This month’s topics:
- Rose petal differential geometry
- Calabi-Yau manifolds and black-hole dynamics
- “Mathematician Solves Algebra’s Oldest Problem”
Rose petal differential geometry.
Rose petals have a characteristic shape: They fold at the edges, forming sharp corners, as in the image below.
A May 1 paper in Science attempts to explain why rose petals fold like this. The authors examine how rose petals curve along two different directions, and find an incompatibility that forces the “distinct sharp tips” along their edges.
To explain the significance of this finding, we need to distinguish between the two kinds of curvature that a surface can have. One, Gaussian curvature, is intrinsic to the surface, in the sense that a geometrically-inclined ant could measure it without ever venturing off the surface. If this curvature is positive at a point, the surface nearby is dome-shaped; if it is negative, that surface will be saddle-shaped. Up to now, the study of irregularities in the shape of leaves has pointed to problems with the Gaussian curvature. In particular, since a surface of negative Gaussian curvature takes up more and more room as it grows, more and more wrinkly shapes occur along the edges of the leaves of kale, escarole and related plants.
A rose petal does not have significant Gaussian curvature; in that way, it is similar to a flat sheet of paper. A zero-Gaussian-curvature surface can bend in certain ways without stretching or compressing—into a cone or a cylinder, for example. This doesn’t change the Gaussian curvature, but it can change the curvature of straight lines drawn on the surface.
From a technical set of equations called the Peterson-Mainardi-Codazzi Equations, one can deduce restrictions on how a surface can bend and stay smooth. One restriction implies that a region on a smooth flat surface described in polar coordinates cannot
uniformly bend in the $r$-direction.
You can see this happen when you put the top sheet on a bed. The sheet has flat intrinsic geometry (Gaussian curvature zero). Along the straight edges of the mattress, the sheet can bend while remaining smooth. But at the corners, where the edge of the mattress curves, there is a geometric incompatibility. Folds must be introduced, which turn into sharp creases when the sheet is tucked in: it is no longer a smooth surface.
A very similar thing happens with rose petals. According to the authors, they want to be flat in the $\theta$-direction. But as they grow, they become more and more curved in the $r$-direction. At the beginning, the elasticity of the tissue can absorb some of the strain caused by this “Peterson-Mainardi-Codazzi incompatibility,” but eventually the geometrical contradiction is too much to bear. The petal can no longer extend smoothly in space: sharp angular cusps form along the edges.
Davide Castelvecchi reported on this work in Nature (May 1, 2025). His subtitle, “Petals’ pointy edges rely on a type of geometric feedback never before seen in nature,” emphasizes the difference between this phenomenon and the Gaussian-curvature-induced wrinkling that had been studied before. Castelvecchi also wonders whether there is some functional reason for the petals’ shape. In fact, the cuspy corners are not the product of natural evolution. Wild roses (botanically, Species roses) never have more than five petals and do not seem to run into problems with Peterson-Mainardi-Codazzi incompatibility.
Calabi-Yau manifolds and black-hole dynamics.
Physicists have used recently discovered mathematical objects to compute the specifics of black holes. The finding was reported in Nature on May 14, 2025. It “echoes Galileo’s insight, expressed in his book The Assayer (1623), that the Universe ‘is written in the language of mathematics,'” wrote Zhengwen Liu (Southeast University, Nanjing) in commentary from the same issue of Nature.
The paper investigates black-hole scattering, the gravitational-wave disturbance caused by a close encounter between two black holes. Such signals are now routinely detected. In theory, one can use the scattering signal to compute the masses and velocities of the black holes, via a set of equations called the Einstein field equations. In practice, those equations are highly non-linear, and consequently onerous to simulate. The authors of the article, a team of eight led by Gustav Mogull and Jan Plefka of Humboldt-Universität, Berlin, followed instead a line of research that constructs high-precision models of scattering by adapting state-of-the-art techniques from high-energy physics (HEP).
HEP studies extremely small particles. So why should it tell us anything about enormous bodies traveling through space? The answer is, as the authors remark, that black holes and elementary particles are both completely characterized by three fundamental properties: mass, spin and charge.
The authors use the HEP technique of perturbation theory: “Picking one or more small parameters and solving the equations order by order in a series expansion.” Earlier work had brought into play 2- and 4-dimensional Calabi-Yau manifolds. These are complex manifolds, so spaces of even real dimension, which can be solutions to Einstein’s equations for the curvature of space-time. The 2- and 4-dimensional ones had earlier names (elliptic curves and K3-surfaces) but inherited the new one from their 6-dimensional cousins, identified by Eugenio Calabi back in 1954. These are algebraic surfaces (i.e. the solution sets of polynomial equations in several complex variables) that, he conjectured, might admit a special kind of geometric structure. His conjecture lasted until its proof by Shing-Tung Yau in 1976. The six-dimensional “Calabi-Yau manifolds” later gained fame as just what string theorists needed to boost our 4D space-time up to the 10 dimensions string theory required. (Their appearance in perturbation theory seems to be an unrelated occurrence). Our authors remark that this is the first use of six-dimensional Calabi-Yau manifolds in predicting observable physical phenomena, and point out that the innovation may “have substantial implications for high-precision predictions in particle physics as well.”
“Mathematician Solves Algebra’s Oldest Problem.”
This, from Newsweek, May 1, 2025, was just one of the startling headlines last month about new work on solutions to polynomial equations like $$c_0 + c_1x + c_2 x^2 + \cdots +c_nx^n = 0.$$Other headlines included “Math shaken as 200-year-old polynomial rule falls to Geode number discovery” (Interesting Engineering) and “Mathematicians Thought This Algebra Problem Was Impossible. Two Geniuses May Have Found a Solution” (Popular Mechanics).
The two geniuses in question are Norman Wildberger and Dean Rubine. Their paper “A Hyper-Catalan Series Solution to Polynomial Equations, and the Geode” appeared in the American Mathematical Monthly on April 8.
To understand what the fuss is about, we need a quick look at the history of polynomial equations and their solutions. Solving the general degree-$n$ polynomial equation means finding an algorithm that inputs the coefficients and outputs a solution of the equation.
As Wildberger and Rubine remind us, solutions to quadratics ($n=2$) date back to the Babylonians. The Babylonian method, using what we now call “completing the square,” evolved into the quadratic formula $$x = \frac{-b\pm\sqrt{b^2-4ac}}{2a},$$ which gives us both of the solutions to $ax^2 + bx + c =0$. This formula became the model for a general solution of degree-$n$ polynomials. It was understood that such a solution would use the operations of arithmetic, along with powers and extraction of roots, to produce $x$ from a polynomial’s coefficients.
In the 16th century Italian algebraists had found solutions along these lines to polynomial equations of degree 3 and 4. But in 1826, Niels Abel wrote “Proof of the impossibility in general of solving algebraic equations of degree higher than the fourth,” which closed the door on finding algorithms that used only arithmetic, powers, and roots.
Wildberger and Rubine propose a different kind of algorithm. They define a collection of power series, one for each $n$. The $n^{\text{th}}$ power series has $n+1$ variables. To solve a polynomial equation of degree $n$, one can plug its coefficients into the $n^{\text{th}}$ power series. There is no contradiction to previous mathematical thought, and the field remains unshaken. (The authors note that while a polynomial equation of degree $n$ has $n$ solutions in general, their algorithm produces only one).
On the other hand, the details of those power series are extremely interesting. The coefficients are Catalan numbers, and generalizations of Catalan numbers called hyper-Catalan numbers. These numbers turn up everywhere in combinatorial mathematics. If you have $n$ left parentheses and $n$ right parentheses, the $n^{\text{th}}$ Catalan number $C_n$ counts how many legal ways you can string them together. For $n=3$ there are $C_3 = 5$ ways: $(~)(~)(~), ((~))(~), (~)((~)), ((~)(~))$ and $(((~)))$. Equivalently, the $n^{\text{th}}$ Catalan number counts the number of ways an $(n+2)$-gon can be subdivided into $n$ triangles.
More about these numbers here and here.
The Catalan numbers can be computed using the formula $$C_n=\frac{(2n)!}{n!(n+1)!}.$$ The first few are 1, 2, 5, 14, 42, and 142.
Wildberger and Rubine give us a peek at the method by calculating a solution to the quadratic equation $tx^2 -x +1=0$ as $x=\sum_{n\geq 0}C_nt^n$. They start with the quadratic formula, which gives the solutions $$x=\frac{1\pm\sqrt{1-4t}}{2t}.$$ Newton’s binomial expansion for $\sqrt{1-a}=(1-a)^{\frac{1}{2}}$ is
$$1+\frac{1}{2}(-a) + \frac{(\frac{1}{2})(\frac{-1}{2})(-a)^2}{2!}
+ \frac{(\frac{1}{2})(\frac{-1}{2})(\frac{-3}{2})(-a)^3}{3!}
+ \frac{(\frac{1}{2})(\frac{-1}{2})(\frac{-3}{2})(\frac{-5}{2})(-a)^4}{4!}
+ \cdots.$$
For $a=4t$ this gives
$$\sqrt{1-4t} = 1-2t + \frac{(\frac{1}{2})(\frac{-1}{2})4^2t^2}{2!}
– \frac{(\frac{1}{2})(\frac{-1}{2})(\frac{-3}{2})4^3 t^3}{3!}
+ \frac{(\frac{1}{2})(\frac{-1}{2})(\frac{-3}{2})(\frac{-5}{2})4^4t^4}{4!}
– \cdots.$$
$$= 1-2t-\frac{2^2t^2}{2!} – \frac{3\cdot 2^3t^3}{3!} – \frac{3\cdot 5\cdot 2^4t^4}{4!} – \cdots .$$Taking the root with the $-$ sign from the quadratic formula gives$$x = 1 + \frac{2t}{2!} + \frac{3\cdot 2^2t^2}{3!} + \frac{3\cdot 5\cdot 2^3t^3}{4!} + \cdots .$$The pattern is clear: the coefficient of $t^n$ is the product of the first $n$ odd numbers times $2^n$ divided by $(n+1)!$. Multiplying this fraction top and bottom by $n!$ gives$$\frac {1\cdot 3\cdot 5\cdot \cdots (2n-1) \cdot 2^n \cdot n!}{n!(n+1)!} = \frac{(2n)!}{n!(n+1)!} = C_n$$since $2^n \cdot n!$ has all the even factors of $(2n)!$, and the odd ones are already there. Magic!
To spell it out, the Catalan-number power series $\sum_{n\geq 0}C_nz^n$ evaluated at $z=t$ gives us a solution to the equation $tx^2-x+1=0$. It remains to note that the general quadratic equation $c_2x^2+c_1x+c_0=0$ can be put into that form by the substitution $y=(-c_1/c_0)x$; in that case $t=c_0c_2/c_1^2$.
The coefficients for a series solution of a polynomial equation of higher degree come from the hyper-Catalan numbers. These count ways of partitioning a $n$-gon, but now the regions can be any combination of polygons. For example, $C[2,0,1]$ is the number of ways a 7-gon can be split into two triangles, zero quadrilaterals and 1 pentagon. (That it must be a 7-gon can be seen by counting the number of free edges: $7=2\cdot 3~ +~ 0\cdot 4~ +~ 1\cdot 5~ -~ 4$.)
The practical consequences of these discoveries are not so clear. The authors use an example that appeared in the works of Newton and John Wallis: The computation of the real solution of $x^3-2x-5=0$. The solution is now called “Wallis’s number,” and its digits are listed in the Online Encyclopedia of Integer Sequences.
Wildberger and Rubine start with the initial guess $x=2$; for a cubic, only the first generation $C[m_1, m_2]$ of hyper-Catalan numbers are needed. They only use those giving terms up to degree 3. Their first run gives $x = 2.0945345708$, which agrees with the OEIS sequence to four decimal places. Using their first answer as a new initial guess, they repeat the process and obtain $x=2.0945514815423265098$, which now has sixteen good decimal places. On the other hand, applying the very elementary Newton’s method with initial guess $x=2$ (and the help of a “full-precision calculator”) gives a first approximation $2.1$, a second approximation $2.094568$ and a third approximation $2.094551481542326591496382$. This last approximation has nineteen decimal places matching the OEIS sequence.
—Tony Phillips, Stony Brook University