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Truchet tiler
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With a new set of 64 identical Truchet tiles, my open-ended sliding tile puzzle (STP) becomes an 8x8 Truchet tiling sketchpad.
About this creation
Please feel free to look over the images and skip the verbiage.

On this page...



Introduction

In this context, to tile is to cover a flat surface without gaps or overlaps using tiles that may or may not differ in size, shape, or decoration. Motif refers to the decoration(s) on individual tiles, whereas pattern refers to the higher-order image that emerges when the motifs are juxtaposed in a tiling.

This MOC tiles an 8x8 square grid with 64 identical square tiles (3068bpb0193) 2x2 LU in size.



Only a handful of decorated square LEGOŽ tiles qualify as Truchet tiles, which are known for the richness of the patterns they form. The tile chosen here (above) is by far the best example of a ready-made LEGOŽ Truchet tile.

Broadly speaking, a Truchet tile is a square tile decorated so as to have
  • Two or more distinct face-up orientations
  • Potential continuity of motifs across at least shared tile edges if not shared corners
  • Mirror symmetry across at least one diagonal.
A Truchet tiling uses identical Truchet tiles juxtaposed in various orientations. The pattern formed depends only on the distribution of tile orientations over the tiled surface.

Such tilings were first described in 1704 by French priest, mathematician, physicist, hydraulic engineer, and typographer Sebastien Truchet.

Removing the dots from the tile chosen here yields the tile originally investigated by Truchet. The fundamentals are the same in both cases: (i) Four distinct face-up orientations, (ii) continuity of motifs across shared edges in some relative orientations but not all, and (iii) mirror symmetry across just one diagonal.



The WIP photo above shows the best-known of all Truchet tiles -- the quarter-circle tile.

Unlike the chosen tile, the quarter-circle tile has (i) 2 distinct orientations, (ii) guaranteed motif continuity across the midpoints of all 4 shared edges, and (iii) mirror symmetry across both diagonals.

Visually, the patterns formed by the chosen tile and the quarter-circle tile are worlds apart.




More photos

Among many other things, Truchet tilings let you fiddle around with the interplay of long- and short-range order -- a very common theme in natural pattern formation.





Having trouble settling on an orientation for a particular grid location?

You could always roll a pair of binary LEGOŽ dice, with one die determining whether the tile's all-black corner points up or down, and the other die, right or left.




Design notes

My original binary sliding-tile puzzle (STP, below) used 64 plain black and white tiles in a particular (50-50) mix.



Making binary patterns was then a matter of redistributing that tile set over the 8x8 grid -- something an STP could easily do.

Once tile orientation enters the picture, however, the full pattern-generating potential of a tiling sketchpad like this can only be realized if tiles can be lifted out and reoriented quickly and easily on a random-access basis.

The STP base earns its keep mainly by providing the random access, as the user can open up a void next to any given tile in just 1 or 2 moves.

Once a tile edge is exposed, the tile can be lifted out, re-oriented, and replaced in seconds. The 64th tile covers the void when the tiling's complete.




Ready-made LEGOŽ Truchet tiles

The vast majority of decorated square LEGOŽ tiles are 2x2 LU in size. BrickLink currently lists 834 different 2x2 tile decorations, but only a handful qualify as Truchet tile markings. There are no Truchet tiles in other square tile sizes.

Most of the bona fide 2x2 Truchet tiles are just more ornate versions (e.g., 3068bpb0392 and 3068bpb0777) of the tile used in this MOC (3068bpb0193). The dot counts vary from 1 to 3, but all of the motifs have 4 distinct orientations and 1 diagonal mirror plane. Relaxing the symmetry requirement a bit adds a few more to the list (e.g., 3068bpb0191 and 3068bpb0393).

None of the bona fide Truchet tiles match Truchet's original tile exactly, but that's probably for the better, as the patterns formed by his stark tiles tend to be lifeless, jarring, or both (examples here).

The closest match to Truchet's original tile is the one chosen for this MOC. It's plentiful and cheap and forms dramatic patterns. The simple addition of 2 black dots on the white half adds an interesting texture to the tilings and a softness that Truchet's patterns sorely lacked.

The 3-dot and 2-dot white 2x2 tiles meant for dice don't quite qualify as Truchet tiles. They have 2 distinct orientations and mirrors along both diagonals, but their motifs never reach a tile edge or corner. On the plus side, they're also plentiful and almost as cheap as the tile used in this MOC.

On an STP base, the 3-dot tile holds some promise as the basis for an adjustable Truchet maze (stay tuned). The 2-dot tile has similar potential, albeit at a much lower dot density that could well dilute out the visual impact.

Several non-Truchet decorated 2x2 LEGOŽ tiles also strike me as having pattern-forming potential, but none qualify as Truchet tiles.




Specifications


Overall dimensions:
144x144x22 mm (LxWxH), excluding stand

Overall mass:
0.250 kg (0.55 lb), excluding stand

Grid:
8x8

Voids:
1

Tile construction:
3-layer

Tile count:
64

Tile type:
Truchet

Tile markings:
See photos

Modified LEGOŽ parts:

None

Non-LEGOŽ parts:
None

Credits:

Peer Krueger for the 3-layer tile design and sliding mechanism; otherwise, original MOC






Comments

Jeremy McCreary
 I like it 
Matt Bace
  January 29, 2015
Very neat. It looks like something that one could purchase in a high-end toy store focused on "thinking games".
 I made it 
  January 29, 2015
Many thanks to all for the kind words and likes. I have 2 more 8x8 STPs with Truchet tiles in the works: A reconfigurable maze based on ready-made 3-dot 2x2 tiles and another "tiling sketchpad" based the DIY 3x3 quarter-circle tiles shown at the top of the page.
 I like it 
  January 29, 2015
Great stuff! I love the idea of throwing the die too! :)
 I like it 
  January 29, 2015
Just wow. Mind boggling!
 I like it 
  January 29, 2015
Very refine work here!
 I like it 
  January 29, 2015
Interesting!
 
By Jeremy McCreary
Add to my favorite builders

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Added January 28, 2015
 


LEGO models my own creation MOCpages toys shop Truchet tilerOther


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