Ice Crystal Matrix measurements of H2O molecules?

I want a mathematically generated [3D STL copy of this image](http://www.snowcrystals.com/science/Animation2.gif), however it's represent algorithmically.


If I do all the H2O's as individual pieces with female connections prepared for chopped bike spokes, are all the H2O's the same and what angles do they have?


Is there a written array of positions and degrees of rotation of the water molecules of a regular ice matrix? 




I decided to summarize all my comments in a form of answer and add some illustrations. Note that I have no experience in 3D printing, so I mainly focus on crystallographic part.


Since you asked for"*a mathematically generated*" 3D structure, you ought to know that water crystallizes in $P\mathrm{6\_3cm}$ (# 185) space group having a hexagonal symmetry. Detailed information about symmetry generators and matrix transformations can be found [online](http://img.chem.ucl.ac.uk/sgp/large/185az1.htm) or in International Tables for Crystallography [1, pp. 582-583]:



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> [![enter image description here](https://i.stack.imgur.com/qqQtK.png)](https://i.stack.imgur.com/qqQtK.png)
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> 


There is *already enough information* to create a 3D model. If you want a more convenient way, you can get a [CIF file for ice (COD-1011023)](http://www.crystallography.net/cod/1011023.html) (which already embeds all the information above), and load it with [Mercury](https://www.ccdc.cam.ac.uk/support-and-resources/Downloads/) (free, available on Windows, Linux, MacOS). From here on it's just a screencast of what's to click to get the desired molecular pattern like [the one on your animated GIF](http://www.snowcrystals.com/science/Animation2.gif):


1. Load the structure (drug-n-drop CIF file, or via `Ctrl`+`O`). What you see now is an [asymmetric unit](http://reference.iucr.org/dictionary/Asymmetric_unit), in simple terms – a seed which an entire crystal structure can be grown from:  

[![enter image description here](https://i.stack.imgur.com/gExb6.png)](https://i.stack.imgur.com/gExb6.png)
2. Let's expand the structure beyond those 6 atoms you've stuck with. Go to `Calculate` > `Packing/Slicing...`. Tick `Pack` option, and click `+ 0.5` boxes next to $a$ and $b$ axis:  

[![enter image description here](https://i.stack.imgur.com/GFMfk.png)](https://i.stack.imgur.com/GFMfk.png)
3. Rotate the grown structure approximately like it's shown below:  

[![enter image description here](https://i.stack.imgur.com/Mexei.png)](https://i.stack.imgur.com/Mexei.png)
4. Note that you have 4 water molecules on each side that you don't need. Those can be deleted by right-clicking on them and selecting `Delete this molecule`:  

[![enter image description here](https://i.stack.imgur.com/jBnCX.png)](https://i.stack.imgur.com/jBnCX.png)
5. At this point you should have exactly the same 3D representation of the crystal structure:  

[![enter image description here](https://i.stack.imgur.com/SNYoi.gif)](https://i.stack.imgur.com/SNYoi.gif)
6. Now you can print a 3D model directly from Mercury (`File` > `Print in 3D...`). The one with supporting framework looks like this:  

[![enter image description here](https://i.stack.imgur.com/KojkF.png)](https://i.stack.imgur.com/KojkF.png) 


6.1. Alternatively, you can save the structure as XYZ (`Ctrl`+`S`, choose `Xmol files`) and do some post-processing work in Blender if needed.


### Reference


1. International Tables for Crystallography: Space-group symmetry, 1st ed.; Hahn, T., Ed.; Fuess, H., Hahn, T., Wondratschek, H., Müller, U., Shmueli, U., Prince, E., Authier, A., Kopský, V., Litvin, D. B., Rossmann, M. G., et al., Series Eds.; International Union of Crystallography: Chester, England, **2006**; Vol. A.


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You can produce STL (and VRML) files directly from Mercury - see <https://www.ccdc.cam.ac.uk/Community/blog/post-56/>


I'm not sure how best you'd go about including female sockets for bike spoke segments, but be aware that for some 3D printing types you're charged for the total volume of material used, so you might be better off trying to create a model with multiple molecules in a smaller area with less empty space.


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