Related papers: The icosahedral line configuration and Waldschmidt…
A polyhedron is Rupert if it is possible to cut a hole in it and thread an identical polyhedron through the hole. It is known that all 5 Platonic solids, 10 of the 13 Archimedean solids, 9 of the 13 Catalan solids, and 82 of the 92 Johnson…
Let $P$ be a set of points and $L$ a set of lines in the (extended) Euclidean plane, and $I \subseteq P\times L$, where $i =(p,l) \in I$ means that point $p$ and line $l$ are incident. The incidences can be interpreted as quadratic…
Recent years have seen a surprising connection between the physics of scattering amplitudes and a class of mathematical objects--the positive Grassmannian, positive loop Grassmannians, tree and loop Amplituhedra--which have been loosely…
In this paper, we study $6$-dimensional GKM manifolds with $4$ fixed points. We classify all possible GKM graphs, and for each type of graph we construct a manifold, proving the existence. We show that six types occur. (P1) complex…
We characterize the topological configurations of points and lines that may arise when placing n points on a circle and drawing the n perpendicular bisectors of the sides of the corresponding convex cyclic n-gon. We also provide exact and…
Let $\mathcal{K}$ denote a nonsingular conic in the complex projective plane. Pascal's theorem says that, given six distinct points $A,B,C,D,E,F$ on $\mathcal{K}$, the three intersection points $AE \cap BF, AD \cap CF, BD \cap CE$ are…
We present a new complementary strategy to quasicrystalline structure determination: The local atomic structure of simple icosahedral (si) Ho11Mg15Zn74 [a(6D)=5.144(3)A] in a sphere of up to r=17A was refined using the atomic pair…
We present two constructions in this paper: (a) A 10-vertex triangulation $\CC P^{2}_{10}$ of the complex projective plane $\CC P^{2}$ as a subcomplex of the join of the standard sphere ($S^{2}_4$) and the standard real projective plane…
The SHGH conjecture proposes a solution to the question of how many conditions a general union of fat points imposes on the complete linear system of curves in $\mathbb P^2$ of fixed degree $d$, and it is known to be true in many cases. We…
We present a symmetry-based exhaustive approach to explore the structural and compositional richness of two-dimensional materials. We use a ``combinatorial engine'' that constructs potential compounds by occupying all possible Wyckoff…
We compute the integral homology and cohomology groups of configuration spaces of two distinct points on a given real projective space. The explicit answer is related to the (known multiplicative structure in the) integral cohomology---with…
In this paper, we study the Lie point symmetry group of a system describing an ideal plastic plane flow in two dimensions in order to find analytical solutions of the system. The infinitesimal generators that span the Lie algebra for this…
This paper continues the investigation of Part I, by studying the conic $\mathcal{C}_P$ on the five points $ABCPQ$, where $ABC$ is a given ordinary triangle and $Q$ is the isotomcomplement of $P$, defined as the complement of the isotomic…
It is shown that the Veldkamp space of the unique generalized quadrangle GQ(2,4) is isomorphic to PG(5,2). Since the GQ(2,4) features only two kinds of geometric hyperplanes, namely point's perp-sets and GQ(2,2)s, the 63 points of PG(5,2)…
Every regular map on a closed surface gives rise to generally six regular maps, its "Petrie relatives", that are obtained through iteration of the duality and Petrie operations (taking duals and Petrie-duals). It is shown that the skeletal…
The general stationary cylindrically symmetric solution of Einstein-massless scalar field system with a non-positive cosmological constant is presented. It is shown that the general solution is characterized by four integration constants.…
By regular tessellation, we mean any hyperbolic 3-manifold tessellated by ideal Platonic solids such that the symmetry group acts transitively on oriented flags. A regular tessellation has an invariant we call the cusp modulus. For small…
We analyze in the context of geometrothermodynamics a Legendre invariant metric structure in the equilibrium space of an ideal gas. We introduce the concept of thermodynamic geodesic as a succession of points, each corresponding to a state…
We find exact solutions of the Einstein equations which describe a black hole pierced by infinitely thin cosmic strings. The string segments enter the black hole along the radii and their positions coincide with the symmetry axes of a…
Symmetry is an implicit objective in structural form-finding that often reconciles efficiency and aesthetics. This paper identifies the symmetry of polyhedral diagrams in three-dimensional graphic statics (3DGS) as point groups and…