arXiv subjects

Peter Doyle

Publications and source records attributed to Peter Doyle.

11 recordsLinked to original sources

arXiv2026

Neoplatonic solids

A \emph{6-net} is a simplicial triangulation of the $2$-sphere with maximum degree $\leq 6$. Experiments suggest that every $6$-net admits a unique realization as an undented Euclidean polyhedron built from unit equilateral triangles, and a unique realization as an ideal equilateral hyperbolic polyhedron. We call these \emph{neoplatonic solids} and \emph{ideal neoplatonics}. A net is \emph{prime} if every 3-cycle bounds a face. A computer-assisted proof shows that every prime $6$-net with $v \leq 50$ has a unique realization as a convex ideal neoplatonic. Numerical homotopy from this realization yields an approximate Euclidean neoplatonic, and a computer-assisted proof shows that a true Euclidean neoplatonic lies nearby, though we do not prove uniqueness. Using the separating-triangle decomposition, we extend Euclidean existence to all $10{,}412{,}340$ $6$-nets with $v\leq50$, counted up to combinatorial isomorphism.

math.MG↗

arXiv2025

Collapsibility and Near Universality for Vertex Minimal Paper Tori

A paper torus is a piecewise linear isometric embedding of a flat torus into $\R^3$. Following up on the $8$-vertex paper tori discovered by the second author, we prove universality and collapsibility results about these objects. One corollary is that any flat torus without reflection symmetry is realized as an $8$-vertex paper torus. Another corollary is that, for any $\epsilon>0$, there is an $8$-vertex paper torus within $\epsilon$ of a unit equilateral triangle in the Hausdorff metric.

math.MG↗

arXiv2025

Taut fillings

Let $\sigma$ be a simplicial triangulation of the 2-sphere, $X$ the associated integral 2-cycle. A filling of $X$ is an integral 3-chain $M$ with $\partial M = X$; a taut filling is one with minimal $L_1$-norm. We show that any taut filling arises from an extension of $\sigma$ to a simplicial complex homeomorphic to the 3-ball. The filling is clean: it has no repeated tetrahedron, and its support complex is a clean simplicial complex. This support complex is shellable and flag: every clique in its 1-skeleton occurs as a simplex. The key to the proof is the general fact that any taut filling of an $n$-cycle splits under disjoint union, connected sum, and more generally what we call almost disjoint union, where summands are supported on sets that overlap in at most $n+1$ vertices. We used AI to formalize and prove in Lean the splitting theorem and the resulting cleanness, shellability, and flagness results.

math.GT↗

arXiv2023

Filling a triangulation of the 2-sphere

Define the tet-volume of a triangulation of the 2-sphere to be the minimum number of tetrahedra in a 3-complex of which it is the boundary, and let $d(v)$ be the maximum tet-volume for $v$-vertex triangulations. In 1986 Sleator, Tarjan, and Thurston (STT) proved that $d(v) = 2v-10$ holds for large $v$, and conjectured that it holds for all $v \geq 13$. Their proof used hyperbolic polyhedra of large volume. They suggested using more general notions of volume instead. In work that was all but lost, Mathieu and Thurston used this approach to outline a combinatorial proof of the STT asymptotic result. Here we use a much simplified version of their approach to prove the full conjecture. This implies STT's weaker conjecture, proven by Pournin in 2014, characterizing the maximum rotation distance between trees.

math.CO↗

arXiv2018

Conway's doughnuts

Morley's Theorem about angle trisectors can be viewed as the statement that a certain diagram `exists', meaning that triangles of prescribed shapes meet in a prescribed pattern. This diagram is the case n=3 of a class of diagrams we call `Conway's doughnuts'. These diagrams can be proven to exist using John Smillie's holonomy method, recently championed by Eric Braude: `Guess the shapes; check the holonomy.' For n = 2, 3, 4 the existence of the doughnut happens to be easy to prove because the hole is absent or triangular.

math.HO↗

arXiv2017

Blowing bubbles on the torus

We consider the regularized trace of the inverse of the Laplacian on a skinny torus. With its flat metric, a skinny torus has large trace, but we show that there are conformally equivalent metrics making the trace close to that of a sphere of the same area. This behavior is in sharp contrast to that of the log-determinant, a well-known spectral invariant which is extremized at the flat metric on any torus. Our examples are bubbled tori, where you take a sphere, discard polar regions, and glue top to bottom. In a addendum, we belatedly notice that our bubbled tori have trace less than the sphere, and outline how to exploit this to get Okikiolu's result that by means of a conformal factor depending only on longitude, any torus can be made to have trace less than the sphere.

math.SP↗

arXiv2015

Changing gears: Isospectrality via eigenderivative transplantation

We introduce a new method for constructing isospectral quantum graphs that is based on transplanting derivatives of eigenfunctions. We also present simple digraphs with the same reversing zeta function, which generalizes the Bartholdi zeta function to digraphs.

math.SP↗

arXiv2011

Shadow movies not arising from knots

A shadow diagram is a knot diagram with under-over information omitted; a shadow movie is a sequence of shadow diagrams related by shadow Reidemeister moves. We show that not every shadow movie arises as the shadow of a Reidemeister movie, meaning a sequence of classical knot diagrams related by classical Reidemeister moves. This means that in Kaufman's theory of virtual knots, virtual crossings cannot simply be viewed as classical crossings where which strand is over has been left `to be determined'.

math.GT↗

arXiv2010

Some planar isospectral domains

We give a number of examples of isospectral pairs of plane domains, and a particularly simple method of proving isospectrality. One of our examples is a pair of domains that are not only isospectral but homophonic: Each domain has a distinguished point such that corresponding normalized Dirichlet eigenfunctions take equal values at the distinguished points. This shows that one really can't hear the shape of a drum.

math.DG↗

arXiv2003

23040 symmetries of hyperbolic tetrahedra

We give a rigorous geometric proof of the Murakami-Yano formula for the volume of a hyperbolic tetrahedron. In doing so, we are led to consider generalized hyperbolic tetrahedra, which are allowed to be non-convex, and have vertices `beyond infinity'; and we uncover a group, which we call 22.5K, of 23040 scissors-class-preserving symmetries of the space of (suitably decorated) generalized hyperbolic tetrahedra. The group 22.5K contains the Regge symmetries as a subgroup of order 144. From a generic tetrahedron, 22.5K produces 30 distinct generalized tetrahedra in the same scissors class, including the 12 honest-to-goodness tetrahedra produced by the Regge subgroup. The action of 22.5K leads us to the Murakami-Yano formula, and to 9 others, which are similar but less symmetrical. From here, we can derive yet other volume formulas with pleasant algebraic and analytical properties. The key to understanding all this is a natural relationship between a hyperbolic tetrahedron and a pair of ideal hyperbolic octahedra.

math.GT↗

arXiv1999

Solving the sextic by iteration: A complex dynamical approach

Recently, Peter Doyle and Curt McMullen devised an iterative solution to the fifth degree polynomial. At the method's core is a rational mapping of the Riemann sphere with the icosahedral symmetry of a general quintic. Moreover, this map posseses "reliable" dynamics: for almost any initial point, the its trajectory converges to one of the periodic cycles that comprise an icosahedral orbit. This symmetry-breaking provides for a reliable or "generally-convergent" quintic-solving algorithm: with almost any fifth-degree equation, associate a rational mapping that has reliable dynamics and whose attractor consists of points from which one computes a root. An algorithm that solves the sixth-degree equation requires a dynamical system with the symmetry of the alternating group on six things. This group does not act on the Riemmann sphere, but does act on the complex projective plane--this is the Valentiner group. The present work exploits the resulting 2-dimensional geometry in finding a Valentiner-symmetric rational mapping whose elegant dynamics experimentally appear to be reliable in the above sense---transferred to the 2-dimensional setting. This map provides the central feature of a conjecturally-reliable sextic-solving algorithm analogous to that employed in the quintic case.

math.DS↗