Mathematician Theorizes a Crystal As Beautiful As A Diamond

Roland Piquepaille writes "Why are diamonds so shiny and beautiful? A Japanese mathematician says it's because of their unique crystal structure and two key properties, called 'maximal symmetry' and 'strong isotropic property.' According to the American Mathematical Society (AMS), he found that out of all the crystals that are possible to construct mathematically, just one shares these two properties with the diamond. So far, his K4 crystal exists only as a mathematical object. And nobody knows if it exists — or if it can be synthesized."

Comments

4 points, in which any two vertices are connected

[Animats]

"4 points, in which any two vertices are connected by an edge." Isn't that a tetrahedron?

There are tetrahedral crystals. The last picture on that page is an unusually nice one.

The possible crystal forms for an element depend on the bond angles, and I don't think carbon will hold a stable tetrahedral lattice. Not sure, though.

Re:4 points, in which any two vertices are connect

[Anonymous Coward]

> I don't think carbon will hold a stable tetrahedral lattice.

Tetrahedral, good call. What do you get when you put carbon atoms into a tetrahedral lattice? Surprise: diamonds!

http://www.iit.edu/~felfkri/report_files/image005.jpg

This article doesn't even say what this new-fangled structure *is*...

Article is complete hogwash

[Anonymous Coward]

To get things out of the way: Yes, I am structural chemist, I did RTFA and I am not a native english speaker, so please bear with my broken english.

I don't want to comment so much on the mathematical part of the paper, which might be interesting, but on the chemistry, which is non-sensical.

First of all the style of the article is very un-scientific. Note how often he mentions how pretty this crystal structure is. This is completely subjective and I don't see how this structure is prettier than many others. There is many fascinating structures and I don't think this or the diamond lattice are the most fascinating ones.

Then the assumption that the prettyness of diamond is a direct result from the crystal structure is silly. Someone else noticed that Silicium (and also Germanium and Tin) have exactly the same crystal structure - and they are not "pretty".

He doesn't mention space group nor atomic positions, which are absolutely fundamental when talking about a crystal structure.

Now even if the crystal would form like he describes (with 1/3rd double bonds), there is just no way this would ever look anything like a diamond. The electronic structure is completely different - diamond is an insulator, a classic dielectric material, whereas this, due to its double bonds and it's extendef pi-electron system, would be a classical conductor. It would probably look like graphite.

But, and this is the worst point, which even someone who only did very basic (highschool?) chemistry should immediately note, the compound can never form in this way. That's the first thing you learn about double bonds: they're flat or nearly flat. Admittedly, in fullerene and carbon nano-tubes, there is a certain curvation (making them not as stable as graphite), but if you look at this crystal structure, the double bonds have a dihedral angle of about 90 degrees. It's totally impossible to obtain this compound and everybody with scientific education should know this. The molecular orbitals can't form this way.

All in all I have no idea how it comes that this non-scientific non-sensical article is published by the AMS. Maybe you could make something out of the math part, but all the babble about prettiness and chemistry has to go.

Re:Article is complete hogwash

[Anonymous Coward]

As a mathematician, let me reply to your comment. We rarely care whether or not what is discussed can exist or not. For example, our universe appears to completely finite, but we discuss orders of infinity constantly. Also, we are completely comfortable with non-scientific words like "pretty". A good example of this may be found at http://www.tcm.phy.cam.ac.uk/~ym101/tie/short/tie_nature4.html discussing the possible nice looking possible tie knots.

Re:Article is complete hogwash

[HalfFlat]

Pretty is being used here not to describe the visual attraction of diamond, but instead to characterise simple but interesting properties of the structure. Quoting Sunada,

The beauty would be more enhanced and its emotional appeal would be raised to a rational one if we would explore the microscopic structure, say the periodic arrangement of carbon atoms, which is actually responsible for the dazzling glaze caused by the effective refraction and reflection of light.

Similarly, the crystal structure being discussed is a mathematical abstraction that captures key aspects of physical crystalline structures, while not purporting to be a complete or even entirely faithful representation of crystals in the real world: for example, real-world crystals are obviously not infinite in extent.

The term pretty, when used in this sort of mathematical context, is not exclusive. Under a different set of criteria, other crystalline structures could well be regarded as being "the prettiest". The properties that Sunada has identified though, are elegant properties from a mathematical viewpoint: they relate the intrinsic symmetries of the structure as a graph with the extrinsic symmetries of the realisation of that graph in a three-dimensional configuration. That the standard realization of a crystal lattice corresponds to a minimal energy configuration (Theorem 1) also demonstrates links to analysis and is an introduction to methods of ab initio calculations of specific heat (see for example the paper of Shubin and Sunada cited in the article.) From considerations of abstract mathematical structure, the diamond crystal is indeed beautiful, and the K4 crystal similarly so.

That the structure may be chemically impossible to realise with carbon atoms is certainly a valid and useful observation, but to criticise the whole article on the basis of 90 words of chemical speculation really is to misunderstand the article's topic and goals.

Re:Article is complete hogwash

[Anonymous Coward]

All the "beatuty" comes from one optical property based on high crystal dentity: high "optical density" which translates into high optical refracrion index - 2.41. This is much higher than glass (about 1.5) and causes larger partial reflection on polished surfaces (shininess), more common total internal reflection within the boundaries, larger "image shift" (due to larger refraction) for objects observed through non-parallel surfaces of the diamond. That's all (not counting color flavors caused by impurities), forget about that mumbo jumbo in the article.
Although diamond has also other very interesting properties coming from the crystalographic structure (ie fabulous thermal conductivity) that has rather nothing to do with its visual perception...

Re:Article is complete hogwash

[HalfFlat]

I have to disagree, it really doesn't try very hard to explain the observed properties of diamond in terms of its elegant abstract structure.

There is a single throw away line in the introduction ascribing the refractive properties of diamond to its particular "periodic arrangement of carbon atoms" (which, essentially is true — other arrangements of carbon atoms certainly do not have the same optical properties.) And then the physical properties of diamond are never mentioned again! This is definitely not an article about the physical properties of crystals.

Yes, the summary is crap — but this is slashdot, after all.

Re:Article is complete hogwash

[Anonymous Coward]

(posting anonymously as I don't want to impact my trolling potential :)

As another mathematician, I have to agree with the OP. Beauty is not the issue. As a community, we often spice up our introductions with vague references to applications in other fields (or even other mathematical areas). This is mostly harmless since technical papers are rarely read by nonmathematicians (although occasionally by physicists), and the actual intended readership are interested in the mathematical parts, not the applications.

On the other hand, mentioning such an application is a way of justifying the research to granting bodies and asserting the relevance to one's broader research project.

But it's easy to get carried away with bogus explanations of phenomena which are typically outside of one's area of competence. There's no excuse for it in papers which are intended for a broad audience, and the OP is quite right in harshly criticizing the chemistry.

What about other single element crystals in A4 ?

[MrMr]

If the symmetry and isotropy give diamonds their shine, why are crystals of
for instance Si, Ge, Sn not as beautiful? They have the same isotropy and crystal structure.
And why is a low-symmetry sapphire prettier than high-symmetry table salt?
I would guess high index of refraction, and the lack of absorption of optical wavelengths are the more relevant properties.

(see any textbook on crystallography, or for instance http://cst-www.nrl.navy.mil/lattice/ )

Re:4 points, in which any two vertices are connect

[rxmd]

Diamond isn't stable, like glass, it just takes billions of years to lose it's shape, or something like that.

Or something.

What makes you think that glass isn't stable? (link, link)

I have an archeologist friend who works with Roman glass found along the Silk Road. Looks perfectly stable to me (well, at least those pieces that aren't smashed to bits).

Demand was inflated through marketing

[aepervius]

It is well known that before the ad-campaign of mid 20th diamonds were not that much in "demand". Heck, it is well known that before the oversupply of the 19th century diamond were relatively rare. Look at wiki for more detail.

quote The De Beers diamond advertising campaign is acknowledged as one of the most successful and innovative campaigns in history. N. W. Ayer & Son, the advertising firm retained by De Beers in the mid-20th century, succeeded in reviving the American diamond market and opened up new markets, even in countries where no diamond tradition had existed before. N.W. Ayer's multifaceted marketing campaign included product placement, advertising the diamond itself rather than the De Beers brand, and building associations with celebrities and royalty. This coordinated campaign has lasted decades and continues today; it is perhaps best captured by the slogan "a diamond is forever". End Quote


Source wiki

Despite being in over surplus from mid 19th to mid 20th, diamond were not that popular and high in demand.
in such context "And diamonds have always been in demand." the always is too much. If you change that to mid 20th century onward, you will be right.

Re:I'm sure...

[Palpitations]

there is also a company out there (sorry, I can't remember the name) that will take the cremated ashes of a relative and turn them into a diamond. LifeGem. They made news a while ago for creating diamonds from locks of Beethoven's hair.

That said, I've seen some very, very high quality diamonds (I forget the correct way to refer to it, but it was around 2 carats, no inclusions, and a D - completely colorless. Essentially, diamonds don't come any higher quality). Side by side with a nice piece of moissanite, I'd take the moissanite. A quick search on it will find better sources and images that show why, exactly, but I don't want to link to a commercial site and seem biased.

Re:I'm sure...

[kryliss]

If I remember correctly, the opal was the traditional wedding stone before all the "marketing" of the diamond.

index of refraction

[Wormholio]

While the beauty of gems is subjective, the one reason that diamonds are attractive is the high index of refraction of the material, which causes total internal reflection. Light from sources around the room can bounce around several times before it exits, giving the gem a "sparkle". The cut of the diamond can enhance this. The crystal structure determines which cuts are possible and which cuts give the best sparkle. But creating artificial gems with the same crystal structure will not give the same "sparkle" effect if the crystal does not have a high enough index of refraction to cause total internal reflection.

Re:Diamonds beauty

[jdschulteis]

Perhaps I have simply never seen a decent, beautifully cut diamond.

Have a look at this one.