785:
243:
162:
716:
815:
128:
800:
148:
155:
2052:
1690:
753:
20:
2314:
731:
275:
1701:
1354:
2047:{\displaystyle \left(2r,r+{\frac {4{\sqrt {3}}r}{3}},r+{\frac {{\sqrt {6}}r2}{3}}\right),\ \left(4r,r+{\frac {4{\sqrt {3}}r}{3}},r+{\frac {{\sqrt {6}}r2}{3}}\right),\ \left(6r,r+{\frac {4{\sqrt {3}}r}{3}},r+{\frac {{\sqrt {6}}r2}{3}}\right),\ \left(8r,r+{\frac {4{\sqrt {3}}r}{3}},r+{\frac {{\sqrt {6}}r2}{3}}\right),\dots .}
1685:{\displaystyle \left(r,r+{\frac {{\sqrt {3}}r}{3}},r+{\frac {{\sqrt {6}}r2}{3}}\right),\ \left(3r,r+{\frac {{\sqrt {3}}r}{3}},r+{\frac {{\sqrt {6}}r2}{3}}\right),\ \left(5r,r+{\frac {{\sqrt {3}}r}{3}},r+{\frac {{\sqrt {6}}r2}{3}}\right),\ \left(7r,r+{\frac {{\sqrt {3}}r}{3}},r+{\frac {{\sqrt {6}}r2}{3}}\right),\dots .}
1237:
2439:
left by hcp and fcc conformations; tetrahedral and octahedral void. Four spheres surround the tetrahedral hole with three spheres being in one layer and one sphere from the next layer. Six spheres surround an octahedral voids with three spheres coming from one layer and three spheres coming from the
837:
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
2443:
Layered structures are formed by alternating empty and filled octahedral planes. Two octahedral layers usually allow for four structural arrangements that can either be filled by an hpc of fcc packing systems. In filling tetrahedral holes a complete filling leads to fcc field array. In unit cells,
745:
is shown in red. The letters indicate which layers are the same. There are two "A" layers in the HCP matrix, where all the spheres are in the same position. All three layers in the FCC stack are different. Note the FCC stacking may be converted to the HCP stacking by translation of the upper-most
1302:
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
270:
on their expedition to
America. 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 â with different orientation to the ground. Hexagonal close-packing would result in a
118:
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
810:. The difference between this stack and the top three tiers of the cannonball stack all occurs in the bottom tier, which is rotated half the pitch diameter of a sphere (60°). Note how the two balls facing the viewer in the second tier from the top do not contact the same ball in the tier below.
2270:
590:
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.
883:
886:
An animation of close-packing lattice generation. Note: If a third layer (not shown) is directly over the first layer, then the HCP lattice is built. If the third layer is placed over holes in the first layer, then the FCC lattice is
1037:
1009:-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
693:
221:. 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
891:
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
2440:
next layer. Structures of many simple chemical compounds, for instance, are often described in terms of small atoms occupying tetrahedral or octahedral holes in closed-packed systems that are formed from larger atoms.
771:
which has an FCC lattice. Note how the two balls facing the viewer in the second tier from the top contact the same ball in the tier below. This does not occur in an HCP lattice (the left organization in
2392:
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
93:
432:
605:
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
2111:
791: Shown here is a modified form of the cannonball stack wherein three extra spheres have been added to show all eight spheres in the top three tiers of the FCC lattice diagramed in
356:
715:
616:
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
511:
473:
2401:. 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
2412:
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
1232:{\displaystyle \left(r,r+{\sqrt {3}}r,r\right),\ \left(3r,r+{\sqrt {3}}r,r\right),\ \left(5r,r+{\sqrt {3}}r,r\right),\ \left(7r,r+{\sqrt {3}}r,r\right),\dots .}
551:
531:
103:
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
626:
565:) 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
107:
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
2810:
P. Krishna & D. Pandey, "Close-Packed
Structures" International Union of Crystallography by University College Cardiff Press. Cardiff, Wales. PDF
2890:
2836:
2886:
561:
In both the FCC and HCP arrangements each sphere has twelve neighbors. For every sphere there is one gap surrounded by six spheres (
2398:
2377:
The FCC and HCP packings are the densest known packings of equal spheres with the highest symmetry (smallest repeat units). Denser
1242:
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.
46:
proved that the highest average density â that is, the greatest fraction of space occupied by spheres â that can be achieved by a
56:
2417:
2406:
2321:
Crystallographic features of HCP systems, such as vectors and atomic plane families, can be described using a four-value
2566:
Cohn, H.; Kumar, A.; Miller, S. D.; Radchenko, D.; Viazovska, M. (2017). "The sphere packing problem in dimension 24".
2265:{\displaystyle {\begin{bmatrix}2i+((j\ +\ k){\bmod {2}})\\{\sqrt {3}}\left\\{\frac {2{\sqrt {6}}}{3}}k\end{bmatrix}}r}
361:
2790:
2765:
2631:
1013:
two spheres, their centers form an equilateral triangle with those two neighbors' centers. The side lengths are all 2
2756:; Cohen-Addad, Sylvie; Elias, Florence; Graner, François; Höhler, Reinhard; Flatman, Ruth; Pitois, Olivier (2013).
2394:
926:-coordinate and z-coordinate will be the same. For simplicity, say that the balls are the first row and that their
1295:-coordinates. Both types of planes are formed using the pattern mentioned above, but the starting place for the
300:
3023:
2829:
2413:
2402:
938:, so that their surfaces rest on the zero-planes. Coordinates of the centers of the first row will look like (2
882:
2950:
2429:
893:
282:
arranged in pyramid shape. The front pyramid is hexagonal close-packed and rear is face-centered cubic.
3044:
2940:
2894:
2882:
606:
2648:
2382:
3054:
2930:
2822:
181:
arrangement can be seen in the triangular orientation, but alternates two positions of spheres, in a
177:
with 12 vertices representing the positions of 12 neighboring spheres around one central sphere. The
702:
is the diameter of a sphere; this follows from the tetrahedral arrangement of close-packed spheres.
173:
arrangement can be oriented in two different planes, square or triangular. These can be seen in the
3049:
2945:
2420:. However, such FCC or HCP foams of very small liquid content are unstable, as they do not satisfy
208:
182:
2473:
191:
There are two simple regular lattices that achieve this highest average density. They are called
2432:
are more stable, having smaller interfacial energy in the limit of a very small liquid content.
2904:
2493:
478:
108:
2809:
437:
23:
Illustration of the close-packing of equal spheres in both HCP (left) and FCC (right) lattices
2935:
710:
2620:
2732:
2679:
2530:
2453:
2386:
2385:. A packing density of 1, filling space completely, requires non-spherical shapes, such as
989:
Now, form the next row of spheres. Again, the centers will all lie on a straight line with
814:
295:
201:
43:
8:
2997:
2859:
2444:
hole filling can sometimes lead to polyhedral arrays with a mix of hcp and fcc layering.
706:
290:
asks which flat square arrangements of cannonballs can be stacked into a square pyramid.
255:
2683:
2534:
1311:
because all of the sides are formed by two spheres touching. The height of which or the
866:
is the radius of the second. In close packing all of the spheres share a common radius,
2611:
2593:
2575:
2497:
688:{\displaystyle {\text{pitch}}_{Z}={\sqrt {6}}\cdot {d \over 3}\approx 0.816\,496\,58d,}
536:
516:
287:
237:
784:
3002:
2899:
2786:
2761:
2729:
2627:
2597:
2548:
2463:
2436:
2421:
1279:
of spheres will have exactly the same coordinates save for a pitch difference in the
741:â The HCP lattice (left) and the FCC lattice (right). The outline of each respective
266:
around 1587, after a question on piling cannonballs on ships was posed to him by Sir
242:
161:
104:
38:
is a dense arrangement of congruent spheres in an infinite, regular arrangement (or
3059:
2845:
2687:
2585:
2538:
2458:
799:
291:
2753:
2707:
742:
100:
47:
39:
2589:
910:
First form a row of spheres. The centers will all lie on a straight line. Their
2981:
2960:
2922:
2909:
2874:
2468:
2378:
1695:
The second row's coordinates follow the pattern first described above and are:
760:
267:
263:
3038:
3018:
2965:
262:
The problem of close-packing of spheres was first mathematically analyzed by
174:
127:
2781:
Woodward, Patrick M.; Karen, Pavel; Evans, John S. O.; Vogt, Thomas (2021).
2552:
2322:
2308:
806: Shown here are all eleven spheres of the HCP lattice illustrated in
553:
is the number of cannonballs along an edge in the flat square arrangement.
1005:-direction so that the center of every sphere in this row aligns with the
147:
16:
Dense arrangement of congruent spheres in an infinite, regular arrangement
2864:
2615:
2425:
2353:
index directions are separated by 120°, and are thus not orthogonal; the
2313:
1304:
822:
225:
154:
2502:
918:
since the distance between each center of the spheres are touching is 2
752:
562:
821: This animated view helps illustrate the three-sided pyramidal (
2737:
2692:
2667:
2543:
2518:
19:
2814:
2580:
279:
218:
111:. Highest density is known only for 1, 2, 3, 8, and 24 dimensions.
28:
533:
is the number of layers in the pyramidal stacking arrangement and
2333:
denotes a degenerate but convenient component which is equal to â
2105:
In general, the coordinates of sphere centers can be written as:
612:
In close-packing, the center-to-center spacing of spheres in the
115:
2727:
1348:-coordinates gives the centers of the first row in the B plane:
730:
274:
34:
2208:
2157:
2752:
88:{\displaystyle {\frac {\pi }{3{\sqrt {2}}}}\approx 0.74048}
2565:
2057:
The difference to the next plane, the A plane, is again
870:. Therefore, two centers would simply have a distance 2
2780:
2668:"Probable Nature of the Internal Symmetry of Crystals"
2120:
2114:
1704:
1357:
1275:
In an A-B-A-B-... stacking pattern, the odd numbered
1040:
713:(APFs) are equal to the number mentioned above, 0.74.
629:
601:
HCP = AB AB AB AB... (every other layer is the same).
539:
519:
481:
440:
364:
303:
59:
598:
FCC = ABC ABC ABC... (every third layer is the same)
1315:-coordinate difference between the two "planes" is
2619:
2610:
2264:
2046:
1684:
1245:The next row follows this pattern of shifting the
1231:
1031:. Thus, this row will have coordinates like this:
763:in ca. 1585 first pondered the mathematics of the
687:
545:
525:
505:
467:
426:
350:
87:
587:for the octahedral, when the sphere radius is 1.
3036:
2496:(1998). "An overview of the Kepler conjecture".
427:{\displaystyle {\frac {1}{6}}N(N+1)(2N+1)=M^{2}}
2665:
2372:
720:
2830:
131:FCC arrangement seen on 4-fold axis direction
434:and conjectured that the only solutions are
2357:component is mutually perpendicular to the
1021:-coordinate difference between the rows is
2837:
2823:
556:
351:{\displaystyle \sum _{n=1}^{N}n^{2}=M^{2}}
271:six-sided pyramid with a hexagonal base.
134:
2785:. Cambridge: Cambridge University Press.
2691:
2579:
2542:
2501:
1340:. This, combined with the offsets in the
675:
671:
2399:gyrated tetrahedral-octahedral honeycomb
2312:
997:, but there will be a shift of distance
881:
813:
798:
783:
751:
746:sphere, as shown by the dashed outline.
273:
241:
126:
18:
2519:"Mathematics: Does the proof stack up?"
122:
3037:
2516:
2418:trapezo-rhombic dodecahedral honeycomb
2407:trapezo-rhombic dodecahedral honeycomb
1299:row's first sphere will be different.
859:is the radius of the first sphere and
825:) shape of the cannonball arrangement.
2818:
2728:
2622:Sphere packings, lattices, and groups
2492:
877:
832:
231:
2844:
2653:The Internet Encyclopedia of Science
2317:MillerâBravais index for HCP lattice
2760:. Oxford: Oxford University Press.
2646:
2397:. The HCP arrangement produces the
2102:-coordinates of the first A plane.
1283:-coordinates and the even numbered
13:
2287:are indices starting at 0 for the
246:Cannonballs piled on a triangular
14:
3071:
2803:
2302:
1307:. All of the sides are equal to 2
2395:tetrahedral-octahedral honeycomb
729:
714:
160:
153:
146:
2783:Solid State Materials Chemistry
2774:
1287:of spheres will share the same
724:Comparison between HCP and FCC
709:of HCP and FCC is 12 and their
2966:Sphere-packing (Hamming) bound
2746:
2721:
2700:
2659:
2640:
2604:
2559:
2510:
2486:
2414:rhombic dodecahedral honeycomb
2403:rhombic dodecahedral honeycomb
2217:
2201:
2166:
2153:
2135:
2132:
2086:-direction and a shift in the
1264:. Add rows until reaching the
408:
393:
390:
378:
294:formulated the problem as the
1:
2758:Foams, Structure and Dynamics
2616:Sloane, Neil James Alexander
2381:are known, but they involve
1272:maximum borders of the box.
993:-coordinate differences of 2
7:
2590:10.4007/annals.2017.185.3.8
2447:
2373:Filling the remaining space
2329:) in which the third index
10:
3076:
2306:
914:-coordinate will vary by 2
594:The most regular ones are
235:
3011:
2990:
2974:
2921:
2873:
2852:
2733:"Hexagonal Close Packing"
2626:. Springer. Section 6.3.
2618:; Bannai, Eiichi (1999).
580:for the tetrahedral, and
506:{\displaystyle N=24,M=70}
168:
137:
2891:isosceles right triangle
2666:Barlow, William (1883).
2479:
934:-coordinates are simply
468:{\displaystyle N=1,M=1,}
183:triangular orthobicupola
2517:Szpiro, George (2003).
2474:Cylinder sphere packing
2435:There are two types of
838:line will therefore be
557:Positioning and spacing
33:close-packing of equal
2905:Circle packing theorem
2383:unequal sphere packing
2318:
2266:
2048:
1686:
1233:
888:
826:
811:
796:
781:
765:cannonball arrangement
711:atomic packing factors
689:
547:
527:
507:
469:
428:
352:
324:
283:
259:
132:
89:
24:
2568:Annals of Mathematics
2316:
2267:
2049:
1687:
1234:
885:
817:
802:
787:
755:
690:
620:(vertical) axis, is:
548:
528:
508:
470:
429:
353:
304:
277:
245:
130:
90:
22:
2887:equilateral triangle
2649:"Cannonball Problem"
2454:Cubic crystal system
2112:
1702:
1355:
1038:
627:
537:
517:
479:
438:
362:
301:
296:Diophantine equation
123:FCC and HCP lattices
57:
44:Carl Friedrich Gauss
3024:SlothouberâGraatsma
2708:"on Sphere Packing"
2684:1883Natur..29..186B
2612:Conway, John Horton
2535:2003Natur.424...12S
1305:regular tetrahedron
1017:, so the height or
986:), ... .
707:coordination number
193:face-centered cubic
2730:Weisstein, Eric W.
2437:interstitial holes
2430:WeaireâPhelan foam
2369:index directions.
2319:
2262:
2253:
2044:
1682:
1229:
889:
878:Simple HCP lattice
833:Lattice generation
827:
812:
797:
782:
685:
543:
523:
503:
465:
424:
348:
288:cannonball problem
284:
260:
238:Cannonball problem
232:Cannonball problem
217:), based on their
133:
119:first principles.
85:
25:
3045:Discrete geometry
3032:
3031:
2991:Other 3-D packing
2975:Other 2-D packing
2900:Apollonian gasket
2464:Random close pack
2405:for FCC, and the
2246:
2240:
2199:
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769:cannonball stack,
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663:
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546:{\displaystyle M}
526:{\displaystyle N}
373:
189:
188:
105:Kepler conjecture
77:
74:
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3055:Packing problems
2913:
2853:Abstract packing
2846:Packing problems
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2754:Cantat, Isabelle
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2693:10.1038/029186a0
2678:(738): 186â188.
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2657:
2656:
2647:Darling, David.
2644:
2638:
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2601:
2583:
2574:(3): 1017â1033.
2563:
2557:
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2514:
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2507:
2505:
2490:
2459:Hermite constant
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905:coordinate space
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3050:Crystallography
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2910:Tammes problem
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2544:10.1038/424012a
2529:(6944): 12â13.
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2379:sphere packings
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2184:
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2116:
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2113:
2110:
2109:
2094:to match those
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2064:
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2011:
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2008:
1983:
1979:
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1961:
1957:
1927:
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1873:
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1303:centers form a
1260:
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1257:-coordinate by
1249:-coordinate by
1199:
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1024:
1022:
880:
865:
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851:
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743:Bravais lattice
655:
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278:Collections of
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101:packing density
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3009:
3008:
3006:
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2988:
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2984:
2982:Square packing
2978:
2976:
2972:
2971:
2969:
2968:
2963:
2961:Kissing number
2958:
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2925:
2923:Sphere packing
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2897:
2879:
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2875:Circle packing
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2307:Main article:
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264:Thomas Harriot
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2951:In a cylinder
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808:Figure 1
805:
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793:Figure 1
790:
786:
779:
778:Figure 4
775:
774:Figure 1
770:
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758:
757:Figure 2
754:
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292:Ădouard Lucas
289:
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253:
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229:
227:
220:
216:
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205:
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185:arrangement.
184:
180:
176:
175:cuboctahedron
172:
167:
163:
159:
156:
152:
149:
145:
144:
140:
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53:
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41:
37:
36:
30:
21:
2955:
2893: /
2889: /
2885: /
2782:
2776:
2757:
2748:
2736:
2723:
2712:. Retrieved
2710:. Grunch.net
2702:
2675:
2671:
2661:
2652:
2642:
2621:
2606:
2571:
2567:
2561:
2526:
2522:
2512:
2494:Hales, T. C.
2488:
2442:
2434:
2411:
2391:
2376:
2366:
2362:
2358:
2354:
2350:
2346:
2342:
2338:
2334:
2330:
2326:
2323:Miller index
2320:
2309:Miller index
2296:
2292:
2288:
2284:
2280:
2276:
2274:
2104:
2099:
2095:
2091:
2087:
2083:
2068:
2056:
1694:
1345:
1341:
1326:
1312:
1308:
1301:
1296:
1292:
1288:
1284:
1280:
1276:
1274:
1269:
1265:
1254:
1250:
1246:
1244:
1241:
1028:
1018:
1014:
1010:
1006:
1002:
998:
994:
990:
988:
983:
979:
975:
971:
967:
963:
959:
955:
951:
947:
943:
939:
935:
931:
927:
923:
919:
915:
911:
909:
902:
898:
894:
890:
871:
867:
860:
853:
846:
839:
836:
828:
818:
807:
803:
792:
788:
777:
773:
768:
764:
756:
738:
704:
699:
697:
617:
613:
611:
604:
593:
589:
560:
285:
261:
251:
247:
214:
211:close-packed
207:
204:close packed
200:
196:
192:
190:
178:
170:
113:
98:
32:
26:
2998:Tetrahedron
2941:In a sphere
2912:(on sphere)
2883:In a circle
2426:Kelvin foam
2325:notation (
823:tetrahedral
776:above, and
254:base, both
226:root system
109:T. C. Hales
50:packing is
3039:Categories
2931:Apollonian
2714:2014-06-12
2581:1603.06518
2387:honeycombs
563:octahedral
3003:Ellipsoid
2946:In a cube
2738:MathWorld
2598:119281758
2409:for HCP.
2039:…
1677:…
1224:…
666:≈
653:⋅
306:∑
280:snowballs
258:lattices.
209:hexagonal
99:The same
80:≈
63:π
2553:12840727
2448:See also
2428:and the
1253:and the
887:created.
819:Figure 5
804:Figure 4
789:Figure 3
739:Figure 1
219:symmetry
29:geometry
3060:Spheres
3012:Puzzles
2680:Bibcode
2531:Bibcode
2082:in the
2080:
2063:√
2059:
1338:
1321:√
1317:
1259:√
1023:√
1001:in the
982:,
978: ,
970:,
966: ,
958:,
954:,
946:,
942:,
780:below).
759:
582:√
573:⁄
567:√
513:. Here
248:(front)
116:crystal
83:0.74048
48:lattice
40:lattice
35:spheres
3019:Conway
2936:Finite
2895:square
2789:
2764:
2672:Nature
2630:
2596:
2551:
2523:Nature
2424:. The
2341:. The
2295:- and
2275:where
2148:
2142:
2098:- and
1955:
1871:
1787:
1596:
1515:
1434:
1291:- and
1285:planes
1277:planes
1177:
1130:
1083:
930:- and
922:. The
852:where
698:where
252:(back)
206:) and
2594:S2CID
2576:arXiv
2498:arXiv
2480:Notes
1297:first
1011:touch
974:), (8
962:), (6
950:), (4
669:0.816
633:pitch
202:cubic
114:Many
2787:ISBN
2762:ISBN
2628:ISBN
2549:PMID
2365:and
2349:and
2327:hkil
2283:and
2090:and
1344:and
1268:and
705:The
475:and
286:The
169:The
141:HCP
138:FCC
2865:Set
2860:Bin
2688:doi
2586:doi
2572:185
2539:doi
2527:424
2416:or
2291:-,
2209:mod
2158:mod
767:or
673:496
358:or
256:FCC
215:HCP
197:FCC
179:HCP
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