Earth. The brown dwarfs describe their overlapping elliptical orbits in about 21 days. As for the planet, one possibility is for it to have mass around 10 times that of the Earth and orbital period around 100 days; in another scenario the numbers are 100 and 400. An animation of a system like this has been prepared by Luis Calçada of the European Southern Observatory. (To make the animation comprehensible, it is not to scale in space or in time: with respect to the orbital motion of the dwarfs, the precession is sped up by a factor of about a thousand.)
Luke Skywalker’s home planet also orbits a binary. The formal resemblance was not lost on Keivan Stassun, whose “Focus” piece in Science Advances, accompanying this article, bore the title “A tilted ‘Tatooine planet’ whose two suns aren’t stars at all.”
Expected and unexpected findings in Math Education research.
An April 17 study in npj Science of Learning examines educational interventions for low achievers in mathematics. The study was done in Danish public schools, where educational inequality is similar to that of the United States, despite Denmark being generally more egalitarian. The interventions specifically address the 20% lowest mathematics achievers.
For this study, 589 second-graders and 440 eighth-graders were randomly assigned to one of three intervention models, or a control group:
- 1-group: A teacher worked with a group of 3 to 4 students.
- 1-group with coaching: Same, but teachers also worked 3-1 with a coach.
- 1-1 with coaching: Teachers worked one-one with students, and teachers were coached 3-1.
The treated students met with teachers in 25-35 minute sessions four times a week, for 12 weeks. They were pre- and post-tested, and tested 14-18 months after the intervention to investigate medium-term effects. The authors remark that theirs is the first such study to systematically investigate fading and cost-effectiveness.
A rough summary of the results:
Second-Grade Effects
Two of the models, the 1-group without coaching and 1-1 with coaching, both produced significant effects (0.5 and 0.8 standard deviations). Why the other listed model was not successful in 2nd grade is an anomaly the authors could not explain.
The effects of the 1-group without coaching persisted into the medium term with 0.5 SD. However, the 1-1 intervention turns out not to be significant in the medium term. The authors suggest that this is because students who worked one-on-one with an instructor had more difficulty transitioning back to a regular classroom.
Eighth-Grade Effects
In this age group, only the 1-group with coaching had the same level of effect (0.55 SD), and none of the interventions showed a significant medium-term effect.
Readers of this article in npj Science of Learning were encouraged to read another, published a year ago (March 15, 2024) with the striking title “Math items about real-world content lower test-scores of students from families with low socioeconomic status”. This one comes from the Netherlands, written by Marjolein Muskens (Maastricht University) and two collaborators. Intuitively, one would think that phrasing a mathematical question in “relevant” terms—the authors mention money, food, social relationships—would engage students and improve their scores. For students from families with low socioeconomic status (SES), quite the contrary: On average, performance goes down substantially, with significant numbers of students dropping as much as 18% from their usual scores.
The authors’ analysis used data from the Trends in International Mathematics and Science Studies (TIMSS) surveys taken in 2007 and 2011, involving over 5 million students in grades 4 and 8. They grouped students into five SES levels based on the reported number of books in their home. This measurement, they found, “captures a key component of SES, namely the position related to the level of education and more specifically, cultural capital.”
They examined students on randomly selected questions, and then tested how facing a “relevant” question affected each student’s performance. Here are the results for grade 8. (The authors state that grade 4 results are completely similar.)
In the Discussion section of the article, the authors look for reasons for these “unexpected” findings. One hypothesis is that low-SES students have developed a scarcity mindset—they react to stressful topics with distracting thoughts. They cite an experimental study that shows that while “more salient everyday examples” can lead to better understanding of a problem, they also lead to more arithmetic errors in its solution. Finally, the authors write that “our results add to evidence that standardized test results can be biased by influences that are unrelated to students’ learning ability.”
—Tony Phillips, Stony Brook University