The densest three-dimensional lattice sphere packing has each sphere touching 12 others, and this is almost certainly true for any arrangement of spheres (the Kepler conjecture). 


12 (twelve ⁱ/ˈtwɛlv/) is the natural number following 11 and preceding 13.
The word "twelve" is the largest number with a single-morpheme name in English. Etymology suggests that "twelve" (similar to "eleven") arises from the Germanic compound twalif "two-leftover", so a literal translation would yield "two remaining [after having ten taken]".^[1] This compound meaning may have been transparent to speakers of Old English, but the modern form "twelve" is quite opaque. Only the remaining tw- hints that twelve and two are related.
A group of twelve things is called a duodecad. The ordinal adjective is duodecimal, twelfth. The adjective referring to a group consisting of twelve things is duodecuple.
The number twelve is often used as a sales unit in trade, and is often referred to as a dozen. Twelve dozen are known as a gross. (Note that there are thirteen items in a baker's dozen.)
As shown below, the number twelve is frequently cited in the Abrahamic religions and is also central to the Western calendar and units of time.

In mathematics

Twelve is a composite number, the smallest number with exactly six divisors, its divisors being 1, 2, 3, 4, 6 and 12. Twelve is also a highly composite number, the next one being twenty four. Twelve is also a superior highly composite number, the next one being sixty. It is the first composite number of the form p²q; a square-prime, and also the first member of the (p²) family in this form. 12 has an aliquot sum of 16 (133% in abundance). Accordingly, 12 is the first abundant number (in fact a superabundant number) and demonstrates an 8 member aliquot sequence; {12,16,15,9,4,3,1,0} 12 is the 3rd composite number in the 3-aliquot tree. The only number which has 12 as its aliquot sum is the square 121. Only 2 other square primes are abundant (18 and 20).
Twelve is a sublime number, a number that has a perfect number of divisors, and the sum of its divisors is also a perfect number. Since there is a subset of 12's proper divisors that add up to 12 (all of them but with 4 excluded), 12 is a semiperfect number.
If an odd perfect number is of the form 12k + 1, it has at least twelve distinct prime factors.
Twelve is a superfactorial, being the product of the first three factorials. Twelve being the product of three and four, the first four positive integers show up in the equation 12 = 3 × 4, which can be continued with the equation 56 = 7 × 8.
Twelve is the ninth Perrin number, preceded in the sequence by 5, 7, 10, and also appears in the Padovan sequence, preceded by the terms 5, 7, 9 (it is the sum of the first two of these). It is the fourth Pell number, preceded in the sequence by 2 and 5 (it is the sum of the former plus twice the latter).
A twelve-sided polygon is a dodecagon. A twelve-faced polyhedron is a dodecahedron. Regular cubes and octahedrons both have 12 edges, while regular icosahedrons have 12 vertices. Twelve is a pentagonal number. The densest three-dimensional lattice sphere packing has each sphere touching 12 others, and this is almost certainly true for any arrangement of spheres (the Kepler conjecture). Twelve is also the kissing number in three dimensions.
Twelve is the smallest weight for which a cusp form exists. This cusp form is the discriminant Δ(q) whose Fourier coefficients are given by the Ramanujan τ-function and which is (up to a constant multiplier) the 24th power of the Dedekind eta function. This fact is related to a constellation of interesting appearances of the number twelve in mathematics ranging from the value of the Riemann zeta function function at −1 i.e. ζ(−1) = −1/12, the fact that the abelianization of SL(2,Z) has twelve elements, and even the properties of lattice polygons.
There are twelve Jacobian elliptic functions and twelve cubic distance-transitive graphs.
There are 12 Latin squares of size 3×3.
The duodecimal system (12₁₀ [twelve] = 10₁₂), which is the use of 12 as a division factor for many ancient and medieval weights and measures, including hours, probably originates from Mesopotamia.
In base thirteen and higher bases (such as hexadecimal), twelve is represented as C. In base 10, the number 12 is a Harshad number.

List of basic calculations

Multiplication	1	2	3	4	5	6	7	8	9	10		11	12	13	14	15	16	17	18	19	20		21	22	23	24	25		50	100	1000
	12	24	36	48	60	72	84	96	108	120		132	144	156	168	180	192	204	216	228	240		252	264	276	288	300		600	1200	12000

Division	1	2	3	4	5	6	7	8	9	10		11	12	13	14	15
	12	6	4	3	2.4	2		1.5		1.2			1			0.8
			0.25			0.5			0.75				1			1.25

Exponentiation	1	2	3	4	5	6	7	8	9	10		11	12	13
	12	144	1728	20736	248832	2985984	35831808	429981696	5159780352	61917364224		743008370688	8916100448256	106993205379072
	1	4096	531441	16777216	244140625	2176782336	13841287201	68719476736	282429536481	1000000000000		3138428376721	8916100448256	23298085122481

In science

• The atomic number of magnesium in the periodic table.
• The human body has twelve cranial nerves.
• The duodenum (from Latin duodecim, "twelve") is the first part of the small intestine, that is about twelve inches (30 cm) long. More precisely, this section of the intestine was measured not in inches but in fingerwidths. In fact, in German the name of the duodenum is Zwölffingerdarm and in Dutch the name is twaalfvingerige darm, both meaning "twelve-finger bowel".
• Force 12 on the Beaufort wind force scale corresponds to the maximum wind speed of a hurricane.

Astronomy

• Messier object M12, a magnitude 8.0 globular cluster in the constellation Ophiuchus
• The New General Catalogue object NGC 12, a magnitude 13.1 spiral galaxy in the constellation Pisces
• The Saros number of the solar eclipse series which began on 2680 BC BC August and ended on 1129 BC BC February. The duration of Saros series 12 was 1550.5 years, and it contained 87 solar eclipses.
• The Saros number of the lunar eclipse series which began on 2360 BC May and ended on 1062 BC July. The duration of Saros series 12 was 1298.1 years, and it contained 73 lunar eclipses.
• The 12th moon of Jupiter is Lysithea.
• The Western zodiac has twelve signs, as does the Chinese zodiac.

In religion and mythology

There are twelve "Jyotirlingas" in Hindu Shaivism. The Shaivites (orthodox devotees of God Shiva) treat them with great respect and they are visited by almost every pious Hindu at least once in a lifetime. The number 12 is very important in many religions, mainly Judaism, Christianity, and Islam, and also found in some older religions and belief systems.
In antiquity there are numerous magical/religious uses of twelves.^[2] Ancient Greek religion, the Twelve Olympians were the principal gods of the pantheon and Heracles enacted out twelve labours. The chief Norse god, Odin, had 12 sons. Several sets of twelve cities are identified in history as a dodecapolis, the most familiar being the Etruscan League. In the King Arthur Legend, Arthur is said to have subdued 12 rebel princes and to have won 12 great battles against Saxon invaders. [source: Benet's Reader's Encyclopedia, 3d ed]
The importance of 12 in Judaism and Christianity can be found in the Bible. The biblical Jacob had 12 sons, who were the progenitors of the Twelve Tribes of Israel, while the New Testament describes twelve apostles of Jesus; when Judas Iscariot was disgraced, a meeting was held (Acts) to add Matthias to complete the number twelve once more. (Today, The Church of Jesus Christ of Latter-day Saints has a Quorum of the Twelve Apostles.)
The Book of Revelation contains much numerical symbolism, and a lot of the numbers mentioned have 12 as a divisor. 12:1 mentions a woman—interpreted as the people of Israel, the Church or the Virgin Mary—wearing a crown of twelve stars (representing each of the twelve tribes of Israel). Furthermore, there are 12,000 people sealed from each of the twelve tribes of Israel, making a total of 144,000 (which is the square of 12 multiplied by a thousand).
In Orthodox Judaism, 12 signifies the age a girl matures (bat mitzvah)
There are 12 days of Christmas. The song Twelve Days of Christmas came from the traditional practice of extending Yuletide celebrations over the twelve days from Christmas day to the eve of Epiphany; the period of thirteen days including Epiphany is sometimes known as Christmastide. Thus Twelfth Night is another name for the twelfth day of Christmas or January 5 (the eve of Epiphany). Similarly, Eastern Orthodoxy observes 12 Great Feasts.
In Twelver Shi'a Islam, there are twelve Imams, legitimate successors of the prophet Muhammad. These twelve early leaders of Islam are—Ali, Hasan, Husayn, and nine of Husayn's descendants.
Imāmah (Arabic: إمامة) is the Shī‘ah doctrine of religious, spiritual and political leadership of the Ummah. The Shī‘ah believe that the A'immah ("Imams") are the true Caliphs or rightful successors of Muḥammad, and Twelver and Ismā‘īlī Shī‘ah further that Imams are possessed of supernatural knowledge, authority, and infallibility (‘Iṣmah) as well as being part of the Ahl al-Bayt, the family of Muhammad.[1] Both beliefs distinguish the Shī‘ah from Sunnis.
In Quran, the Sura number 12 is Sura Yusuf (Joseph), and it is located in Juz'a (Arabic : الجزء) number 12. This Sura narrates the story of Prophet Yusuf and his 12 brothers.
In Hinduism, the sun god Surya has 12 names. Also, there are 12 Petals in Anahata (Heart Chakra.)

In time

• Most calendar systems have twelve months in a year.
• The Chinese use a 12-year cycle for time-reckoning called Earthly Branches.
• There are twenty-four hours in a day in all, with twelve hours for a half a day. The hours are numbered from one to twelve for both the ante meridiem (a.m.) half of the day and the post meridiem (p.m.) half of the day. 12:00 after a.m. and before p.m. (in the middle of the day) is midday or noon, and 12:00 after p.m. and before a.m. (in the middle of the night) is midnight. A new day is considered to start with the stroke of midnight. The basic units of time (60 seconds, 60 minutes, 24 hours) can all perfectly divide by twelve.

Kepler conjecture
 As a graduate student in mathematics, I often found that lectures by visiting speakers exercised my eyelids more than my brain. I'd struggle to understand the subject for five minutes, fail, then struggle to stay awake for 55 minutes longer. But one talk was decidedly different. The speaker walked in, emptied his pockets of a large quantity of ball bearings, which rolled with a tremendous clatter all over the desk at the front of the room, and asked, "What's the best way to pack these things together?"
The speaker was Neil Sloane of Bell Laboratories (now AT&T Research), and his question—how to pack balls together in the densest possible way—was one of the oldest unsolved problems in mathematics. In 1611, the German physicist Johannes Kepler stated what he felt to be the obvious solution: You make a triangular array, then fit another layer into the interstices between the balls in the first layer, and so on. In this arrangement, called the face-centered cubic lattice, just over 74 percent of the volume of the space is taken up by balls, and 26 percent by the spaces between the balls. Kepler never even tried to prove that this was the densest packing. But later mathematicians questioned his assumption, now called the Kepler Conjecture. For all Sloane knew, his ball bearings might one day settle into a configuration with only 25 percent empty space.
Now, it appears, Sloane can put his ball bearings away. Tom Hales of the University of Michigan claims to have a proof that no sphere packing can be denser than the face-centered cubic lattice. Like the proof of the Four-Color Conjecture, another notorious problem that was solved in the 1970s, his argument relies heavily on computer calculations: roughly 100,000 of them, virtually all of them too lengthy to do by hand. Consequently, mathematicians are unlikely to pronounce the problem solved for at least several months.

One way to understand what makes the sphere-packing problem so hard is to contrast it with the problem of finding the densest lattice packing of spheres. Lattices, the most regular arrangement of spheres, can be constructed by choosing a single "crystal" shape, placing a sphere at each of the imaginary crystal's corners and repeating that motif ad infinitum. Because the positions of all the other crystals depend on the first one, the entire lattice is determined by the position of the first set of spheres. Finding the maximum density then becomes a calculus problem with a finite number of variables. "It's not exactly a trivial calculus problem," says John Conway of Princeton University. "But it's something that could be done in 1831," which is when Karl Friedrich Gauss solved it.
By contrast, in the full sphere-packing problem, the position of every sphere becomes a variable—and it takes infinitely many spheres to fill up all of abstract mathematical space. A problem with a finite number of variables can be solved by hand if the number is small enough, or by computer otherwise. No computer, though, can solve unaided a problem with infinitely many variables, because it would require the storage of an infinite amount of data.
The trick, then, is to reduce the problem somehow to a finite one—to divide space up into small regions or "cells" and analyze each one separately. If the face-centered cubic pattern could be proved superior or equal to any challenger in every single cell, then it would be best in all of space, too. Unfortunately, this is a hard way to defend a championship—analogous to requiring the champion to win or at least draw every round of a boxing match. And, indeed, early attempts to prove Kepler's Conjecture this way fell short.
In the 1950s, Hungarian mathematician Lászlo Fejes Tóth suggested defining a cell as the set of points that are closer to one particular sphere's center than any other. Each cell thus defined (called a Voronoi cell) contains one spherical "nucleus." In the face-centered cubic packing, every Voronoi cell is a rhombic dodecahedron, whose nucleus fills up 74 percent of the cell. But some less-regular packings can contain denser Voronoi cells—for example, regular dodecahedra, whose nuclei occupy 75 percent of the volume. Thus these packings would beat the face-centered cubic pattern in some regions.
In 1990, Wu-Yi Hsiang of the University of California at Berkeley attempted to show that the Voronoi cells surrounding any extra-dense cell would have to compensate by containing more empty space. (In our analogy, the champion would be allowed to lose a round if he could win the rounds before and after.) Although his argument aroused spirited discussion, most mathematicians who read it felt that it fell short of being a convincing proof of the conjecture. Fejes Tóth's son Gábor wrote in Mathematical Reviews, "The greater part of the work has yet to be done."
Hales kept the rule that the champion has to win every round, and changed the scoring system instead. He found a delicate "splicing" of the Voronoi cells with a second type of decomposition, called the Delaunay decomposition. The shapes of the spliced regions depend on the positions of no more than 50 spheres, so the problem is still finite and solvable by computer. Although the computation was automatic, Conway emphasizes that the outcome—that the face-centered cubic is densest in every one of Hales's regions—never was. "I found it very surprising that a scheme like this could work," Conway says.
Since Wolfgang Haken and Kenneth Appel's proof of the Four Color Theorem rocked the mathematical world in 1976, mathematicians have gotten a little more used to the idea of computer proofs. Still, checking a "mega-proof" like Hales's will be a serious challenge. Robert Connelly of Cornell University suggests that it may require something more like the replication of an experiment than a line-by-line scrutiny. In the long run, Conway believes that such computer proofs are "not part of the permanent furniture of mathematics." Someday, he predicts, mathematicians will find a computer-free proof of the Kepler Conjecture—although it might take another 400 years.


The Kepler conjecture, named after the 17th-century German mathematician and astronomer Johannes Kepler, is a mathematical conjecture about sphere packing in three-dimensional Euclidean space. It says that no arrangement of equally sized spheres filling space has a greater average density than that of the cubic close packing (face-centered cubic) and hexagonal close packing arrangements. The density of these arrangements is slightly greater than 74%.
In 1998 Thomas Hales, following an approach suggested by Fejes Tóth (1953), announced that he had a proof of the Kepler conjecture. Hales' proof is a proof by exhaustion involving the checking of many individual cases using complex computer calculations. Referees have said that they are "99% certain" of the correctness of Hales' proof, so the Kepler conjecture is now very close to being accepted as a theorem. In 2014, the Flyspeck project team, headed by Hales, announced the completion of a formal proof of the Kepler conjecture using a combination of the Isabelle and HOL Light proof assistants.

Background

Imagine filling a large container with small equal-sized spheres. The density of the arrangement is equal to the collective volume of the spheres divided by the volume of the container. To maximize the number of spheres in the container means to create an arrangement with the highest possible density, so that the spheres are packed together as closely as possible.
Experiment shows that dropping the spheres in randomly will achieve a density of around 65%. However, a higher density can be achieved by carefully arranging the spheres as follows. Start with a layer of spheres in a hexagonal lattice, then put the next layer of spheres in the lowest points you can find above the first layer, and so on. At each step there are two choices of where to put the next layer, so this natural method of stacking the spheres creates an uncountably infinite number of equally dense packings, the best known of which are called cubic close packing and hexagonal close packing. Each of these arrangements has an average density of

The Kepler conjecture says that this is the best that can be done—no other arrangement of spheres has a higher average density.

Origins

The conjecture was first stated by Johannes Kepler (1611) in his paper 'On the six-cornered snowflake'. He had started to study arrangements of spheres as a result of his correspondence with the English mathematician and astronomer Thomas Harriot in 1606. Harriot was a friend and assistant of Sir Walter Raleigh, who had set Harriot the problem of determining how best to stack cannonballs on the decks of his ships. Harriot published a study of various stacking patterns in 1591, and went on to develop an early version of atomic theory.

Nineteenth century

Kepler did not have a proof of the conjecture, and the next step was taken by Carl Friedrich Gauss (1831), who proved that the Kepler conjecture is true if the spheres have to be arranged in a regular lattice.
This meant that any packing arrangement that disproved the Kepler conjecture would have to be an irregular one. But eliminating all possible irregular arrangements is very difficult, and this is what made the Kepler conjecture so hard to prove. In fact, there are irregular arrangements that are denser than the cubic close packing arrangement over a small enough volume, but any attempt to extend these arrangements to fill a larger volume always reduces their density.^{[citation needed]}
After Gauss, no further progress was made towards proving the Kepler conjecture in the nineteenth century. In 1900 David Hilbert included it in his list of twenty three unsolved problems of mathematics—it forms part of Hilbert's eighteenth problem.

Twentieth century

The next step toward a solution was taken by Hungarian mathematician László Fejes Tóth. Fejes Tóth (1953) showed that the problem of determining the maximum density of all arrangements (regular and irregular) could be reduced to a finite (but very large) number of calculations. This meant that a proof by exhaustion was, in principle, possible. As Fejes Tóth realised, a fast enough computer could turn this theoretical result into a practical approach to the problem.
Meanwhile, attempts were made to find an upper bound for the maximum density of any possible arrangement of spheres. English mathematician Claude Ambrose Rogers (1958) established an upper bound value of about 78%, and subsequent efforts by other mathematicians reduced this value slightly, but this was still much larger than the cubic close packing density of about 74%.
Wu-Yi Hsiang (1993, 2001) claimed to prove the Kepler conjecture using geometric methods. However Gábor Fejes Tóth (the son of László Fejes Tóth) stated in his review of the paper "As far as details are concerned, my opinion is that many of the key statements have no acceptable proofs." Hales (1994) gave a detailed criticism of Hsiang's work, to which Hsiang (1995) responded. The current consensus is that Hsiang's proof is incomplete.^[1]

Hales' proof

Following the approach suggested by Fejes Tóth (1953), Thomas Hales, then at the University of Michigan, determined that the maximum density of all arrangements could be found by minimizing a function with 150 variables. In 1992, assisted by his graduate student Samuel Ferguson, he embarked on a research program to systematically apply linear programming methods to find a lower bound on the value of this function for each one of a set of over 5,000 different configurations of spheres. If a lower bound (for the function value) could be found for every one of these configurations that was greater than the value of the function for the cubic close packing arrangement, then the Kepler conjecture would be proved. To find lower bounds for all cases involved solving around 100,000 linear programming problems.
When presenting the progress of his project in 1996, Hales said that the end was in sight, but it might take "a year or two" to complete. In August 1998 Hales announced that the proof was complete. At that stage it consisted of 250 pages of notes and 3 gigabytes of computer programs, data and results.
Despite the unusual nature of the proof, the editors of the Annals of Mathematics agreed to publish it, provided it was accepted by a panel of twelve referees. In 2003, after four years of work, the head of the referee's panel Gábor Fejes Tóth (son of László Fejes Tóth) reported that the panel were "99% certain" of the correctness of the proof, but they could not certify the correctness of all of the computer calculations.
Hales (2005) published a 100-page paper describing the non-computer part of his proof in detail. Hales & Ferguson (2006) and several subsequent papers described the computational portions. Hales and Ferguson received the Fulkerson Prize for outstanding papers in the area of discrete mathematics for 2009.




In geometry, a kissing number is defined as the number of non-overlapping unit spheres that can be arranged such that they each touch another given unit sphere. For a lattice packing the kissing number is the same for every sphere, but for an arbitrary sphere packing the kissing number may vary from one sphere to another. Other names for kissing number that have been used are Newton number (after the originator of the problem), and contact number.
In general, the kissing number problem seeks the maximum possible kissing number for n-dimensional spheres in (n + 1)-dimensional Euclidean space. Ordinary spheres correspond to two-dimensional closed surfaces in three-dimensional space.
Finding the kissing number when centers of spheres are confined to a line (the one-dimensional case) or a plane (two-dimensional case) is trivial. Proving a solution to the three-dimensional case, despite being easy to conceptualise and model in the physical world, eluded mathematicians until the mid-20th century.^[1][2] Solutions in higher dimensions are considerably more challenging, and only a handful of cases have been solved exactly. For others investigations have determined upper and lower bounds, but not exact solutions.^[3]

Known greatest kissing numbers[edit]

In one dimension, the kissing number is 2:

In two dimensions, the kissing number is 6:

Proof: Consider a circle with center C that is touched by circles with centers C₁, C₂, .... Consider the rays C C_i. These rays all emanate from the same center C, so the sum of angles between adjacent rays is 360°.
Assume by contradiction that there are more than 6 touching circles. Then at least two adjacent rays, say C C₁ and C C₂, are separated by an angle of less than 60°. The segments C C_i have the same length - 2r - for all i. Therefore the triangle C C₁ C₂ is isosceles, and its third side - C₁ C₂ - has a side length of less than 2r. Therefore the circles 1 and 2 intersect - a contradiction.^[4]
In three dimensions, the kissing number is 12, but the correct value was much more difficult to establish than in dimensions one and two. It is easy to arrange 12 spheres so that each touches a central sphere, but there is a lot of space left over, and it is not obvious that there is no way to pack in a 13th sphere. (In fact, there is so much extra space that any two of the 12 outer spheres can exchange places through a continuous movement without any of the outer spheres losing contact with the center one.) This was the subject of a famous disagreement between mathematicians Isaac Newton and David Gregory. Newton correctly thought that the limit was 12; Gregory thought that a 13th could fit. Some incomplete proofs that Newton was correct were offered in the nineteenth century, but the first correct proof did not appear until 1953.^[1][2]
The twelve neighbors of the central sphere correspond to the maximum bulk coordination number of an atom in a crystal lattice in which all atoms have the same size (as in a chemical element). A coordination number of 12 is found in a cubic close-packed or a hexagonal close-packed structure.
In four dimensions, it was known for some time that the answer is either 24 or 25. It is easy to produce a packing of 24 spheres around a central sphere (one can place the spheres at the vertices of a suitably scaled 24-cell centered at the origin). As in the three-dimensional case, there is a lot of space left over—even more, in fact, than for n = 3—so the situation was even less clear. In 2003, Oleg Musin proved the kissing number for n = 4 to be 24, using a subtle trick.^[5][6]
The kissing number in n dimensions is unknown for n > 4, except for n = 8 (240), and n = 24 (196,560).^[7][8] The results in these dimensions stem from the existence of highly symmetrical lattices: the E₈ lattice and the Leech lattice.
If arrangements are restricted to regular arrangements, in which the centres of the spheres all lie on points in a lattice, then this restricted kissing number is known for n = 1 to 9 and n = 24 dimensions.^[9] For 5, 6 and 7 dimensions the arrangement with the highest known kissing number is the optimal lattice arrangement, but the existence of a non-lattice arrangement with a higher kissing number has not been excluded.

A formal proof

In January 2003, Hales announced the start of a collaborative project to produce a complete formal proof of the Kepler conjecture. The aim is to remove any remaining uncertainty about the validity of the proof by creating a formal proof that can be verified by automated proof checking software such as HOL and Isabelle. This project is called Flyspeck – the F, P and K standing for Formal Proof of Kepler. Hales estimated that producing a complete formal proof would take around 20 years of work. The project was announced completed on August 10, 2014.^[2]

Related problems

Thue's theorem

The regular hexagonal packing is the densest sphere packing in the plane. (1890)

The 2-dimensional analog of the Kepler conjecture; the proof is elementary. Henk and Ziegler attribute this result to Lagrange, in 1773 (see references, pag. 770).

The hexagonal honeycomb conjecture

The most efficient partition of the plane into equal areas is the regular hexagonal tiling. Hales' proof (1999).

Related to Thue's theorem.

The dodecahedron conjecture

The volume of the Voronoi polyhedron of a sphere in a packing of equal spheres is at least the volume of a regular dodecahedron with inradius 1. McLaughlin's proof, for which he received the 1999 Morgan Prize.

A related problem, whose proof uses similar techniques to Hales' proof of the Kepler conjecture. Conjecture by L. Fejes Tóth in the 1950s.

The Kelvin problem

What is the most efficient foam in 3 dimensions? This was conjectured to be solved by the Kelvin structure, and this was widely believed for over 100 years, until disproved by the discovery of the Weaire–Phelan structure. The surprising discovery of the Weaire–Phelan structure and disproof of the Kelvin conjecture is one reason for the caution in accepting Hales' proof of the Kepler conjecture.

Sphere packing in higher dimensions

The optimal sphere packing question in dimensions bigger than 3 is still open.

Ulam's packing conjecture

It is unknown whether there is a convex solid whose optimal packing density is lower than that of the sphere.

References