Exploring Crystal Lattices

Introduction

In early 2012, I was trying to teach myself some crystallography and crystal chemistry beyond what I had picked up in passing from mindat and other sources while my mineral collection had been growing. Although I'm coming from a mathematical background and work with complex software for a living, and thus am not afraid of high levels of abstraction, the textbooks and 2D/3D graphics I was looking at were leaving me wishing for more.

German has that lovely word begreifen, roughly synonymous with verstehen (to understand) but with tactile roots: its etymology suggests to grab something, handle it, balance it on your palm, turn it around, touch its corners and its edges, feel whether it is rigid or flexible... I wanted to stick pieces together to learn how they fit together.

I looked at magnets-and-balls kits, but found them too limited. Other people have done wonderful things with wooden balls and rods, but being less mechanically adept, I foresaw that anything I tried in this direction would end up warped and messy. I started playing around with Zometool^[1][2]. This, too, has some obvious shortcomings for this particular purpose; but it was easy enough to get going with some simple examples from the isometric crystal system. Soon, little lattice models started to litter my living room: diamond and fluorite, followed by a not-so-little spinel. (I skipped halite.)

The breakthrough moment came in March 2013 when twelve nodes and twelve struts first closed up into an oddly kinked ring... satisfactorily approximating a small fragment of the quartz lattice.

The International Year of Crystallography 2014 provided additional motivation.

And now (heh, I'm only three years late for that pageant) I'll go and spoil it all and resort to a visual medium to tell you about some of my exploits. Text and pictures will fall a bit short of the true first-hand experience. Yet I hope they will be of interest to mindat participants and guests and encourage you, gentle reader, to embark on some exploration of your own.

This is not going to be another textbook on crystallography: I won't hesitate to toss in a technical term whenever it fits, but I'll tone down the mathematics. It's not meant to be product placement either; I'll be singing the praise of cardboard and scissors and yarn and glue in due course. (The French have a lovely word for that: bricolage).

Enjoy!

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Contents

The chapters will be added as separate articles, progressing as time allows.
Below the table of contents I'll include a little Zometool glossary, and an index follows at the bottom of the page.

• 1: Starting Small
• 2: Enter the Model Universe
• 3: Taking Three as the Subject...
• 4: Glides and Screws
• 5: A Skeleton with Cleavage
• 6: Holes for Giants
• 7: The Mother of all Silicates
• 8: Forever Failing to Meet
• tbd: The Cherry on the Cake
• tbd: High Field Strength
• tbd: ...And Bright Plastic Chains...
• tbd: Quasi!
• tbd: Islands and Chains
• tbd: A Lattice that Nature Missed
• tbd: Sawdust and Glue
• ...

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Appendix - Glossary: What's what in the Zometool universe

A Zometool node (sometimes called a connector or a ball) is a fancy icosahedron-dodecahedron-triacontahedron with little holes along each of its 31 rotational symmetry axes: A pair of rectangular holes for each of the 15 two-fold axes, at right angles to the 15 mirror planes; a pair of triangular holes for each of the 10 three-fold axes; and a pair of pentagonal holes for each of the 6 five-fold axes. The stubby ends of the struts plug into these holes. All Zometool parts - the nodes and all the various struts - have at least a central inversion symmetry. This automatically forces any two nodes joined by a strut to be parallel to each other: One and the same set of 62 fundamental strut directions always applies to an entire model.

Strut lengths within each direction are quantized in powers of the golden ratio (taking the node diameters into account).

In the basic kits, all nodes are white, and struts are colour-coded according to type: Blue (B) along 2-fold axes, Yellow (Y) along 3-fold axes (a Y strut represents half the body diagonal of a cube built with the corresponding B struts), and Red (R) along 5-fold ones.

Struts in non-standard colours exist, e.g. "B struts in red/yellow/green/...", "Y struts in white", "R struts in black", etc. The following picture shows some of these at the bottom, and some more serving as the "petals" of the 3-sided and 5-sided "flowers" at top right:

In order of increasing lengths, we speak of B0, B1, B2 struts, and similarly for the other directions (replacing older notions of "supershort", "short", and "medium" struts; the original "long" B3s having been abandoned many years ago). The shorter-than-R0 (formerly "hypershort") R strut, fourth from the right in the middle row in the picture, ought to be called "R minus 1", but is sold as "R00", which makes the mathematician in me itch.

Here's a stereo view:

For scale, nodes have diameter 1.77cm = .696in in the B and Y directions (slightly larger along R), and a B0 strut has a shoulder to shoulder length of 28.6mm = 1.126in, resulting in a distance of 46.3mm = 1.821in between node midpoints. B1 is 1.618... times as long centre-to-centre and exactly twice as long as a B0 shoulder-to-shoulder. A "B minus 1" would have shoulder-to-shoulder length equal to the node diameter, and a "B -2" would amount to two nodes glued directly to each other along a B direction.

The Zometool node is a little engineering and manufacturing miracle, but it took a second miraculous idea to make the kit really useful for crystallographic purposes.

Green (G) struts add another 60 directions. Regarded as crystallographic individuals, they are monoclinic, with a mirror plane along their length and a 2-fold rotation axis at right angles to it through their midpoint. Their kinked ends "borrow" the nearest 5-sided holes of the nodes, requiring some care in aligning them correctly. Use them for diagonals of squares and the edges of regular tetrahedra and octahedra, and use their "half-green" (HG) siblings, natively coming in a teal hue (but now also available in green), for half-diagonals:

Nodes in various hues other than white (black, grey, red, orange, yellow, green, teal, blue, purple) are available separately or as parts of various project kits. Some of them are featured in the title picture at the top of this article. Model-builder's acrylic paints can be used to obtain additional hues - and of course we want quite a palette to represent different species of atoms or ions! Be sure to paint the side walls of the holes, too; but don't lay it on too thick or the struts will no longer fit.



Zometool was created with 3D projections of 4D polyhedra in mind, which use lots of nodes and a moderate number of struts of all kinds of directions and lengths. The basic "Creator" kits are taylored to these needs. Crystal lattice models tend to use lots of struts of only a few types. So you'll always run out of struts long before you would run out of nodes. Additional struts are available in bulk when you need them.

Zometool re-did their injection-mold tooling at one point. If you (like me) happen to possess struts cast before the change, as well as recent ones, then for best effects, don't mix old and new struts of the same type in the same model. The look and feel is very slightly different (NB the above pictures show the old versions); and there are slight differences in shrinkage after casting, thus a big octahedron with a central node, built from old B2 and new G2 struts, won't close up properly! This is less of a problem with the shorter strut lengths.

Several CAD-savvy people have created additional 3D-printable parts to extend the Zometool kit: there's the aforementioned "B-1" strut, and struts in yet more directions with more elaborately kinked ends, and "bobs" which can be used to join two struts together where a node would be distracting. Search for "zome" and "vzome" on Shapeways^[3]. Order them as "White Strong & Flexible," unpolished, and apply acrylic paint afterwards. (Polishing the struts would result in too loosely fitting stubs.) The holes of the bobs (especially of the Y bobs with their acute angles) tend to come with some unconsolidated plastic powder trapped in them; carefully clean them out with a needle before pressing a strut into them.

Index of minerals and key concepts featured

brunogeierite - 4
calcite - 3
carrollite - 4
cassiterite - 2
chalcopyrite - 2
chromite - 4
corundum - 3
ccp (cubic close-packing) (=fcc) - 2, 4, 6
diamond - 1
dolomite - 3
fcc (face-centred cubic) lattice (= ccp) - 2, 4, 6
feldspar group - 5
fluorite - 1
franklinite - 4
graphite - 1
hausmannite - 4
hcp (hexagonal close-packing) - 3
hematite - 3
ilmenite - 3
jacobsite - 4
londonite - 6
magnesite - 3
magnetite - 4
orthoclase - 5
polytypes - 1
quartz - 7
rhodizite - 6
rhodochrosite - 3
ringwoodite - 4
rutile - 2
sanidine - 5
siderite - 3
smithsonite - 3
spinel - 4
spinel group - 4
thiospinel group - 4

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1. The author is not affiliated with Zometool Inc. except as a somewhat addicted customer.

2. All trademarks mentioned in these pages are owned by their respective owners.

3. The author is not affiliated with Shapeways either, except... (see above).
- Gerhard Niklasch ©2017