Related papers: Radial Density in Apollonian Packings
In the early 1990s, A. Bezdek and W. Kuperberg used a relatively simple argument to show a surprising result: The maximum packing density of circular cylinders of infinite length in $\mathbb{R}^3$ is exactly $\pi/\sqrt{12}$, the planar…
We provide, for any $r\in (0,1)$, lower and upper bounds on the maximal density of a packing in the Euclidean plane of discs of radius $1$ and $r$. The lower bounds are mostly folk, but the upper bounds improve the best previously known…
We estimated the scale-length of the thin disc with the J and W1 magnitudes of the most probable Red Clump (RC) stars in the Galactic plane, $-0\overset{^\circ}.5 \leq b \leq +0\overset{^\circ}.5$, in 19 equal sized fields with consecutive…
We prove that given a fixed radius $r$, the set of isometry-invariant probability measures supported on ``periodic'' radius $r$-circle packings of the hyperbolic plane is dense in the space of all isometry-invariant probability measures on…
We consider circle packings in the plane with circles of sizes $1$, $r\simeq 0.834$ and $s\simeq 0.651$. These sizes are algebraic numbers which allow a compact packing, that is, a packing in which each hole is formed by three mutually…
We consider packings of the plane using discs of radius 1 and r=0.545151... . The value of r admits compact packings in which each hole in the packing is formed by three discs which are tangent to each other. We prove that the largest…
Motivated by the relation between particle shape and packing, we measure the volume fraction $\phi$ occupied by the Platonic solids which are a class of polyhedron with congruent sides, vertices and dihedral angles. Tetrahedron, cube,…
The disk surface density of the nearby spiral galaxy M33 is estimated assuming that it is marginally stable against gravitational perturbations. For this purpose we used the radial profile of line-of-sight velocity dispersion of the disk…
Suppose one has a collection of disks of various sizes with disjoint interiors, a packing in the plane, and suppose the ratio of the smallest radius divided by the largest radius lies between $1$ and $q$. In his 1964 book Regular Figures…
This study explores the density profile of the stellar disk, radially and azimuthally, based on approximately 8.4 million red clump stars selected from Gaia Bp/Rp spectra. After correcting for selection effects and distance uncertainties,…
A compact circle-packing $P$ of the Euclidean plane is a set of circles which bound mutually disjoint open discs with the property that, for every circle $S\in P$, there exists a maximal indexed set $\{A_{0},\ldots,A_{n-1}\}\subseteq P$ so…
Three colour photometry on CCD frames in the Special Area SpA23 provides a deep probe of the galactic disc in a low absorption window towards the anticenter. Magnitudes to better than 10% at V = 25 and B-V colour down to V = 23 have been…
We have re-estimated the surface density of the Galactic disk in the solar neighborhood within $\pm$ 0.4 kpc of the Sun using parallaxes and proper motions of a kinematically and spatially unbiased sample of 1476 old bright red giant stars…
We present new multi-band imaging data in the optical (BVRI and Halpha) and near infrared bands (JHK) of 15 candidate ring galaxies from the sample of Appleton & Marston (1997). We use these data to obtain color composite images, global…
A disc packing in the plane is compact if its contact graph is a triangulation. There are $9$ values of $r$ such that a compact packing by discs of radii $1$ and $r$ exists. We prove, for each of these $9$ values, that the maximal density…
In "Dense Sphere Packings: A Blueprint for Formal Proofs" Hales proves that for every packing of unit spheres, the density in a ball of radius $r$ is at most $\pi/\sqrt{18}+c/r$ for some constant $c$. When $r$ tends to infinity, this gives…
An earlier paper describes a program to prove the Kepler conjecture on sphere packings. This paper carries out the second step of that program. A sphere packing leads to a decomposition of $R^3$ into polyhedra. The polyhedra are divided…
This is the sixth in a series of papers giving a proof of the Kepler conjecture, which asserts that the density of a packing of congruent spheres in three dimensions is never greater than $\pi/\sqrt{18}\approx 0.74048...$. This is the…
This is the fifth in a series of papers giving a proof of the Kepler conjecture, which asserts that the density of a packing of congruent spheres in three dimensions is never greater than $\pi/\sqrt{18}\approx 0.74048...$. This is the…
We propose that the lower bound of the stellar radial velocity dispersion c_r of an equilibrium stellar disk is determined by the gravitational stability condition. We compared the estimates of stellar velocity dispersion at radii r>(1.5 -…