Plus-Plus, according to the Plus-Plus website, allows you to "Unleash the power of your imagination with a unique design in a single shape. Plus-Plus offers unlimited opportunities for creativity. From a single repeated element, the possibilities are endless as you bring your ideas to life in both 2D and 3D forms."
Plus-Plus (or sometimes just PlusPlus) is a toy made up of small plastic pieces similar to jigsaw puzzle pieces, which are flexible enough to interlock in either two or three dimensions, and strong enough to stand up and make small towers or other 3D models.
My interest in Plus-Plus is purely geometrical. I was most interested to discover if it's possible to produce a repeating pattern of PlusPlus pieces (which I think the manufacturers refer to as 'hexels') in a 2D surface with perfect tesselation?
As an aside, I guess they call them hexels because they approximate a hexagonal shape? I looked at them and decided they were made up of nine mini-squares all combined together, but a nonel sounds daft. Either way, they are shaped like two plus symbols that have been overlapped ++
After some logical puzzling with a large pile of hexels, I was able to confirm that it's possible to make a tessellating pattern with them. I don't know if anybody else has named the pattern, so I'm going to go with HV, for Horizontal Vertical. Each hexel is connected to another in a straight line, with the orientation for the adjacent hexel being rotated 90 degrees (in the same plane) to the first. The repeating pattern is Horizontal, Vertical, Horizontal, Vertical, and so on. The picture shows how it looks in a line, and then connects multiple lines to cover a 2D surface.
Interestingly, HV is not the only way to connect a series of hexels: they can be connected HHH, with a slight diagonal offset (green lines below), or connected VVV (orange lines below). I want to mention at this point that the two are not the same: it's possible to have multiple HH lines tessellate by lining them up together (there's no interlocking between adjacent lines, so the structure is physically weak, but it still tessellates).
However, HH and VV cannot tessellate together. The two patterns are incompatible. Rotating a HH line will not make it fit into a VV line. Another way of thinking of it (through rotation) is that one line steps 'upwards' while another steps 'downwards' - see below.
Something I found particularly interesting is that it's possible to combine HH with VH, and also to combine VV with VH. In the diagram below, the grey hexels are in VH configuration. The white hexels are in HH, with the lines of HH running from top left to bottom right (it's a short line, and the interface between the patterns runs perpendicular to it). The blue hexels fit into both patterns, and show the interface or the border between the two.
And here is proof that you can combine VH, HH and VV: the grey is VH as before; the white is HH, while the orange is VV. The blues show the interface between VH and HH; the yellow shows the interface between VV and VH. The bands of white, grey and orange can be of any fixed width, but they don't get narrower or wider.
This reminded me of the studying I did at Cambridge University, looking at phases of iron-carbon alloys. Each 'grain' of a particular alloy would have a certain area (or volume, in 3D) where it was made entirely of a particular phase of iron/carbon, and the composition would be consistent throughout the grain. There would be a grain boundary where it touched another grain of the same composition, but where the grain had been oriented differently.
Diagram from http://www.gowelding.com/met/carbon.htm
I have not yet considered 3D shapes - these are clearly possible, and I shall be looking at if it's possible to produce regular polyhedra such as cubes and cuboids, and what limitations there are.
Other Geometry Articles I've Written:
Calculating the tetrahedral bond angle
The angle of elevation of a geostationary satellite
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