Featured Research

from universities, journals, and other organizations

One Crystal That Nature May Have Missed

Date:
January 8, 2008
Source:
American Mathematical Society
Summary:
Some secrets of the beauty of a diamond can be uncovered by a mathematical analysis of its microscopic crystal structure. This structure has some very special, and especially symmetric, properties. Out of an infinite universe of mathematical crystals, only one other, the "K4 crystal", shares these properties with the diamond. It is not known whether the K4 crystal exists in nature or could be synthesized.

K4 crystal.
Credit: Image created by Hisashi Naito

For centuries, human beings have been entranced by the captivating glimmer of the diamond. What accounts for the stunning beauty of this most precious gem? As mathematician Toshikazu Sunada explains in an article appearing in the Notices of the American Mathematical Society, some secrets of the diamond's beauty can be uncovered by a mathematical analysis of its microscopic crystal structure. It turns out that this structure has some very special, and especially symmetric, properties. In fact, as Sunada discovered, out of an infinite universe of mathematical crystals, only one other shares these properties with the diamond, a crystal that he calls the "K4 crystal". It is not known whether the K4 crystal exists in nature or could be synthesized.

One can create an idealized mathematical model of a crystal by focusing on its main features, namely, the atoms and the bonds between them. The atoms are represented by points, which we will call "vertices", and the bonds are represented as lines, which we will call "edges". This kind of network of vertices and edges is called a "graph". A crystal is built up by starting with a building-block graph and joining together copies of itself in a periodic fashion. Thus there are two patterns operating in a crystal: The pattern of edges connecting vertices in the building-block graphs (that is, the pattern of bonding relations between the atoms), and the periodic pattern joining the copies of the graphs. One can create infinitely many mathematical crystals this way, by varying the graphs and by varying the way they are joined periodically.

The diamond crystal has two key properties that distinguish it from other crystals. The first, called "maximal symmetry", concerns the symmetry of the arrangement of the building-block graphs. Some arrangements have more symmetry than others, and if one starts with any given arrangement, one can deform it, while maintaining periodicity and the bonding relations between the atoms, to make it more symmetrical. For the diamond crystal, it turns out that no deformation of the periodic arrangement can make it any more symmetrical than it is. As Sunada puts it, the diamond crystal has maximal symmetry.

Any crystal can be deformed into a crystal with maximal symmetry, so that property alone does not distinguish the diamond crystal. But the diamond crystal has a second special property, called "the strong isotropic property". This property resembles the rotational symmetry that characterizes the circle and the sphere: No matter how you rotate a circle or a sphere, it always looks the same. The diamond crystal has a similar property, in that the crystal looks the same when viewed from the direction of any edge. Rotate the diamond crystal from the direction of one edge to the direction of a different edge, and it will look the same.

It turns out that, out of all the crystals that are possible to construct mathematically, just one shares with the diamond these two properties. Sunada calls this the K4 crystal, because it is made out of a graph called K4, which consists of 4 points, in which any two vertices are connected by an edge.

"The K4 crystal looks no less beautiful than the diamond crystal," Sunada writes. "Its artistic structure has intrigued me for some time." He notes that, although the K4 crystal presently exists only as a mathematical object, it is tempting to wonder whether it might occur in nature or could be synthesized. This is not so far-fetched as it may sound: The Fullerene, which has the structure of a soccer ball (technically called a truncated icosahedron), was identified as a mathematical object before it was found, in 1990, to occur in nature as the C60 molecule.

Sunada's article, "Crystals That Nature Might Miss Creating", is appearing in the February 2008 issue of the AMS Notices and will be posted online January 3.


Story Source:

The above story is based on materials provided by American Mathematical Society. Note: Materials may be edited for content and length.


Cite This Page:

American Mathematical Society. "One Crystal That Nature May Have Missed." ScienceDaily. ScienceDaily, 8 January 2008. <www.sciencedaily.com/releases/2008/01/080103101134.htm>.
American Mathematical Society. (2008, January 8). One Crystal That Nature May Have Missed. ScienceDaily. Retrieved September 3, 2014 from www.sciencedaily.com/releases/2008/01/080103101134.htm
American Mathematical Society. "One Crystal That Nature May Have Missed." ScienceDaily. www.sciencedaily.com/releases/2008/01/080103101134.htm (accessed September 3, 2014).

Share This



More Matter & Energy News

Wednesday, September 3, 2014

Featured Research

from universities, journals, and other organizations


Featured Videos

from AP, Reuters, AFP, and other news services

Halliburton Reaches $1B Gulf Spill Settlement

Halliburton Reaches $1B Gulf Spill Settlement

AP (Sep. 2, 2014) Halliburton's agreement to pay more than $1 billion to settle numerous claims involving the 2010 BP oil spill could be a way to diminish years of costly litigation. A federal judge still has to approve the settlement. (Sept. 2) Video provided by AP
Powered by NewsLook.com
Google Teases India Event, Possible Android One Reveal

Google Teases India Event, Possible Android One Reveal

Newsy (Sep. 1, 2014) Google has announced a Sept. 15 event in India during which they're expected to reveal their Android One phones. Video provided by Newsy
Powered by NewsLook.com
Australian Airlines Relax Phone Ban Too

Australian Airlines Relax Phone Ban Too

Reuters - Business Video Online (Aug. 26, 2014) Qantas and Virgin say passengers can use their smartphones and tablets throughout flights after a regulator relaxed a ban on electronic devices during take-off and landing. As Hayley Platt reports the move comes as the two domestic rivals are expected to post annual net losses later this week. Video provided by Reuters
Powered by NewsLook.com
Hurricane Marie Brings Big Waves to California Coast

Hurricane Marie Brings Big Waves to California Coast

Reuters - US Online Video (Aug. 26, 2014) Huge waves generated by Hurricane Marie hit the Southern California coast. Rough Cut (no reporter narration). Video provided by Reuters
Powered by NewsLook.com

Search ScienceDaily

Number of stories in archives: 140,361

Find with keyword(s):
Enter a keyword or phrase to search ScienceDaily for related topics and research stories.

Save/Print:
Share:

Breaking News:
from the past week

In Other News

... from NewsDaily.com

Science News

Health News

Environment News

Technology News



Save/Print:
Share:

Free Subscriptions


Get the latest science news with ScienceDaily's free email newsletters, updated daily and weekly. Or view hourly updated newsfeeds in your RSS reader:

Get Social & Mobile


Keep up to date with the latest news from ScienceDaily via social networks and mobile apps:

Have Feedback?


Tell us what you think of ScienceDaily -- we welcome both positive and negative comments. Have any problems using the site? Questions?
Mobile: iPhone Android Web
Follow: Facebook Twitter Google+
Subscribe: RSS Feeds Email Newsletters
Latest Headlines Health & Medicine Mind & Brain Space & Time Matter & Energy Computers & Math Plants & Animals Earth & Climate Fossils & Ruins