Related papers: The Bring sextic of equilateral pentagons
Bring's curve, the unique Riemann surface of genus-4 with automorphism group $S_5$, has many exceptional properties. We review, give new proofs of, and extend a number of these including giving the complete realisation of the automorphism…
Bring's curve is the genus 4 Riemann surface with automorphism group of maximal size, $S_5$. Riera and Rodr\'iguez have provided the most detailed study of the curve thus far via a hyperbolic model. We will recover and extend their results…
An equilateral pentagon is a polygon in the plane with five sides of equal length. In this paper we classify the central configurations of the $5$-body problem having the five bodies at the vertices of an equilateral pentagon with an axis…
The Fano models of Enriques surfaces produce a family of tens of mutually intersecting planes in $\mathbf P^5$ with a $10$-dimensional moduli space. The latter is linked to several 10-dimensional moduli spaces parametrizing other types of…
It is shown that in dimension at least three a local diffeomorphism of Euclidean n-space into itself is injective provided that the pull-back of every plane is a Riemannian submanifold which is conformal to a plane. Using a similar…
A group-theoretical approach to the construction of quasiperiodic tilings of a Euclidean plane, possessing five-fold symmetry, is applied. Of the infinitely many of variants of quasiperiodic partitions of the plane, possessing the dihedral…
The modular group $\Gamma$ (which is the Hecke group $H_3$) can be used to study triangular maps. Here we use the Hecke group $H_4$ to study the regular map that underlies Bring's surface of genus 4. Our main result is the determination of…
Which surfaces can be realized with two-dimensional faces of the five-dimensional cube (the penteract)? How can we visualize them? In recent work, Aveni, Govc, and Roldan, show that there exist 2690 connected closed cubical surfaces up to…
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…
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…
The combinatorial structure of the realization space of the euqilateral pentagon linkage is closely related to a tiling of the hyperbolic plane by right-angled pentagons. In this correspondence lower dimensional faces of the tiling…
Geometric Langlands duality is usually formulated as a statement about Riemann surfaces, but it can be naturally understood as a consequence of electric-magnetic duality of four-dimensional gauge theory. This duality in turn is naturally…
This article deals with the construction of surfaces that are suitable for representing pentachords or 5-pitch segments that are in the same $T/I$ class. It is a generalization of the well known \"Ottingen-Riemann torus for triads of…
We construct positive-genus analogues of Welschinger's invariants for many real symplectic manifolds, including the odd-dimensional projective spaces and the renowned quintic threefold. In some cases, our invariants provide lower bounds for…
An almost forgotten gem of Gauss tells us how to compute the area of a pentagon by just going around it and measuring areas of each vertex triangles (i.e. triangles whose vertices are three consecutive vertices of the pentagon). We give…
We propose an analytical approach to the conformal mapping of (rectangular) polygons based on the theory of Riemann surfaces and theta functions.
The paper develops the fundamentals of quaternionic holomorphic curve theory. The holomorphic functions in this theory are conformal maps from a Riemann surface into the 4-sphere, i.e., the quaternionic projective line. Basic results such…
A genus one curve of degree 5 is defined by the 4 x 4 Pfaffians of a 5 x 5 alternating matrix of linear forms on P^4. We describe a general method for investigating the invariant theory of such models. We use it to explain how we found our…
The Hodge decomposition is a fundamental result in differential geometry and algebraic topology, particularly in the study of differential forms on a Riemannian manifold. Despite extensive research in the past few decades,…
We study framed translation surfaces corresponding to meromorphic differentials on compact Riemann surfaces, for which a horizontal separatrix is marked for each pole or zero. Such geometric structures naturally appear when studying flat…