The gaps are filled by subsequent soddy spheres thus forming a solid object at infinity. Software for generating and visualization of the apollonian sphere packing. Fractal dimension of apollonian packing of spherical. If we apply this information to our program picture of packing we can still generate a circle packing.
Apollonion sphere packing is a recursive algorithm to fill a hollow. Apollonian packing model is usually used to study the fractal properties of packing objects 36 37 38. Given 4 mutually tangent circles in a packing whose curvatures are a1,a2,a3,a4. A theorem on apollonian circle packings for every integral apollonian circle packing there is a uniqueminimalquadrupleofintegercurvatures,a,b,c,d,satisfyinga. Zero curvature gives a line circle with infinite radius. I am trying to find a way, but cannot figure it out now. Apollonian sphere packing or soddy spheres by leisink. We introduce the notion of a crystallographic sphere packing, defined to. Positive curvature indicates that all other circles are externally. In the plane, the inverse of a point p in respect to a circle of center o and radius r is a point p such that p and p are on the same ray going from o, and op times op equals. Apollonion sphere packing is a recursive algorithm to fill a hollow ball with spheres of different diameters. Modulating the positioning of a apollonian gasket effect. An apollonian circle packing acp is an ancient greek construction which is made by repeatedly inscribing circles into the triangular interstices in a descartes con. Slices of an apollonian sphere packing as we rotate around the zaxis by 16 of a revolution duration.
In order to construct an apollonian circle packing, we begin with four mutually tangent circles in the plane see figure 2 for possible con gurations and keep adding newer circles tangent to three of the previous circles provided by theorem 1. A outer soddy sphere is added that encloses the 4 spheres. Number of inner spheres maximum radius of inner spheres packing density optimality diagram exact form approximate 1 1. Thanks for contributing an answer to mathematics stack exchange. Asking for help, clarification, or responding to other answers. The university of tennessee, knoxville knoxville, tennessee 37996 865974. The resulting graph illustrates the first and third columns of the table of the main theorem. Sphere packing and kissing numbers problems of arranging balls densely arise in many situations, particularly in coding theory the balls are formed by the sets of inputs that the errorcorrection would map into a single codeword. Apollonian coronas and a new zeta function 19 pages, arxiv. In two dimensions, the equivalent problem is packing circles on a plane. R 1, which consist of the orbits of the four circles in d under the action of a discrete group g ad of mobius transformations inside the conformal group mob2. The method used to create the apollonian gasket is based on circle inversion, which is a geometrical transformation acting with a reference circle that modifies points. Based on the 31 944 875 541 924 spheres of radius greater than. The result is often called an apollonian sphere packing.
Based on the 31 944 875 541 924 spheres of radius greater than 219contained in the apollonian packing of the unit sphere, we obtained an estimate of 2. We used a program written in matlab to calculate the fractal dimension. Sep 30, 2005 apollonian circle packings arise by repeatedly filling the interstices between four mutually tangent circles with further tangent circles. Apollonian sphere packing apollonian sphere packing is the threedimensional equivalent of the apollonian gasket. Apollonian sphere packing or soddy spheres by leisink thingiverse.
Based on the 31 944 875 541 924 spheres of radius greater than 2. In dimensions higher than three, the densest regular packings of hyperspheres are known up to 8 dimensions. Sphere packing is the problem of arranging nonoverlapping spheres within some space, with the goal of maximizing the combined volume of the spheres. A geometric algorithm based on tetrahedral meshes to generate a. A list of conjectural best packings in dimensions less than 10 can be found in 6. Mar 15, 2011 this is an updated version of slices of a gm sphere packing a generalization of gmcircle packings. Continuing this process inde nitely, we arrive at an in nite circle packing, called an apollonian circle packing. It is the threedimensional equivalent of the circle packing in a circle problem in two dimensions. Amirjanow and sobolev 38,39 established a dense packing fractal model of spherical. The most important question in this area is keplers problem. Based on the 31 944 875 541 924 spheres of radius greater than 219 contained in the apollonian packing of the unit sphere, we obtained an estimate of 2. Learn more before you buy, or discover other cool products in mathematical art.
At infinity there is no empty space inside the ball. Pdf the fractal dimension of the apollonian sphere packing has been computed. The algorithm starts with four balls at the vertices of a tetrahedron. Negative curvature indicates that all other circles are internally tangent to that circle. Sphere packing in a sphere is a threedimensional packing problem with the objective of packing a given number of equal spheres inside a unit sphere. The apollonian group a deeper study of an apollonian packing is facilitated by introducing the symmetry group awhich is called the apollonian group. In this version we take slices rotating through the zaxis as we vary the angle in a range of. In 2d packing,the ceter is straightforward and can be thought of as a point in the 2dplane and be described by a complex number. For generic domains, the generation of a sphere pack may be complex and. Generalized apollonian gasket fractal python recipes.
When run, that effect setting generated the second image of a black sphere on a white background. Sphere packing in a cuboid algorithm mathematics stack exchange. Based on the 31 944 875 541 924 spheres of radius greater than 2 19 contained. Before now, the exact values of the sphere packing constants in all dimensions greater than 3 have been unknown. In the classical case, the spheres are all of the same sizes, and the space in question is threedimensional space e. Fractal properties of apollonian packing of spherical. Apollonian circle packings arise by repeatedly filling the interstices between four mutually tangent circles with further tangent circles.
The maximum packing value of randomly packed equallysized spheres was found. The apollonian palace of an apollonian circle packing is the graph having one vertex for each circle of the packing, and having an edge joining each pair of vertices whose circles are tangent. The goal of this program is to understand the basic nature of the classical. Curvature the curvature of a circle bend is defined to be the inverse of its radius. In spite of that, this week has been the week of the spheres, after looking at how to generate spherical solids, display them using avf, and use them for geometrical proximity filtering. The sensual apollonian circle packing sciencedirect. Do you have any idea on how to represent the center of sphere in 3d packing. For generic domains, the generation of a sphere pack may be complex and timeconsuming, especially if the pack must comply with a prescribed sphere size distribution and the stability requirements of the simulation. I started with the sphere multiplier image operation settings as shown in the 1st gallery image. The algorithm was described in details in this paper 2004 random close packing of spheres in a round cell. Integral apollonian packings peter sarnak maa lecture. Aste showed that for apollonian packing, where a dense sphere packing is generated by filling the interstitial spaces with non overlapping spheres of maximum sizes, the only relevant metric parameter is the ratio between the external radii of the spheres that generate an interstice and the internal radius of the sphere that fill this interstice. Department of mathematics the university of tennessee. Sphere packing project gutenberg selfpublishing ebooks.
Pdf the fractal dimension of the apollonian sphere packing. Counting problems for apollonian circle packings an apollonian circle packing is one of the most of beautiful circle packings whose construction can be described in a very simple manner based on an old theorem of apollonius of perga. An efficient algorithm to generate random sphere packs in. Apollonian gasket, a fractal circle packing formed by packing smaller circles into. The sphere packing problem is the threedimensional version of a class of ball packing problems in arbitrary dimensions. This value coincides with that for a random apollonian packing of spheres, for which the most commonly cited value is 2. Multisized sphere packing shuji yamada1 jinko kanno2 miki miyauchi3 1department of computer science kyoto sangyo university, japan 2mathematics and statistics program louisiana tech university, u. Apollonian sphere packing is the threedimensional equivalent of the apollonian gasket.
Doremus, a geometric algorithm based on tetrahedral meshes to generate a dense polydisperse sphere packing, granular matter, 11 2009 4352. The fractal dimension of the 3d apollonian has been calculated as 2. The idea is based on tangency spinors defined for pairs of tangent disks. Geometry and arithmetic of crystallographic sphere packings ncbi. Construction of apollonian circle packings beginning with4 mutually tangent circles, we can keep adding newer circles tangent to three of the previous circles, provided by the apollonius theorem. We can take a random apollonian circle like one with curvatures 2, 3, 6, and 7, and perform some calculations using simple geometry to find the coordinates of each circle. The image gallery above shows off a new variation of building a spatially modulated close packing effect using the msg preset we built above. An apollonian circle packing is one of the most of beautiful circle packings whose construction can be described in a very simple manner based on an old theorem of apollonius of perga. From apollonian circle packings to fibonacci numbers.
An efficient algorithm to generate random sphere packs in arbitrary. The next few posts will look at solving this problem in 3d. This thing includes five stages of the apollonion sphere packing. Apollonian packing characterization1 geometric characterization of apollonian packings i an apollonian packing p d is a set of circles in the riemann sphere c. Optimization of a computer simulation model for packing of. The fractal dimension of the apollonian sphere packing has been computed numerically up to six trusty decimal digits. A key property enjoyed by the classical apollonian circle packing and. No spheres are intersecting, they touch each other. Hi all, i have some question of my current project of the sphere packing issue. Spheres are not as common as planar faceted objects in the architectural domain.772 923 358 82 6 963 769 1526 1549 585 924 1318 313 587 1451 1021 150 964 1291 374 1214 362 205 1324 1539 910 394 81 934 1052 752 1207 922 37