This month’s topics:
Half a Möbius strip?
In a March 5 Science article, an international team of chemists report on “A molecule with half-Möbius topology.” The twisting is electronic rather than physical. The authors, led by Igor Rončević (University of Manchester), worked with ring-shaped molecules made up of 13 carbon atoms, with 2 chlorine atoms on opposite sides of the ring. One of those chlorines is tilted slightly up with respect to the plane of the ring, the other slightly down.
The eccentric orientations of the chlorine atoms induce a kind of torque on the electron clouds of the carbons in the ring. The 11 carbons which are not connected to a chlorine each have two electrons in one-dimensional orbits (“$p$-orbitals“). Those two orbits are orthogonal, creating an ${\mathsf X}$-like configuration around each molecule. The chains of carbons then form chains of ${\mathsf X}$-configurations. Each chain runs along one semicircle of the carbon ring (one chain has length 5, the other has length 6). The other two carbons have just one $p$-orbital each, but that is enough to link the 5 and 6 into a continuous field of 13 ${\mathsf X}$s defined on the entire molecule.
The torque creates a quarter-turn rotation in the ${\mathsf X}$-configuration as you circle the ring. This is the “half-Möbius” topology, since the twist is half of what one sees in a Möbius strip: One circuit around a Möbius strip produces a $180^{\circ}$ twist, which would require two circuits around the C$_{13}$Cl$_2$ molecule.
The authors go on to describe the family of conceivable topologies for this type of molecule. The family includes all possible twists of the ${\mathsf X}$-shaped space, four possibilities in all. Two of those are mirror images—different molecules, but topologically identical.
These structures are unusual “real-life” occurrences of the mathematical structures called fiber bundles. Here, the fiber bundle base is the circle, the fiber is the ${\mathsf X}$-space and the group is the cyclic group with four elements, acting as rotational symmetries of ${\mathsf X}$. Fiber bundles have been an essential element of recent progress in topology, geometry and in theoretical physics.
Calculating $\pi$ in Scientific American.
The best known irrational number is probably $\pi = 3.14159 \dots$, the ratio of a circle’s circumference to its diameter. Since $\pi$ is irrational, its decimal expansion never repeats; since it is an intrinsic feature of the mathematical universe, mathematicians want to know as much as possible about it, and have been working for centuries at computing more and more of those decimals.
Those calculations, at least recently, involve open-ended formulas: the evaluation of an infinite series, an infinite product, or an infinite continued fraction. Dozens of such formulas for $\pi$ have been discovered. For Scientific American (March 13, 2026), Lyndie Chiou reports on an effort to determine how many of these formulas are really unique—i.e., not related by elementary manipulations.
The research was first presented in an arXiv posting last fall. It involves extensive use of AI, both to comb the published literature for appropriate formulas and to wrangle the formulas into a standardized format. The authors, a team of seven from the Technion in Haifa, discover that many formulas for $\pi$ have “a kind of mathematical common ancestor,” as Chiou puts it. This “ancestor” is a Conservative Matrix Field, or CMF (definition below).
To get an idea of how a CMF works, we have to start with the formulas themselves. All the formulas in the new paper are or are derived from generalized continued fractions (GCFs), expressions of the form $$y = b_0+\cfrac{a_1}{b_1+\cfrac{a_2}{b_2+\cfrac{a_3}{b_3+\dots}}}~~~~(*)$$(A 1748 theorem of Euler’s shows that most interesting infinite series can be repackaged as GCFs, so there is no great loss of generality.)
In the expression for a GCF, the $b_i$ and $a_i$ are integers, and the “$\dots$” means that $y$ is the limit of the infinite sequence of ordinary fractions:
$$ b_0~,~~~ b_0+\frac{a_1}{b_1}~,~~~ b_0+\frac{a_1}{b_1+\frac{a_2}{b_2}}~,~~~ \dots~,$$which can be rewritten
$$\frac{b_0}{1}~,~~~ \frac{b_0b_1 + a_1}{b_1}~,~~~ \frac{b_2(b_0b_1+a_1)+b_0a_2}{b_1b_2+a_2}~,~~~ \dots$$ These fractions, with the above numerators and denominators, are known as the convergents of the GCF. They are usually represented as $\frac{P_0}{Q_0}, \frac{P_1}{Q_1},
\frac{P_2}{Q_2},$ and so on.
The CMF that is the “common ancestor” of many of the formulas for $\pi$ is defined on the nodes of a 3-dimensional integer lattice, as shown schematically in this illustration.
To understand the definition of a CMF and how it works to produce GCFs, we will focus on the one for $e = 2.71828\dots$, the base of the natural logarithms; it works exactly like the CMF for $\pi$ but is two-dimensional instead of three, which makes the presentation much simpler.
The CMF for $e$.
A 2-dimensional CMF consists of two matrix-valued functions, ${\bf M}_x$ and ${\bf M}_y$, defined on the nodes $(i,j)$ of the integer lattice in the plane. Specifically, ${\bf M}_x$ and ${\bf M}_y$ are $(2\times 2)$ matrices with integer entries. The matrix ${\bf M}_x(i,j)$ is associated to the movement across from $(i,j)$ to $(i+1,j)$ and ${\bf M}_y(i,j)$ is associated to the movement up from $(i,j)$ to $(i,j+1)$. In the illustration below, the matrices are written on the relevant lattice edges. To qualify as a CMF, the functions ${\bf M}_x$ and ${\bf M}_y$ must satisfy the following “conservation law”:
$${\bf M}_x(i,j)\cdot {\bf M}_y(i+1,j)={\bf M}_y(i,j)\cdot {\bf M}_x(i,j+1).$$Geometrically, this means that if you move from $(i,j)$ to $(i+1,j+1)$, it doesn’t matter which way you get there—moving horizontally, then vertically is the same as moving vertically, then horizontally.
For $e$ the authors give the following CMF:
To derive a GCF from a CMF, the authors choose a starting point, an initial matrix, and a direction going to infinity, as illustrated above. As you move along the lattice in this direction, you compute the GCF step by step, multiplying successive matrix values as they occur. The conservation law means that only the direction matters—not the precise path.
How are GCFs related to matrices? The connection has been known for some years. (A. J. van der Poortman’s Introduction surveys the case of simple CFs, those where all the $a_i = 1$, while John Cook’s blog describes how the process extends to general GCFs.) The link is the sequence of convergents $(\frac{P_0}{Q_0}, \frac{P_1}{Q_1}, \frac{P_2}{Q_2},\dots)$ associated to a GCF. If we additionally define $P_{-1}=1$ and $Q_{-1}=0$, then $P_k$ and $Q_k$ satisfy the recurrence relations$$P_k = b_kP_{k-1}+a_kP_{k-2}~, ~~~~Q_k = b_kQ_{k-1}+a_kQ_{k-2}~~~ (**)$$ for any $k \geq 1$.
These relations can be written in matrix form:
An analogous computation happens along a direction in a CMF. In this case, because of the matrix-fraction nature of the algorithm, the formulas resulting from the CMF for a constant $x$ turn out to equal an expression of the form $\frac{ax+b}{cx+d}$.
Here’s an example: Suppose we start at the origin and move along the line $j=0$ in the CMF for $e$, multiplying the matrices we encounter. This gives the sequence of matrices
As $k$ increases, the right-hand column of ${\mathcal M}_k$ gives the numerator and denominator of the fractions $~\frac{1}{2},~$$\frac{-4}{-3},~$ $\frac{-18}{~31},~$ $\frac{-96}{~165},~$ $\frac{-600}{~1031},~$ evaluating to $0.5,~1.33,~-.5806,~-.5818,~-.58196,~ \dots~$, a sequence converging to $\frac{1}{1-e} = -.581976\dots~$. (For a slightly different example and more details, you can watch this video illustrating the CMF for $e$, produced by the Technion team.) These fractions can be interpreted as giving the numerators and denominators of convergents $\frac{P_i}{Q_i}$ for a GCF giving $\frac{1}{1-e}$. To reconstruct the GCF, note that the recurrence relations $(**)$ are a set of linear equations that be solved for $a_k$ and $b_k$ in terms of $P_{k-1}, P_{k-2}, Q_{k-1}$ and $Q_{k-2}$.
Getting back to $\pi$.
The authors used large language models to scour 455,050 arXiv preprints for equations containing $\pi$. They ended up with 385 distinct approximation formulas for $\pi$. The team found relations between 360 of those formulas; their CMF for $\pi$ contained 166 of them. These last included formulas from across the mathematical ages: Some were discovered by Euler and Gauss; others were published as recently as 2022.
The occurrence of these publications so close to “$\pi$-day” (3/14) is presumably a coincidence.
—Tony Phillips, Stony Brook University