Rob Reijerkerk

 

 

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Kepler’s snow.

 

Kepler’s renown first and foremost is based on the three laws that gave a scientific underpinning to the heliocentric world picture of Copernicus. But he also occupied himself with very small things: he was one of the first to wonder if the shape of crystals could be explained by the arrangement of the particles.

 

In the sixteenth and seventeenth century the custom to give presents at Christmas did not yet exist; presents were given at the start of the New Year instead. Of course it was a sensible thing not to skip your benefactor at this occasion. At the start of the year 1611 Johannes Kepler gladdened his friend and benefactor Wackher von Wackenfels with a small booklet, entitled Strena, seu de nive sexangula, in English: A Present, or Concerning the Six Cornered Snowflake.

 

Snow flake by Wilson Bentley, 1902

 

The booklet opens with a dedication in which Kepler with a masterly show of erudition puns on the different ways the word “nothing” can be used. What shall he give to his friend who is amused by nothing? After ample considerations he concludes that it should be a snowflake, for it is almost nothing.

 

Nam si Germano quaeras, nix quid sit, respondebit Nihil, siquidem Latine possit.

For if you ask a German the meaning of  “nix”, he will answer: Nihil, if he knows a little Latin.

 

Apparently Kepler supposes that the average German knows enough Latin to be able to tell that the Latin word for the German “nix” is “nihil”, but to little to know that Latin word “nix” means “snow” (Schnee). Apart from the jokes the dedication is also interesting because of the fact that Kepler (in addition to Plato and Socrates) also names Epicure. For ages Epicures theory of atoms had been denounced by the Church, but among Kepler’s acquaintances it still was so well known that he could make a passing reference to it. The word “atom” (atomus) even occurs a few times in De nive sexangula. Maybe equally meaningful is the fact that he does not name Aristotle in his introduction. Aristotle had gone out of fashion for quite some time already ­– which does not imply that the influence of Aristotle cannot be traced in the text.

 

Once the dedication is finished, joking is over. There is work to be done. The question Kepler wants to answer is this one: why do snowflakes always have a six-fold symmetry, and never a five-fold or a seven-fold? Snow comes from vapour, and vapour has no structure, so the symmetry must originate elsewhere. To answer his question Kepler examines examples of six-fold symmetry he has encountered in nature. First he takes a look at the bees. It is of course generally known that honeycombs have cells with six corners, and that they do fill the plane. But Kepler goes one step further: he takes a look at the bottom of the cells, and to his great enjoyment he observes that it consists of three rhombuses, which protrude a little to the other side. It looks like this, more or less (in three cells I left out the rhombuses).

 

 

fig. 1

 

If you look into the cells, then the black dot is in the position that is furthest away. But one can turn things around. When you look from the other side, from the side where the cells are closed, then the dots indicate the places where the closed sides protrude like the roof of a tent.

Between three dots there is a hexagon now that in the middle is lower, and the bottom of another cell would nicely fit into it. The bees have seen that too and they build a honeycomb out of two layers of cells that are positioned back to back.

 

fig. 2

 

The dots are now on the top of the tiny roofs and the middle of the coloured hexagon is the lowest point (figure 2, left). The two layers of cells that are back-to-back fit into each other with a small shift (figure 2, right).

 

This gave Kepler the idea that he could construct a polyhedron with rhombuses, in such a way that the polyhedron could fill the three-dimensional space, if repeated again and again. In his mind he equipped a cell of a honeycomb with a top that had exactly the same shape as the bottom. If he placed the top in the right position, and kept the right distance, he obtained a solid with twelve identical rhombuses as faces, the rhombic dodecahedron.

fig.3

 

This is how the rhombic dodecahedron looks. It is not easy to draw, nor is it easy to get a mental picture of the solid just by looking at a picture in perspective. One of the reasons is that our eye has the not easy to suppress inclination to interpret a rhombus as a square. I think it is easier to imagine the rhombic dodecahedron as a kind of zigzagging hexagonal cylinder that consists of six rhombuses, with two lids, each made out of three rhombuses, one on top and one under. One such lid is coloured in figure 3. When looking from above the rhombuses of the top lid have an orientation that is just opposite to the orientation of the rhombuses of the bottom. A planar net of the three parts should approximately look like this (figure 4, left).

 

 

fig. 4

 

The red lines in the top view (figure 4, right) indicate the orientation of the rhombuses of the bottom lid. Now it is easy to have good idea about the way these dodecahedrons can fill three-dimensional space: first make a layer of hexagons that fit a plane, as the hexagons in figure 2, and then the bottoms of the dodecahedrons of the next layer will fit into the small indentations that arise. The coloured part if figure 3 will fit into the colour of figure 2. This process can be repeated indefinitely.

 

Kepler took great pleasure in this discovery. He was very fond of regular polyhedrons. Following Plato he was convinced that behind each natural phenomenon a mathematical truth was hidden. In his Timaeus Plato had proposed that the smallest particles of the four elements fire, air, water and earth were formed as the four regular polygons tetrahedron, octahedron, icosahedron and cube, respectively. To Plato there remained one regular polyhedron: de dodecahedron, with regular pentagons as faces, which he attributed to the cosmos as a whole. Ever since the five regular polyhedrons are known as the “Platonic Solids”, although it almost sure that Plato did not discover them. Kepler wanted to do for the world of the very big things what Plato had done for the world of the very small. This led him to a result that at first glance is rather bizarre: his well-known and much reproduced model of the solar system. It was based on the five Platonic solids and their inscribed en circumscribed spheres (with the sun in the centre, as Copernicus had put forward).

 

fig. 5 from Keplers Mysterium Cosmographicum (1596).

 

Here the innermost part of the model is shown, with the sun surrounded by an octahedron, an icosahedron and a dodecahedron, with sphere shells in between. Around this inner part a tetrahedron and a cube are added (again with a sphere shell in between). The intervening sphere shells establish the distances of the planets to the sun and the circumscribed sphere of the cube gives the position of the fixed stars. In fact this is a mixture of the Platonic Solids with the spheres that Aristotle assigned to the planets. Kepler was convinced he could explain the ratios of the distances of the planets to the sun in this way. In hindsight this seems a rather silly attempt, of course, but in the end Kepler’s search led him to a beautiful result: in his Harmonices Mundi, published in 1619, he gives a simple relation between the mean distances of the planets and there periods. This mathematical relation however, was not based on geometrical speculation, but on astronomical observations – Brahe’s and his own. A few years later it would inspire Newton to formulate his famous gravitation law.

 

Back to the rhombic dodecahedron. This polyhedron is not one of the Platonic Solids, because the vertices are not always surrounded by the same number of faces. At some vertices (six out of fourteen) four faces meet and at other vertices only three. Nor does it belong to the Archimedean Solids that had Kepler’s special interest (Archimedean Solids are composed of two different kinds of regular convex polygons). And yet the rhombic dodecahedron has a certain charm: at first sight it is clear it has a high degree of symmetry, but one does not see immediately which symmetry exactly. At a closer look it turns out to have the same symmetry as the cube.

 

Kepler knew of an example of a rhombic dodecahedron that occurs in nature: the seeds (kernels) of a pomegranate. I did cut open a pomegranate, not so long ago, but to be honest, I did not notice that the kernels had the shape of rhombic dodecahedrons. But maybe the pomegranate was not old enough, for Kepler tells us that the kernels only get their dodecahedron shape when the rind of the pomegranate has hardened and the kernels, that keep on growing, do lack room. And now Kepler makes a brilliant move: he connects these considerations to a contemplation of the ways spheres can be stacked.

fig. 6

 

He starts with spheres on a level plain. Now there are two possible regular distributions; these are the distributions one can see in figure 6. It is clear that distribution A utilizes space less well than distribution B: the space between the spheres is much bigger in the square packing A than in the “triangular” packing B. First Kepler examines how to stack more layers, starting with distribution A. It is possible to have a second layer in such a way that every sphere of the second layer is exactly on top of a sphere of the first layer. If you go on like this every sphere will be surrounded by six other spheres: four in the same layer, one on top and one under. If the spheres in this packing would push against each other, like the growing kernels of the pomegranate, they would turn into cubes.

 

It is clear that this is not an economical way to fill space. In modern crystallography this way of stacking is known as simple cubic or primitive cubic. A rather straightforward calculation shows that in this way of stacking the spheres use only 52,4% of the available space; or as crystallographers put it: the packing efficiency is 52,4%.

 

But how is it when you put the spheres of the second layer not directly on top of the spheres of the first layer, but rather in the indentations? This should give a more economical way of packing.

 

fig. 7

 

It certainly does. When you keep on stacking in this way each sphere (with the exception of the spheres at the outside, of course) is in contact with twelve other spheres: four in the same layer, four under and four on top. Among crystallographers this way of packing is called cubic close packed and it is schematically pictured by the following unit cell; red dots give the positions of the centres of the spheres.

 

fig. 8

 

If you are aware of the fact that this unit cell is repeated again and again in all directions, then it is easy to see that in this packing each sphere is surrounded by twelve other spheres in the way described above. The base of the unit cell can be seen in picture A of figure 6 or figure 7; there it is rotated over an angle of 45 degrees. Because of the fact that the centres of the spheres are on the vertices of a cube, and also in the middle of the faces, this way of packing is also called face centered cubic.

 

Now we find a much higher packing efficiency. It can be calculated as follows. First is important to see that the spheres that constitute the body diagonal do touch each other. If the spheres have a radius r, then the length of the body diagonal is 4r and the unit cube has an edge 2 r √2. Now the volume of the unit cell is 16 r3 √2. The unit cell contains four spheres; this can be seen as follows. A sphere on a vertex belongs only for on eighth part to the unit cell we are considering, and because there are eight vertices, this gives us one sphere. The spheres in the middle of the faces count only half, but as there are six faces, this gives us another three spheres, bringing the total to four. Four spheres together have a volume of 4 x 4/3 π r3. The packing efficiency can now be calculated as 4 x 4/3 π r3 divided by 16 r3 √2. This equals π divided by 3 √2 and that amounts to about 74,05%. This is much better than the packing efficiency of the simple cubic packing. Kepler even assured it was the best result possible (he did not himself do the calculation).

 

Coaptatio fiet arctissima, ut nullo praeterea ordine plures globuli in idem vas compingi queant.

 

This packing is the closest possible, and with no other stacking more spheres can be brought together in the same room.

 

This assertion is known as the “Kepler Conjecture”. Kepler did not give a proof for this proposition – he did not even try. By intuition he knew it to be true and most of the scholars who were occupied with the question afterwards, agreed with him, but it was surprisingly difficult to give a proof for something so obvious.

 

The closest packing provided Kepler with an explanation for the shape of the pomegranate kernels. The kernels arrange themselves in such a way that they use the narrow room inside the rind as well as possible, and when they keep on growing, the originally sphere-shaped kernels are deformed to dodecahedrons by the pressure of their neighbours. The twelve faces of these regular polyhedrons correspond to the twelve neighbours of each kernel. Mathematicians would say that the rhombic dodecahedron is the Voronoi cell of the cubic close packed lattice – but in fact that is just another way to say the same thing. Thus Kepler found that the dodecahedron shape is a consequence of a material necessity (necessitas materialis) and does not arise from the nature of the tree (essentia animae in hac arbore).

 

And how about the bees? With the bees it is similar. The bees work next to each other with their round bodies and they try to utilize the plane as well as possible. Of all geometric figures that can fill the plane, the hexagon has the largest area and the rhombuses in the bottom of the cell give de possibility to save on wax by making two-sided honeycombs (see figure 2). Here also Kepler finds a material necessity.

 

 

Another possibility?

Next Kepler considers the different ways of packing that are possible when you start with a layer of spheres that are arranged in triangles as shown in picture B of figure 6 or figure 7. Each sphere in the first layer is now surrounded by sic others. This gives this layer a six-fold (hexagonal) symmetry.

 

Of course one can put the spheres of the second layer right on top of the spheres of the first layer; this gives a stacking that is called simple hexagonal. Each sphere is in contact with eight other spheres: six in the same layer, one on top and one under. As far as the efficient use of space is concerned this packing does better than simple cubic (as was to be expected), but less well than cubic close packed; the packing efficiency  for simple hexagonal is 60,5%.

 

Kepler has another possibility: start with a triangular layer and place the spheres of the second layer in the indentations of the first layer. To show what he means Kepler builds a three-sided pyramid. I suppose that the majority of his contemporaries immediately would have recognized the stacking of cannonballs. Have a look at the picture.

fig. 9

 

He starts with a triangle made out of 15 spheres (layer E). In the indentations of that layer 10 other spheres are placed, that form a triangle again (layer D) and the 6 and then 3 and than1. The numbers 1, 3, 6, 10, 15 … are also known as triangular numbers. The nth triangular number equals the sum of the first n natural numbers. It is not difficult to prove that the nth triangular number equals ½ n (n +1).

 

This stacking of cannonballs is closest packing again! It looks as if Kepler has found a new way to use space as efficiently as possible; but this is not the case. On closer inspection it appears that this way of stacking spheres is identical to the most efficient stacking we have seen before: cubic close packed with a  packing efficiency of 74,05%. Kepler saw this too.

 

Ita in solida coaptatione arctissima non potest ordo triangularis sine quadrangulari nec vicissim.

 

Thus in a packing that is as close as possible, there can be no triangular arrangement without a square arrangement, and vice versa.

 

This is remarkable anyway. Apparently it does not matter whether you build a triangular stack of cannonballs, like the one in figure 9, or whether you build a four-sided pyramid – the packing you get is the same in both cases, the orientation relative to the horizontal plane left aside. At first this seems very strange, because the number of cannonballs in a horizontal layer differs in the two cases: in the triangular stack the numbers are given by the triangular numbers, and in the square stack by the square numbers (there is a certain logic in this). But when you build a square pyramid with marbles, you see immediately that in the faces of the pyramid the triangular arrangement of figure 9E appears, with each marble in that layer surrounded by six others in the same layer. It is slightly more difficult to find a square arrangement in one of the layers of the three-side pyramid, like the arrangement in figure 6A or 7A, but if you take your time, you will see that such a layer does exist at an angle with the faces of the pyramid.

 

There is a numerical relationship between the number of marbles or cannonballs in a horizontal layer of one pyramid and the number in a horizontal layer of the other after all: if you take the number in the nth layer of the triangular pyramid (counted from the top), and you add to it the number in preceding layer, you find the number in the nth layer of the four-sided pyramid (also counted from the top, of course). To prove this is your homework for the next time.

 

 

Kepler was not aware of it, but another method for packing spheres as close as possible, which differs from cubic close packed, really does exist. Nowadays this is an elementary fact in crystallography. We start again with a triangular arrangement on a level plane, as in figure B, and in the indentations we place a second layer as we did before (red circles).

 

fig. 10

 

In the second layer there are indentations with a hole underneath (number 1 in the picture at the right) and indentations with a sphere under it (number 2). You cannot use both types of indentations at the same time, for if you choose for the indentations number 1, the indentations number 2 are blocked and vice versa. (Try this yourself with marbles or cannonballs or whatever is at hand). If you decide for the indentations number 1, you do essentially the same thing as when you build the triangular stacking of cannonballs, and you will have cubic close packing again. But the choice for the indentations number 2 gives a new possibility: the spheres of the third layer are right above the spheres of the first layer and you have a stacking that is called hexagonal close packed.

 

This is the unit cell for a hexagonal close packing lattice.

 

fig. 11

 

Beware: perspective may deceive your eye. The base of the unit cell is not a square, as you might think, but a rhombus with angles of 60 and 120 degrees. The edges of the rhombus are equal, of course (otherwise it would not be a rhombus), but the height of unit cell is different. Assuming that the unit cell is filled with spheres with radius r that are touching each other, the edges of the base will have a length of 2r and the height of the unit cell will be 4/3 r √6. This means that the height of the unit cell is about 1.63 times the length of an edge of the basis. In the unit cell one can clearly see the three layers, and it is evident that the spheres of the third layer are right above the spheres of the first layer. It may appear as if the spheres of the first and third layer are overrepresented, but this is not the case. The four spheres of the bottom layer are partly outside the unit cell, and together they count just for one, and the same applies to the spheres of the third layer, and so the three layers will have the same number of spheres.

 

It is clear that hexagonal close packing has the same packing efficiency as cubic close packed; the fact that the third layer has shifted up a little does not change the way it makes use of space. So hexagonal close packed also has a packing efficiency of 74,05%.

 

If the soft kernels in a hexagonal close packing keep on growing and push against each other, they will not become rhombic dodecahedrons, but a “twisted” or “squashed” version of the dodecahedron, which also is space-filling. This twisted dodecahedron does not only have rhombuses as faces – there are six isosceles trapeziums (trapezoids) as well.

 

fig. 12

 

It is hard again to have mental image of this solid just by looking at a picture in perspective. To make it clearer I took off the two lids that consist of rhombuses, until there remained a six-sided cylinder that is made out of trapeziums. The three parts of the twisted rhombic dodecahedron then give the following planar nets (fig. 13, left).

 

fig.13

 

The red lines in the top view on the right side of figure 13 show that the rhombuses of the bottom lid in this case have exactly the same orientation as the rhombuses of the top lid. In the non-twisted rhombic dodecahedron they are reversed (compare fig. 4, right side). The twisted dodecahedron is space filling essentially in the same way as the non-twisted is space filling. First fill a layer with hexagons (seen from above) and then the lids of the next layer will fit nicely into the indentations, and so on. The symmetry of the twisted dodecahedron is much easier to survey than the symmetry of the normal rhombic dodecahedron: there are no more four-fold axes of symmetry, there is only one three-fold axis left and three two-fold axes.

 

Close packing, both cubic and hexagonal, plays an important role in crystallography. Because the atoms of metals can be considered as spheres to a good approximation, many crystals of metals do show one of these ways of stacking. Silver and gold are cubic close packed (face centered cubic); zinc and cadmium are hexagonal close packed.

 

 

Snow flake by Wilson Bentley, 1902

 

 

The snowflake.

For the explanation of the shape of the snow crystals Kepler uses a Platonic Solid, in this case the octahedron. First he tells us that six-sides snowflakes are not planar at the moment of their origin, but that in the beginning they have a three-dimensional shape. He has a valid argument for this: snowflakes come into being in space, and not on a surface. He assumes that a snow crystal is formed by the action of cold on wet air, or by the withdrawal of heat to the centre of the crystal – which is exactly the same thing, according to Kepler. Here one can recognize the influence of the idea of Aristotle that heat and cold are the active principles that govern the changes in matter. In space this process can take place from all sides, and this means that a snow crystal, when it is formed, should have a tree-dimensional shape. In other words: symmetry considerations forced Kepler to conclude that snow crystals cannot be planar.

 

Kepler believed that a snow crystal, when it originated, had the shape of three feathered rods, which were joined in the middle at right angles. These three rods represent the three dimensions of space. Only when such a crystal touches the ground, its legs do sag and it becomes flat. The geometrical foundation of this construction is laid by the three long axes of the octahedron. The shape of the snow crystal originates from the fact that the freezing vapour forms small spheres that stack in a simple cubic packing. It is easy to see that such packing can result in the shape of an octahedron. The cube and the octahedron are cognate shapes (“they are dual polyhedrons,” is the way mathematicians express this nowadays – “they are like husband and wife,” says Kepler – see figure 14). Kepler adds: it is not surprising that the cube and the octahedron give rise to shapes in matter, for, like the other Platonic Solids, they do exist in the mind of the creative God (in Dei creatoris mente).

 

fig. 14

 

 

It seems as if Kepler the scientist gives way to Kepler the mystic. In fact he does not. For a real mystic one look inside the mind of God is sufficient to acquire certainty that has no connection with the real world, and that is beyond the realm of doubt. This is not the case here. First, Kepler tries to infer the geometrical shape of a snow crystal from the regular packing of the particles that constitute it. This is a revolutionary step that is far from any mystical considerations. Secondly, to Kepler observations are always decisive, also in this case. Observation even forces him to abandon his theory.

 

Dum enim ista scribo, rursum ninxit et confertius quam nuper.

 

While I am writing this, it did snow again, and even thicker then some time ago.

 

He has another close look and concludes that snow crystals with six beams are always flat, even when they are still whirling in mid air. There also are tiny granules, but these are almost round and do not have the beautiful dendritic projections of the flat crystals. So the octahedron cannot be the shape that brings about the six-sided star like crystals.

 

Then what does bring them about? Kepler cannot find a definitive answer. He assumes it has something to do with the fact that hexagons can fill the plane with no overlap. But there must be more to it, for squares and isosceles triangles do have the same property. It might be because you can make three-dimensional bodies out of squares and triangles, but not out of hexagons. Kepler surmises that there must be a formative principle in the matter itself formatrix facultas). But this is a question he leaves to the chemists (chymici).

 

The chemists found out, but it took them more than three centuries. The hexagons do play a role in this, and there is a formatrix facultas. There are even two kinds of formatrix facultas: the covalent bond and the hydrogen bond. The structure of ice can be described as layers of hexagons, which are connected. One such layer is shown in figure 15, left.

 

fig. 15

 

In this picture the centres of the oxygen atoms in water are indicated by red dots, and the hydrogen atoms by white dots. Between two oxygen atoms there is always one hydrogen atom, and each hydrogen atom is always bound to two oxygen atoms, but in different ways. At one side de hydrogen atom is connected to an oxygen atom by a covalent bond (short line) and at the other side by a hydrogen bond (longer line). Each oxygen atom is bound to two hydrogen atoms by covalent bonds; together this one oxygen atom and these two hydrogen atoms constitute a water molecule. It seems as if in the picture at the left side there are some oxygen atoms with only one hydrogen attached to it. This cannot be. Each oxygen atom has a fourth bond, either upwards or downwards and these bonds do link the net of oxygen atoms in ice together. So altogether each oxygen atom has four bonds: two covalent bonds and two hydrogen bonds with hydrogen atoms in neighbouring molecules. These four bonds are divided more or less evenly into space. This means that the hexagons in figure 15 (left) are not flat, but puckered. The picture at the right side of figure 15 shows how the layers of hexagons are linked together. To make things more surveyable I left out the hydrogen atoms in this picture.

 

It is apparent that the structure of ice has a hexagonal symmetry – but ice certainly does not have close packing. On the contrary, because of the large cage-like holes between the layers, ice has a relatively low density. Ice has a lower density than water, which is exceptional: normally the solid has a higher density than the fluid. This anomaly is the reason that ice cubes are floating in a glass of water. When the ice melts the hydrogen bonds in the ice network can be broken (the covalent bond of course can not) and the water molecules in the fluid can approach each other more closely.

 

It is the hexagonal symmetry of the ice network that determines the hexagonal symmetry of the snow crystals. In the centre of each crystal there is a small six-sided prism; if the conditions are favourable, there can be a fast crystal growth at the corners of this prism, for diffusion can supply water molecules much quicker there. When the circumstances in the atmosphere change quickly, many different shapes can arise that still maintain a six-fold symmetry.

 

 

The Kepler Conjecture.

In all those years that past since 1611 the mathematicians did not succeed in finding an analytic proof of Kepler’s conjecture that his way of packing spheres gives the best result possible. One reason for this is the fact that there is still a little room left when you surround one sphere with twelve other spheres (but not enough for a thirteenth sphere) – the situation is not entirely fixed. In theory there could be a non-regular packing that is just a little more efficient than cubic close packing – though no one really believes this. In the past century the mathematicians did find a gradually lower upper limit for the best possible packing efficiency. In 1958 Rogers did find that maximum attainable efficiency must be lower than 77,964% and in 1988 someone else proved that it could not be better than 77,836%. This almost as low as the 74,05% of cubic close packing, but this is still not a proof, of course.

 

In 1998 Tom Hales declared that he had proven the Kepler Conjecture. To do this he used a very extensive computer program that needs a few months of computer time to do all the calculations. Hales sent the proof of more than 250 pages to Annals of Mathematics and the referees, who have spent four years on it, are almost convinced that the proof is right, but they are not completely sure. It is impossible to check the elaborate computer proof “by hand”. Annals of Mathematics will only publish the theoretical part of the proof (or have they have already published it?) and the computer part will go to another journal.

 

Many mathematicians have mixed feelings about this whole story. In fact no one is really sure whether the Kepler Conjecture has been proven or not: it cannot be excluded that in the future someone find a flaw in the computer program. This is a situation mathematicians are not accustomed to; they are used to have old-fashioned proofs, with old-fashioned certainty. In the meantime Hales has started a project to check the computer proof step by step, using more sophisticated computer software. This is a project that will take many years (maybe twenty). It looks as if the mathematicians will have to wait a long time for old-fashioned certainty on the Kepler Conjecture.

 

 

N.B. 1. The first computer-aided proof was published many years ago. In 1976 Appel and Haken proved the four-colour conjecture using a computer that to nowadays standards was very simple. It took the computer 1500 hours to inspect all 1482 possible configurations.

 

N.B. 2. In De nive sexangula Kepler mentions the numbers that now are known as the Fibinacci numbers. He is aware of the fact that the ratio of successive Fibonacci numbers approaches the Golden Ratio.

 

N.B. 3. Kepler’s third law states that the square of the period of the orbit of a planet is proportional to the cube of the mean distance of the planet to the sun. The first law says that the orbits of the planets are ellipses with the sun at one focus and the second law states that the line connecting a planet with the sun sweeps equal areas in equal times.

 

 

Rob Reijerkerk

 

 

SOURCES.

 

The Latin text of De nive sexangula can be found at:

http://www.fh-augsburg.de/~harsch/Chronologia/Lspost17/Kepler/kep_stre.html

or at:

http://www.thelatinlibrary.com/kepler/strena.html

 

A photographical reproduction of the 1611 edition can be found at:

http://fotothek.slub-dresden.de/digisamm/buch000058.html

 

I found a German translation from the series “Ostwalds Klassiker der exakten Wissenschaften”:

Johannes Kepler Von sechseckigen Schnee, translation, introduction and notes by Dorothea Goetz, Geest und Portig, Leipzig 1987

 

An English translation (1966) by Colin Hardie  does exist, but I could not get hold of it.

 

On Kepler:

Arthur Koestler  The Sleepwalkers Penguin Arkana, Penguin Books, London 1989 (first published in 1959). Also on Copernicus and Brahe and Galileo, but most of all on Kepler.

 

Images of dodecahedons, which can be rotated in virtual space, can be found at:

http://mathworld.wolfram.com/HexagonalClosePacking.html

 

On this beautiful website something on the Kepler Conjecture can be found too:

http://mathworld.wolfram.com/SpherePacking.html

 

On Hales’ proof: see the article in the New York Times of april 6 2004, which can be found at:

http://www.math.binghamton.edu/zaslav/Nytimes/+Science/+Math/sphere-packing.20040406.html

 

A website on elementary crystallography:

http://www.chem.ox.ac.uk/icl/heyes/structure_of_solids/Lecture1/Lec1.html

 

And for snowflakes there is also of course:

http://www.its.caltech.edu/~atomic/snowcrystals/

 

Figures 6, 7, 9 en 10 are edited images from the original edition of De nive sexangula. Figure 5 was taken from Wikipedia (but it can be found at many places on the internet).

 

 

© Rob Reijerkerk

 

Dutch version finished 4-10-2005

English version finished 4-24-2005

 

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