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   Close-packing of equal spheres explained
  </title><link href=../explained.today.css?v=1 rel=stylesheet><meta content="row, centers, first, hcp, left, right, packing, two, coordinates, fcc, coordinate, plane, arrangement, close-packing, equal, lattice"name=keywords><meta content="What is Close-packing of equal spheres? Explaining what we could find out about Close-packing of equal spheres."name=description><script>(function(i,s,o,g,r,a,m){i['GoogleAnalyticsObject']=r;i[r]=i[r]||function(){(i[r].q=i[r].q||[]).push(arguments)},i[r].l=1*new Date();a=s.createElement(o),m=s.getElementsByTagName(o)[0];a.async=1;a.src=g;m.parentNode.insertBefore(a,m)})(window,document,'script','https://www.google-analytics.com/analytics.js','ga');ga('create', 'UA-101684966-1', 'auto');ga('send', 'pageview');</script><link href=https://cdn.jsdelivr.net/npm/cookieconsent@3/build/cookieconsent.min.css rel=stylesheet><script async src=//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js></script><script>(adsbygoogle = window.adsbygoogle || []).push({  google_ad_client: "ca-pub-5994403859070052",  enable_page_level_ads: true });</script><body><div class=Header><div class=BreadCrumb><ul><li class=First>You are here<li><a href=http://everything.explained.today> Everything </a> <a href=http://explained.today> Explained.Today </a><li><a href=../contents/list.htm> A-Z Contents </a><li><a href=../contents/list_C.htm> C </a><li><a href=../contents/list_CL.htm> CL </a><li><a href=../contents/list_CLO.htm> CLO </a><li><a href=../contents/list_CLOS.htm> CLOS </a><li><a href=../contents/list_CLOSE.htm> CLOSE </a><li>Close-packing of equal spheres</ul></div></div><div class=SideBar><p class=logo><img alt="Explained Today"title="The Information and Knowledge Portal"height=125px src=../images/brain.gif width=200px><form action=http://www.google.com/cse id=cse-search-box target=_blank><div><input name=cx type=hidden value=partner-pub-5994403859070052:sf3q7o-pgfc><input name=ie type=hidden value=UTF-8><input name=q size=24><input name=sa type=submit value=Search></div></form><script src=http://www.google.com/coop/cse/brand?form=cse-search-box&lang=en></script><p> <div class=HideIfFull><script async src=//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js></script><ins class=adsbygoogle data-ad-client=ca-pub-5994403859070052 data-ad-format=auto data-ad-slot=6382878523 style=display:block> </ins><script>(adsbygoogle = window.adsbygoogle || []).push({});</script></div></div><div class=HeaderBar><script async src=//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js></script><ins class=adsbygoogle data-ad-client=ca-pub-5994403859070052 data-ad-format=auto data-ad-slot=4896110924 style=display:block> </ins><script>(adsbygoogle = window.adsbygoogle || []).push({});</script></div><div class=Body><h1>Close-packing of equal spheres explained</h1><p>In <a title="Geometry is called a geometer."href=../geometry/> geometry </a> , <b> close-packing of equal <a title="A sphere is a geometrical object in three-dimensional space that is the surface of a ball."href=../sphere/> sphere </a> s </b> is a dense arrangement of congruent spheres in an infinite, regular arrangement (or <a href=../Lattice_(group)/> lattice </a> ). <a title="Carl Friedrich Gauss was a German mathematician and physicist who made significant contributions to many fields in ..."href=../Carl_Friedrich_Gauss/> Carl Friedrich Gauss </a> proved that the highest average density – that is, the greatest fraction of space occupied by spheres – that can be achieved by a <a href=../Lattice_(group)/> lattice </a> packing is<div class=math><table cellpadding=0 cellspacing=0 class=math style=float:left><tbody><tr><td rowspan=2><td style="border-bottom:1px solid black"><span> <i> \pi </i> </span><tr><td><span> 3 <i> \sqrt </i> 2 </span></table><p style=padding-top:0.5em;float:left><span> ≈ </span> <span> 0 <i> . </i> 74048 </span></div> .The same <a title="A packing density is the fraction of the space filled by the figures making up the packing."href=../packing_density/> packing density </a> can also be achieved by alternate stackings of the same close-packed planes of spheres, including structures that are aperiodic in the stacking direction. The <a title="The Kepler conjecture is a mathematical theorem about sphere packing in three-dimensional Euclidean space."href=../Kepler_conjecture/> Kepler conjecture </a> states that this is the highest density that can be achieved by any arrangement of spheres, either regular or irregular. This conjecture was proven by <a title="Thomas Callister Hales is an American mathematician working in the areas of representation theory, discrete geometry, and ..."href=../Thomas_Callister_Hales/> T. C. Hales </a> . <sup> <a href=#Ref-1> [1] </a> </sup><sup> <a href=#Ref-2> [2] </a> </sup> Highest density is known only in case of 1, 2, 3, 8 and 24 dimensions. <sup> <a href=#Ref-3> [3] </a> </sup><p style=clear:left>Many <a title="A crystal is a solid material whose constituents are arranged in a highly ordered microscopic structure, forming a crystal ..."href=../crystal/> crystal </a> structures are based on a close-packing of a single kind of atom, or a close-packing of large ions with smaller ions filling the spaces between them. The cubic and hexagonal arrangements are very close to one another in energy, and it may be difficult to predict which form will be preferred from first principles.<h2>FCC and HCP lattices</h2> There are two simple regular lattices that achieve this highest average density. They are called <b> face-centered cubic </b> ( <b> FCC </b> ) (also called <b> <a href=../cubic_crystal_system/> cubic </a> close packed </b> ) and <b> <a title="The hexagonal crystal family is the union of the hexagonal crystal system and the trigonal crystal system."href=../Hexagonal_crystal_system/> hexagonal </a> close-packed </b> ( <b> HCP </b> ), based on their <a title="Symmetry is usually used to refer to an object that is invariant under some transformations ; including translation, ..."href=../symmetry/> symmetry </a> . Both are based upon sheets of spheres arranged at the vertices of a triangular tiling; they differ in how the sheets are stacked upon one another. The FCC lattice is also known to mathematicians as that generated by the A <sub> 3 </sub><a title="Root system is a configuration of vector s in a Euclidean space satisfying certain geometrical properties."href=../root_system/> root system </a> . <sup> <a href=#Ref-4> [4] </a> </sup><h3>Cannonball problem</h3><p>See main article: <a href=../Cannonball_problem/> Cannonball problem </a> . The problem of close-packing of spheres was first mathematically analyzed by <a title="Thomas Harriot was also recognized for his contributions in navigational techniques, working closely with John White to ..."href=../Thomas_Harriot/> Thomas Harriot </a> around 1587, after a question on piling cannonballs on ships was posed to him by Sir <a title="Walter Raleigh was an English statesman, soldier, spy, writer, poet, explorer and landed gentleman."href=../Walter_Raleigh/> Walter Raleigh </a> on their expedition to America. <sup> <a href=#Ref-5> [5] </a> </sup> Cannonballs were usually piled in a rectangular or triangular wooden frame, forming a three-sided or four-sided pyramid. Both arrangements produce a face-centered cubic lattice – with different orientation to the ground. Hexagonal close-packing would result in a six-sided pyramid with a hexagonal base.<div class=math><table cellpadding=0 cellspacing=0 class=math style=float:left><tbody><tr><td><td class=.smaller><span> <i> N </i> </span><tr><td><i> \sum </i><td><tr><td><td class=.smaller><span> <i> n </i> =1 </span></table><p style=padding-top:0.5em;float:left><span> <i> n </i> <sup> 2 </sup> </span> <span> = </span> <span> <i> M </i> <sup> 2 </sup> </span></div> or <div class=math><table cellpadding=0 cellspacing=0 class=math style=float:left><tbody><tr><td rowspan=2><td style="border-bottom:1px solid black"><span> 1 </span><tr><td><span> 6 </span></table><p style=padding-top:0.5em;float:left><span> <i> N(N </i> +1 <i> )( </i> 2 <i> N </i> +1 <i> ) </i> </span> <span> = </span> <span> <i> M </i> <sup> 2 </sup> </span></div> and conjectured that the only solutions are <div class=math><p style=padding-top:0em;float:left><span> <i> N </i> </span> <span> = </span> <span> 1 <i> , </i> </span> <span> <i> M </i> </span> <span> = </span> <span> 1 <i> , </i> </span></div> and <div class=math><p style=padding-top:0em;float:left><span> <i> N </i> </span> <span> = </span> <span> 24 <i> , </i> </span> <span> <i> M </i> </span> <span> = </span> <span> 70 </span></div> . Here <div class=math><p style=padding-top:0em;float:left><span> <i> N </i> </span></div> is the number of layers in the pyramidal stacking arrangement and <div class=math><p style=padding-top:0em;float:left><span> <i> M </i> </span></div> is the number of cannonballs along an edge in the flat square arrangement. <h3>Positioning and spacing</h3><p>In both the FCC and HCP arrangements each sphere has twelve neighbors. For every sphere there is one gap surrounded by six spheres ( <a title="Octahedron is a polyhedron with eight faces, twelve edges, and six vertices."href=../octahedron/> octahedral </a> ) and two smaller gaps surrounded by four spheres (tetrahedral). The distances to the centers of these gaps from the centers of the surrounding spheres is for the tetrahedral, and for the octahedral, when the sphere radius is 1.<p>Relative to a reference layer with positioning A, two more positionings B and C are possible. Every sequence of A, B, and C without immediate repetition of the same one is possible and gives an equally dense packing for spheres of a given radius.<p>The most regular ones are<ul><li>FCC = ABC ABC ABC... (every third layer is the same)<li>HCP = AB AB AB AB... (every other layer is the same).</ul><p>There is an uncountably infinite number of disordered arrangements of planes (e.g. ABCACBABABAC...) that are sometimes collectively referred to as "Barlow packings", after crystallographer <a title="William Barlow was an English amateur geologist specialising in crystallography."href=../William_Barlow_(geologist)/> William Barlow </a> <sup> <a href=#Ref-6> [6] </a> </sup><p>In close-packing, the center-to-center spacing of spheres in the <i> xy </i> plane is a simple honeycomb-like tessellation with a pitch (distance between sphere centers) of one sphere diameter. The distance between sphere centers, projected on the <i> z </i> (vertical) axis, is:<div class=math><p style=padding-top:0em;float:left><span> <span> pitch </span> <sub> <i> Z </i> </sub> </span> <span> = </span> <span> <i> \sqrt{ </i> 6 <i> } </i> </span> <span> ⋅ </span> <span> <i> {d\over </i> </span> <span> 3 <i> } </i> ≈ 0 <i> . </i> 81649658 </span> <span> <i> d, </i> </span></div><p style=clear:left>where <i> d </i> is the diameter of a sphere; this follows from the tetrahedral arrangement of close-packed spheres.<p>The <a title="Coordination number is the number of atoms, molecules or ions bonded to it."href=../coordination_number/> coordination number </a> of HCP and FCC is 12 and their <a title="Atomic packing factor is the fraction of volume in a crystal structure that is occupied by constituent particles."href=../atomic_packing_factor/> atomic packing factor </a> s (APFs) are equal to the number mentioned above, 0.74.<h2>Lattice generation</h2><p>When forming any sphere-packing lattice, the first fact to notice is that whenever two spheres touch a straight line may be drawn from the center of one sphere to the center of the other intersecting the point of contact. The distance between the centers along the shortest path namely that straight line will therefore be <i> r </i> <sub> 1 </sub>  +  <i> r </i> <sub> 2 </sub> where <i> r </i> <sub> 1 </sub> is the radius of the first sphere and <i> r </i> <sub> 2 </sub> is the radius of the second. In close packing all of the spheres share a common radius, <i> r </i> . Therefore two centers would simply have a distance 2 <i> r </i> .<h3>Simple HCP lattice</h3><p>To form an A-B-A-B-... hexagonal close packing of spheres, the coordinate points of the lattice will be the spheres' centers. Suppose, the goal is to fill a box with spheres according to HCP. The box would be placed on the <a title="A Cartesian coordinate system is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, ..."href=../Cartesian_coordinate_system/> <i> x </i> - <i> y </i> - <i> z </i> coordinate space </a> .<p>First form a row of spheres. The centers will all lie on a straight line. Their <i> x </i> -coordinate will vary by 2 <i> r </i> since the distance between each center of the spheres are touching is 2 <i> r </i> . The <i> y </i> -coordinate and z-coordinate will be the same. For simplicity, say that the balls are the first row and that their <i> y </i> - and <i> z </i> -coordinates are simply <i> r </i> , so that their surfaces rest on the zero-planes. Coordinates of the centers of the first row will look like (2 <i> r </i> ,  <i> r </i> ,  <i> r </i> ), (4 <i> r </i> ,  <i> r </i> ,  <i> r </i> ), (6 <i> r </i>  , <i> r </i> ,  <i> r </i> ), (8 <i> r </i>  , <i> r </i> ,  <i> r </i> ), ... .<p>Now, form the next row of spheres. Again, the centers will all lie on a straight line with <i> x </i> -coordinate differences of 2 <i> r </i> , but there will be a shift of distance <i> r </i> in the <i> x </i> -direction so that the center of every sphere in this row aligns with the <i> x </i> -coordinate of where two spheres touch in the first row. This allows the spheres of the new row to slide in closer to the first row until all spheres in the new row are touching two spheres of the first row. Since the new spheres <i> touch </i> two spheres, their centers form an equilateral triangle with those two neighbors' centers. The side lengths are all 2 <i> r </i> , so the height or <i> y </i> -coordinate difference between the rows is <i> r </i> . Thus, this row will have coordinates like this:<div class=math><p style=padding-top:0em;float:left><span> <i> \left(r, </i> </span> <span> <i> r </i> </span> <span> + </span> <span> <i> \sqrt{ </i> 3 <i> }r, </i> </span> <span> <i> r\right), </i>   <i> \left( </i> 3 <i> r, </i> </span> <span> <i> r </i> </span> <span> + </span> <span> <i> \sqrt{ </i> 3 <i> }r, </i> </span> <span> <i> r\right), </i>   <i> \left( </i> 5 <i> r, </i> </span> <span> <i> r </i> </span> <span> + </span> <span> <i> \sqrt{ </i> 3 <i> }r, </i> </span> <span> <i> r\right), </i>   <i> \left( </i> 7 <i> r, </i> </span> <span> <i> r </i> </span> <span> + </span> <span> <i> \sqrt{ </i> 3 <i> }r, </i> </span> <span> <i> r\right), </i> </span> <span> ... <i> . </i> </span></div><p style=clear:left>The first sphere of this row only touches one sphere in the original row, but its location follows suit with the rest of the row.<p>The next row follows this pattern of shifting the <i> x </i> -coordinate by <i> r </i> and the <i> y </i> -coordinate by . Add rows until reaching the <i> x </i> and <i> y </i> maximum borders of the box.<p>In an A-B-A-B-... stacking pattern, the odd numbered <i> planes </i> of spheres will have exactly the same coordinates save for a pitch difference in the <i> z </i> -coordinates and the even numbered <i> planes </i> of spheres will share the same <i> x </i> - and <i> y </i> -coordinates. Both types of planes are formed using the pattern mentioned above, but the starting place for the <i> first </i> row's first sphere will be different.<p>Using the plane described precisely above as plane #1, the A plane, place a sphere on top of this plane so that it lies touching three spheres in the A-plane. The three spheres are all already touching each other, forming an equilateral triangle, and since they all touch the new sphere, the four centers form a <a title="Tetrahedron is the simplest of all the ordinary convex polyhedra and the only one that has fewer than 5 faces."href=../tetrahedron/> regular tetrahedron </a> . <sup> <a href=#Ref-7> [7] </a> </sup> All of the sides are equal to 2 <i> r </i> because all of the sides are formed by two spheres touching. The height of which or the <i> z </i> -coordinate difference between the two "planes" is . This, combined with the offsets in the <i> x </i> and <i> y </i> -coordinates gives the centers of the first row in the B plane:<div class=math><p style=padding-top:0.5em;float:left><span> <i> \left(r, </i> </span> <span> <i> r </i> </span> <span> + </span><table cellpadding=0 cellspacing=0 class=math style=float:left><tbody><tr><td rowspan=2><td style="border-bottom:1px solid black"><span> <i> \sqrt{ </i> 3 </span><tr><td><span> <i> r}{ </i> 3 <i> }, </i> </span></table><p style=padding-top:0.5em;float:left><span> <i> r </i> </span> <span> + </span><table cellpadding=0 cellspacing=0 class=math style=float:left><tbody><tr><td rowspan=2><td style="border-bottom:1px solid black"><span> <i> \sqrt{ </i> 6 </span><tr><td><span> <i> r </i> 2 <i> }{ </i> 3 <i> }\right), </i>   <i> \left( </i> 3 <i> r, </i> </span></table><p style=padding-top:0.5em;float:left><span> <i> r </i> </span> <span> + </span><table cellpadding=0 cellspacing=0 class=math style=float:left><tbody><tr><td rowspan=2><td style="border-bottom:1px solid black"><span> <i> \sqrt{ </i> 3 </span><tr><td><span> <i> r}{ </i> 3 <i> }, </i> </span></table><p style=padding-top:0.5em;float:left><span> <i> r </i> </span> <span> + </span><table cellpadding=0 cellspacing=0 class=math style=float:left><tbody><tr><td rowspan=2><td style="border-bottom:1px solid black"><span> <i> \sqrt{ </i> 6 </span><tr><td><span> <i> r </i> 2 <i> }{ </i> 3 <i> }\right), </i>   <i> \left( </i> 5 <i> r, </i> </span></table><p style=padding-top:0.5em;float:left><span> <i> r </i> </span> <span> + </span><table cellpadding=0 cellspacing=0 class=math style=float:left><tbody><tr><td rowspan=2><td style="border-bottom:1px solid black"><span> <i> \sqrt{ </i> 3 </span><tr><td><span> <i> r}{ </i> 3 <i> }, </i> </span></table><p style=padding-top:0.5em;float:left><span> <i> r </i> </span> <span> + </span><table cellpadding=0 cellspacing=0 class=math style=float:left><tbody><tr><td rowspan=2><td style="border-bottom:1px solid black"><span> <i> \sqrt{ </i> 6 </span><tr><td><span> <i> r </i> 2 <i> }{ </i> 3 <i> }\right), </i>   <i> \left( </i> 7 <i> r, </i> </span></table><p style=padding-top:0.5em;float:left><span> <i> r </i> </span> <span> + </span><table cellpadding=0 cellspacing=0 class=math style=float:left><tbody><tr><td rowspan=2><td style="border-bottom:1px solid black"><span> <i> \sqrt{ </i> 3 </span><tr><td><span> <i> r}{ </i> 3 <i> }, </i> </span></table><p style=padding-top:0.5em;float:left><span> <i> r </i> </span> <span> + </span><table cellpadding=0 cellspacing=0 class=math style=float:left><tbody><tr><td rowspan=2><td style="border-bottom:1px solid black"><span> <i> \sqrt{ </i> 6 </span><tr><td><span> <i> r </i> 2 <i> }{ </i> 3 <i> }\right), </i> </span></table><p style=padding-top:0.5em;float:left><span> ... <i> . </i> </span></div><p style=clear:left>The second row's coordinates follow the pattern first described above and are:<div class=math><p style=padding-top:0.5em;float:left><span> <i> \left( </i> 2 <i> r, </i> </span> <span> <i> r </i> </span> <span> + </span><table cellpadding=0 cellspacing=0 class=math style=float:left><tbody><tr><td rowspan=2><td style="border-bottom:1px solid black"><span> 4 <i> \sqrt{ </i> 3 </span><tr><td><span> <i> r}{ </i> 3 <i> }, </i> </span></table><p style=padding-top:0.5em;float:left><span> <i> r </i> </span> <span> + </span><table cellpadding=0 cellspacing=0 class=math style=float:left><tbody><tr><td rowspan=2><td style="border-bottom:1px solid black"><span> <i> \sqrt{ </i> 6 </span><tr><td><span> <i> r </i> 2 <i> }{ </i> 3 <i> }\right), </i>   <i> \left( </i> 4 <i> r, </i> </span></table><p style=padding-top:0.5em;float:left><span> <i> r </i> </span> <span> + </span><table cellpadding=0 cellspacing=0 class=math style=float:left><tbody><tr><td rowspan=2><td style="border-bottom:1px solid black"><span> 4 <i> \sqrt{ </i> 3 </span><tr><td><span> <i> r}{ </i> 3 <i> }, </i> </span></table><p style=padding-top:0.5em;float:left><span> <i> r </i> </span> <span> + </span><table cellpadding=0 cellspacing=0 class=math style=float:left><tbody><tr><td rowspan=2><td style="border-bottom:1px solid black"><span> <i> \sqrt{ </i> 6 </span><tr><td><span> <i> r </i> 2 <i> }{ </i> 3 <i> }\right), </i>   <i> \left( </i> 6 <i> r, </i> </span></table><p style=padding-top:0.5em;float:left><span> <i> r </i> </span> <span> + </span><table cellpadding=0 cellspacing=0 class=math style=float:left><tbody><tr><td rowspan=2><td style="border-bottom:1px solid black"><span> 4 <i> \sqrt{ </i> 3 </span><tr><td><span> <i> r}{ </i> 3 <i> }, </i> </span></table><p style=padding-top:0.5em;float:left><span> <i> r </i> </span> <span> + </span><table cellpadding=0 cellspacing=0 class=math style=float:left><tbody><tr><td rowspan=2><td style="border-bottom:1px solid black"><span> <i> \sqrt{ </i> 6 </span><tr><td><span> <i> r </i> 2 <i> }{ </i> 3 <i> }\right), </i>   <i> \left( </i> 8 <i> r,r </i> </span></table><p style=padding-top:0.5em;float:left><span> + </span><table cellpadding=0 cellspacing=0 class=math style=float:left><tbody><tr><td rowspan=2><td style="border-bottom:1px solid black"><span> 4 <i> \sqrt{ </i> 3 </span><tr><td><span> <i> r}{ </i> 3 <i> }, </i> </span></table><p style=padding-top:0.5em;float:left><span> <i> r </i> </span> <span> + </span><table cellpadding=0 cellspacing=0 class=math style=float:left><tbody><tr><td rowspan=2><td style="border-bottom:1px solid black"><span> <i> \sqrt{ </i> 6 </span><tr><td><span> <i> r </i> 2 <i> }{ </i> 3 <i> }\right), </i> ... <i> . </i> </span></table></div><p style=clear:left>The difference to the next plane, the A plane, is again in the <i> z </i> -direction and a shift in the <i> x </i> and <i> y </i> to match those <i> x </i> - and <i> y </i> -coordinates of the first A plane.<p>In general, the coordinates of sphere centers can be written as:<div class=math><p style=padding-top:0.5em;float:left><span> <i> \begin{bmatrix} </i> </span> <span> 2 <i> i </i> </span> <span> + </span> <span> <i> ((j </i>  +  <i> k) </i> </span> <span> <i> \bmod </i> </span> <span> 2 <i> )\\ </i> </span> <span> <i> \sqrt{ </i> 3 <i> }\left[j </i> </span> <span> + </span><table cellpadding=0 cellspacing=0 class=math style=float:left><tbody><tr><td rowspan=2><td style="border-bottom:1px solid black"><span> 1 </span><tr><td><span> 3 </span></table><p style=padding-top:0.5em;float:left><span> <i> (k </i> </span> <span> <i> \bmod </i> </span> <span> 2 <i> )\right]\\ </i> </span><table cellpadding=0 cellspacing=0 class=math style=float:left><tbody><tr><td rowspan=2><td style="border-bottom:1px solid black"><span> 2 <i> \sqrt{ </i> 6 </span><tr><td></table></div> }k\endr <p style=clear:left>where <i> i </i> , <i> j </i> and <i> k </i> are indices starting at 0 for the <i> x </i> -, <i> y </i> - and <i> z </i> -coordinates.<h2>Miller indices</h2><p>See main article: <a href=../Miller_index/> Miller index </a> . Crystallographic features of HCP systems, such as vectors and atomic plane families, can be described using a four-value <a href=../Miller_index/> Miller index </a> notation ( <i> hkil </i> ) in which the third index <i> i </i> denotes a convenient but degenerate component which is equal to − <i> h </i>  −  <i> k </i> . The <i> h </i> , <i> i </i> and <i> k </i> index directions are separated by 120°, and are thus not orthogonal; the <i> l </i> component is mutually perpendicular to the <i> h </i> , <i> i </i> and <i> k </i> index directions.<h2>Filling the remaining space</h2><p>The FCC and HCP packings are the densest known packings of equal spheres with the highest symmetry (smallest repeat units).Denser <a title="Sphere packing is an arrangement of non-overlapping sphere s within a containing space."href=../sphere_packing/> sphere packing </a> s are known, but they involve unequal sphere packing.A packing density of 1, filling space completely, requires non-spherical shapes, such as <a title="Honeycomb is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps."href=../honeycomb_(geometry)/> honeycombs </a> .<p>Replacing each contact point between two spheres with an edge connecting the centers of the touching spheres produces tetrahedrons and octahedrons of equal edge lengths.The FCC arrangement produces the <a href=../tetrahedral-octahedral_honeycomb/> tetrahedral-octahedral honeycomb </a> .The HCP arrangement produces the gyrated tetrahedral-octahedral honeycomb.If, instead, every sphere is augmented with the points in space that are closer to it than to any other sphere, the duals of these honeycombs are produced: the <a href=../rhombic_dodecahedral_honeycomb/> rhombic dodecahedral honeycomb </a> for FCC, and the trapezo-rhombic dodecahedral honeycomb for HCP.<p>Spherical bubbles appear in soapy water in a FCC or HCP arrangement when the water in the gaps between the bubbles drains out. This pattern also approaches the <a href=../rhombic_dodecahedral_honeycomb/> rhombic dodecahedral honeycomb </a> or trapezo-rhombic dodecahedral honeycomb. However, such FCC or HCP foams of very small liquid content are unstable, as they do not satisfy <a href=../Plateau%27s_laws/> Plateau's laws </a> . The Kelvin foam and the <a href=../Weaire%e2%80%93Phelan_structure/> Weaire–Phelan foam </a> are more stable, having smaller interfacial energy in the limit of a very small liquid content. <sup> <a href=#Ref-8> [8] </a> </sup><h2>See also</h2><ul><li><a href=../Cubic_crystal_system/> Cubic crystal system </a><li><a href=../Hermite_constant/> Hermite constant </a><li><a title="Random close pack is an empirical parameter used to characterize the maximum volume fraction of solid objects obtained when ..."href=../Random_close_pack/> Random close pack </a><li><a title="Sphere packing is an arrangement of non-overlapping sphere s within a containing space."href=../Sphere_packing/> Sphere packing </a><li><a href=../Cylinder_sphere_packing/> Cylinder sphere packing </a></ul><h2>External links</h2><ul><li><a href=http://www.iucr.org/__data/assets/pdf_file/0015/13254/5.pdf rel=nofollow> P. Krishna & D. Pandey, "Close-Packed Structures" International Union of Crystallography by University College Cardiff Press. Cardiff, Wales. PDF </a></ul><h2>Notes and References</h2><ol><li><a name=Ref-1> </a> Hales . T. C. . Thomas Callister Hales . math/9811071v2 . An overview of the Kepler conjecture . 1998.<li><a name=Ref-2> </a> Mathematics: Does the proof stack up? . 424. 6944 . 10.1038/424012a . Nature . 12–13. 2003Natur.424...12S. Szpiro . George . 2003 . 12840727. free .<li><a name=Ref-3> </a> H. . Cohn . A. . Kumar . S. D. . Miller . D. . Radchenko . M. . Viazovska . The sphere packing problem in dimension 24 . Annals of Mathematics . 185 . 3 . 2017 . 1017–1033 . 10.4007/annals.2017.185.3.8 . 1603.06518. 119281758 .<li><a name=Ref-4> </a> Book: John Horton Conway . Conway . John Horton . Neil Sloane . Sloane . Neil James Alexander . Bannai . Eiichi . <a href=https://archive.org/details/spherepackingsla0000conw_b8u0 rel=nofollow> Sphere packings, lattices, and groups </a> . registration . Springer . 1999 . 9780387985855 . Section 6.3.<li><a name=Ref-5> </a> Encyclopedia: <a href=http://www.daviddarling.info/encyclopedia/C/Cannonball_Problem.html rel=nofollow> Cannonball Problem </a> . The Internet Encyclopedia of Science . David . Darling .<li><a name=Ref-6> </a> Barlow, William. <a href=https://zenodo.org/record/1429283 rel=nofollow> Probable Nature of the Internal Symmetry of Crystals </a> . Nature. 1883. 29. 738. 186–188. 10.1038/029186a0. 1883Natur..29..186B. free.<li><a name=Ref-7> </a> Web site: <a href=http://www.grunch.net/synergetics/sphpack.html rel=nofollow> on Sphere Packing </a> . Grunch.net . 2014-06-12.<li><a name=Ref-8> </a> Book: Foams, Structure and Dynamics. Isabelle. Cantat. Sylvie. Cohen-Addad. Florence. Elias. François. Graner. Reinhard. Höhler. Ruth. Flatman. Olivier. Pitois. Oxford University Press. Oxford. 2013. 9780199662890.</ol><div class=Footer><div class=robots-noindex><p>This article is licensed under the <a href=http://www.gnu.org/copyleft/fdl.html rel=nofollow> GNU Free Documentation License </a> . It uses material from the Wikipedia article " <a href=http://en.wikipedia.org/wiki/Close-packing_of_equal_spheres rel=nofollow> Close-packing of equal spheres </a> ".</div><p>Except where otherwise indicated, Everything.Explained.Today is © Copyright 2009-2021, A B Cryer, All Rights Reserved. <a href=http://explained.today/cookie_policy.htm> Cookie policy </a> .</div></div><script data-cfasync=false src=https://cdn.jsdelivr.net/npm/cookieconsent@3/build/cookieconsent.min.js></script><script>window.cookieconsent.initialise({  "palette": {    "popup": {      "background": "#000"    },    "button": {      "background": "#f1d600"    }  },  "content": {    "href": "explained.today/cookie_policy.htm"  }});</script>