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Quarter-circle Truchet tiler
Another Truchet tiling sketchpad, this time with 3x3 quarter-circle tiles. The resulting patterns are quite unique.
About this creation
Please feel free to look over the images and skip the verbiage.

You really need to watch the patterns evolve to appreciate how the new quarter-circle Truchet tiles presented here work their magic.



Like my original Truchet tiler below, this MOC is an 8x8 Truchet tiling sketchpad built on a sliding-tile puzzle (STP) base.



This time, however, the Truchet tiles are identical 3x3 quarter-circle tiles, each with a DBG background and a raised motif consisting of paired white "macaroni" bricks (2x2 round corners, 85080).



In this context, to tile is to cover a flat surface without gaps or overlaps using tiles that may or may not vary 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.

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 some shared tile edges if not shared corners
  • Mirror symmetry across at least one diagonal.
A Truchet tiling covers a surface with identical Truchet tiles. The pattern formed depends only on the tile orientations assigned to each cell.

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

As seen below, the patterns generated by this new motif and larger tile size are 50% larger, much more fluid and, in many ways, much more intriguing. They also have a 3-dimensional character best appreciated in the oblique shot below.





Compared to the 3-layer construction used in previous tilers, the new 2-layer tile design used here saved a lot of weight and money but left the sliding action a bit more fiddly and the tiles a little more prone to fall out when pressed too hard or tilted.

On this page:


The quarter-circle tile

When C.S. Smith revived interest in Truchet's by then long-forgotten work in 1987, he added a twist of his own: A clever new pattern-forming motif that quickly became the Truchet tile motif. Smith's tile, the quarter-circle tile (QCT), is a square tile of side length L decorated with 2 separate quarter circles (QCs) of radius L/2.

Importantly, the QCs are centered at opposite ends of the same tile diagonal and join the midpoints of adjacent tile edges. The 2 QCs collectively make up the tile's motif.

The other QCT diagonal follows the center of a "channel" running between the QCs. If the DBG tile areas within the concavities of the white QCs are the "banks" of the channel, the QCs are its "shorelines".

The QCT qualifies as a Truchet tile by virtue of having 2 distinct orientations, guaranteed continuity of motifs accross all 4 tile edges, and mirror symmetry along both diagonals.



The slash characters "/" and "\" make a convenient shorthand for QCT orientations when taken to represent the channel diagonals. The tile pair above is then written "\ /". The DBG 1x1 round plates are there only to mark the channel diagonals in this photo.



The tile pairs "\ \", "\ /", "/ \", and "/ /" above represent all 4 possible QCT pairings. Note the kink-free linkage of motifs across the shared edge in each case.

When identical QCTs are used to tile a flat surface with their orientations free to change from cell to cell, remarkable curvilinear patterns emerge.



The genius of Smith's choice of motif is now apparent: Every tile's motif links up smoothly -- i.e., without kinks -- with the motifs of all 4 nearest (edge-sharing) neighbors, regardless of the relative tile orientations involved.

I think of the sinuous white features formed by these linkages as "walls" and the pinching and swelling DBG area flanking them as "paths". Which is figure and which is ground is in the eye of the beholder.

To appreciate just how special the QCT is in this regard, take another look at the 2x2 Truchet tile (3068bpb0193) used in my original Truchet tiler.



The tile cluster above includes a central tile and all 4 of its "edge-sharing neighbors". The 4 additional "corner-sharing neighbors" touching the central tile only at a corner are absent.

Note that none of the motifs link up across shared edges when all five tiles happen to have the same orientation.

The tiling below contains many 5-tile clusters of the same shape, but none of them happen to have all 5 tiles in the same orientation. Hence, their motifs link up across shared edges to varying degrees.






Connectedness

Two wall or path segments in a tiling are "connected" if a minifig could walk from one to the other without leaving the original wall or path.

"Wall connectedness" refers to the extent to which a wall segment of interest connects to other wall segments. It can often be judged at a glance in this MOC. "Path connectedness" is harder to see but worth looking for, as paths often ramify through QCT tilings in truly surprising ways.



Quick, which connected path covers the largest chunk of real estate in the tiling above, and exactly where does it go?

Truchet tiling computer programs often use color to make path connectedness more apparent. The user typically applies a color at a single point in a path segment of interest. The software then floods the rest of the path with that color, including all its branches.

Dropping colored "breadcrumbs" into the 2x2 "nodes" (swellings) encountered as one traces out a path of interest accomplishes much the same thing in this MOC, though admittedly with more work.



Same tiling as before, now with red breadcrumbs of various types in all nodes belonging to the connected path with the largest footprint. If you didn't catch all of it without the aid of the breadcrumbs, don't feel bad. I didn't either.

Colored 2x2 domes, cones, dishes, and plates are my favorite breadcrumbs, but anything you can fit into the nodes and fish out afterward will do.

Warning: Turning the sketchpad upside down to remove breadcrumbs will also remove the tiles.



Same tiling again, this time with all remaining connected paths longer than a node or two marked with breadcrumbs of the same color and type. My lovely assistant Anna stands on a path just one whole node long. Only 2 walls form closed loops in this particular tiling. Minifigs other than Anna stand in the resulting "enclosed paths".

Truchet tilings are remarkable for their sensitivity to small changes in the distribution of tile orientations. In fact, flipping just a few tiles can trigger dramatic changes in wall and path connectedness and overall pattern. Because paths can branch and cross themselves but walls can't, path connectedness is generally more sensitive than wall connectedness.




Editing

Once tile orientation enters the picture, as it does in Truchet tilings by definition, the full pattern-generating potential of a tiling sketchpad like this can only be realized if cell orientations can be edited quickly and easily on a random-access basis.

This tiler offers at least 3 ways to change the orientation of a cell occupied by a tile:
  • Macaroni method: Remove the tile's macaronis and put them back in the other orientation.
  • STP method: Vacate the cell using the STP feature and shift a tile with the desired orientation into the cell from another location.
  • Tile rotation method: Lift the tile out of the target cell, rotate it 90° in either direction, relocate the guide tab accordingly, and put the tile back.
The last method uses the STP feature only to expose an edge of the target tile for removal purposes. It was a viable editing option in my original Truchet tiler, but it's generally more trouble than it's worth in this tiler due to differences in tile construction.

The macaroni and STP methods have their pros and cons. Reorienting only the macaronis guarantees the desired result with no collateral damage. This method shines if you have just 1 tile to flip but gets old fast in extensive edits.

The STP method is the one demonstrated in the video. Changing just 1 tile orientation this way is a mathematical impossibility. Arranging a particular pattern with the STP method can be quite challenging, but that's part of the fun. The big payoff is getting to watch the pattern evolve rapidly as you work. The mechanical sound and feel of moving the tiles around is also enjoyable.

The video nicely demonstrates another major advantage of the STP method: The ability to shift an entire row or column of tiles by one cell in a single move. There's something very satifying about this maneuver, and the impact on the overall pattern can be quite dramatic. A dozen or so well-chosen row and column shifts can be the equivalent of shuffling a deck of cards.

The 64th "finishing tile" (the one sans guide tab in the photo below) covers the void when the tiling's complete.



The finishing tile in this MOC was easily tagged with an inconspicuous DBG 1x1 round plate. Those in previous STPs with 2x2 tiles were hard to mark and easy to lose track of.



For massive tiling changes, a dump might be your best bet: (i) Remove one side of the frame and dump out all the tiles. (ii) Reload the existing tiles in the desired orientations, starting at the side opposite the one removed. (iii) Reconfigure individual tiles only if you run out of a particular orientation before the tiling's complete. (iv) Replace the side of the frame previously removed.




Design notes

The definition of a QCT and my desire for a hand-held or laptop 8x8 sketchpad with high-contrast tiles drove the entire design.

Since none of the 834 available decorated 2x2 tiles even remotely resembles a QCT, I had to build my own. The 2x2 LU maraconis were the only quarter-circle elements small enough to meet the overall size constraint. How they'd look in the completed MOC, I had no idea.

To intersect the midpoints of adjacent tile edges, the macaronis could only ride on 3x3 plates, which only come in LBG and DBG colors. High-contrast tiles meant light-colored macaronis on DBG 3x3 LU plates. With characteristic blandness, I chose white.

The fate of the entire project then hinged on getting a 3x3 LU plate to work as a sliding tile base. To keep bulk, weight, and cost down, I decided to abandon Peer Krueger's 3-layer tile design (used in my binary sliding-tile puzzle and original Truchet tiler) for a 2-layer construction.

The 6 dark red 2x2 tiles in the mock-up below show the difference between 3-layer and 2-layer tiles. Their top layers have been removed.



The 2-layer tiles on the right save additional bulk and weight by allowing the frame to be lowered by 1 plate relative to the floor (sliding surface).

The next 2 photos show the only way I could think of to turn a 3x3 LU plate into a 2-layer sliding tile. As in the 3-layer case, a 2x2 LU jumper serves as guide tab. The 2x2 LU corner plate keeps the tile level and checks guide tab misalignment.





That all of these rather narrow windows of opportunity eventually aligned is a small miracle. The only flies in the ointment:
  • Sliding action took a small step backward in smoothness.
  • Tiles will buckle up if the corner opposite the guide tab is pressed too hard.
  • Tiles are more likely to fall out when the sketchpad is tilted beyond 45° or so.
The first 2 issues are evident in the video.

An 8x8 array of 3x3 LU tiles dictated a 26x26 LU sketchpad frame enclosing a 24x24 LU floor. Keeping bulk, weight, and cost down, overall sturdiness high, and the sliding action as smooth as possible meant a floor consisting mainly of 8x16 LU tiles with bottom tubes applied directly to a 32x32 LU baseplate subfloor.




A closer look at QCT tiling patterns



Pattern scales

The patterns generated by QCT tilings are typically patchworks of sub-patterns. Like the small white circles in the upper tiling below, many recurring sub-patterns have characteristic scales. The lower tiling is just for comparison.





It's useful, then, to think of a sub-pattern as occupying a "patch" of at least 2 contiguous tiles but not all the tiles. Patches have patterns in the sense defined above; individual tiles do not.

The largest possible scale, of course, is that of the tiling as a whole. Call it "global". Patch-scale phenomena are then "local". Below patch scale is "tile scale" -- the scale of the motif as a whole.

"Tile grid" refers to the global square array (here, 8x8) of cells populated by the tiles. The rows and columns of the tile grid parallel the rows and columns of studs on the baseplate.



Walls

The 2 patch-scale features common to every QCT tiling pattern are walls and paths. Higher-order patterns emerge from their interaction.

The macaroni bricks used as QCs here automatically link up across shared tile edges to form undulating white "walls" of constant height and width but of varying length and layout. The natural unit of wall length is the "macaroni".

Two wall segments are "connected" if and only if a minifig could start walking along the top of one and eventually end up on the other without having to jump a path along the way. Whether the 2 macaronis on a given tile are connected will vary from tiling to tiling.

Eventually, all walls either (i) close on themselves within the grid to form "loops" or (ii) dead-end against the sketchpad frame. They can't (a) pass through tile centers or corners, (b) branch, (c) peter out short of the frame, or (d) dead-end against themselves or other walls.



The shortest possible loop is, of course, a circle 4 macaronis in circumference. Since this is also the largest possible true circle, I'll just call it a "circle" hereafter. There are 4 circles in the upper left half of the tiling above.

As non-circular walls undulate across the grid like sine waves, they follow a diagonal "wall grid" in the same way that cars follow city streets. The wall grid is essentially a slightly offset copy of the tile grid rotated by 45°.



Paths

Just as the motif-forming macaronis link up across shared tile edges to form white walls, the flanking tile channels and banks link up across shared tile corners to form DBG "paths" complementary to the walls.

Like walls, paths are patch-scale phenomena of varying length meandering along a diagonal grid. However, the "path grid" is slighly offset from the wall grid, and the natural unit of path length is the length of a tile diagonal (33.6 mm = 4.2 LU).

Unlike walls, paths pinch and swell at regular itervals. At tile centers, where the macaronis nearly kiss, they narrow to just 2 mm in width. However, at the next shared corner, where 2 channels and 2 banks coalesce, they swell to a maximum width of 16 mm (2 LU). I think of the swellings as "nodes".

Importantly, paths can do all the things walls can't. They pass through tile centers and corners and can branch and cross themselves any number of times.

Paths that close on themselves occur only inside loops formed by walls. All other paths eventually end on the sketchpad frame, but path branches can dead-end within the grid. All branching and turning points along a path occur at nodes.

Two path segments are "connected" if and only if a minifig could walk from one to the other without climbing a wall or setting foot on the sketchpad frame. Various red "breadcrumbs" mark a connected path covering much of the tiling below.



Since walls can't end within the grid, the channel and banks on any given tile must lie on 3 unconnected paths. Ultimately, this fact underlies a characteristic feature of QCT tilings -- namely, the ability to produce dramatic changes in path connectedness and average length with just a few tile orientation flips.

The "path pattern" formed collectively by the paths within a given patch is complementary to the "wall pattern" there, and vice versa. The various path and wall patterns in a QCT tiling combine to form the "overall pattern" seen by the viewer.

For a variety of reasons having to do with things like wall height, thickness, and relative brightness, the wall pattern usually dominates the overall pattern in this sketchpad. However, the path pattern can steal the show on a patch-by-patch basis. Due to the raised motifs, lighting is often the deciding factor.

The resulting figure-ground ambiguity is one of the things that makes QCT tilings so fascinating.



Landscapes

The contrived pattern below consists of 3 distinct patch-scale "landscapes" -- one at upper left, another at lower right, and one in between.



The triangular landscape at upper left consists entirely of loops. Call it "Loopland". Loopland contains wall loops of 3 different shapes and sizes. Smallest of all are the circles discussed above. Next up are the "dumbbells", which can be thought of as a pair of adjacent circles fused along a diagonal.

For lack of a better term, the largest loop in Loopland is the wavy 4-leaf "clover", which necessarily encloses a circle at its center. Larger wavy loops come in a variety of shapes.

Note that each loop requires precise coordination of all tile orientations in a patch of a particular size and shape.

The triangular landscape at bottom right is a meandering "maze" devoid of loops. (Hereafter, "maze" used by itself will imply an absence of loops unless otherwise noted.) Call it "Mazeland".

"Free-form mazes" -- i.e., mazes with no particular configuration or connectedness -- take a lot less coordination of neighoring tiles than do loops. It takes even less coordination to form a patchwork of free-form mazes and loops of various sizes.

"Organized mazes" -- i.e., mazes with a particular geometry or with a high degree of connectedness throughout -- are a different story.

The linear diagonal region separating the loop and maze region is a large, highly organized maze connected throughout its length. It represents the careful coordination of no less than [] tiles in a large diagonal cluster of very specific size and shape.
The odds of getting an organized maze like this by chance are virtually nil.



Random tilings

If all 64 QCT orientations were determined by flipping a coin or tossing a binary LEGOŽ die like the one below, the result would be a random QCT tiling.



Random tilings are patchworks of largely disconnected free-form mazes of various sizes dotted with much smaller organized mazes and loops.

Circles, the smallest of loops, appear only if all 4 tiles in a 2x2 patch are oriented just so. They emerge spontaneously at an average rate of one for every 18 tiles in random QCT tilings.

For dumbbells, which require precise coordination of 3 additional neighboring tile orientations, the spontaneous emergence rate plummets to 1 for every 80 tiles.

A tiling with a lot more circles and dumbbells than these rates would suggest is unlikely to be truly random. In fact, QCT tilings with suitably encoded tile orientations have been used to dectect departures from randomness in large data sets by visual means.




Specifications

Overall dimensions:256x256x14 mm (LxWxH), including baseplate
Sketchpad dimensions:208x208x13 mm (LxWxH), excluding baseplate
Overall mass:[] kg, including baseplate
Sketchpad editing:Sliding-tile random access
Tile grid:8x8
Tile count:64 with finishing tile
Voids:1
Tile size:3x3 LU
Tile type:Quarter-circle Truchet
Tile motif:2 diagonally opposed raised quarter-circles
Tile construction:Macaroni brick motif on new 2-layer base
Modified LEGOŽ parts:
None
Non-LEGOŽ parts:None
Credits:
Original MOC




Comments

 I made it 
  June 28, 2017
Quoting AL X So interesting! And inspirative, too
Very kind!
 I like it 
  June 28, 2017
So interesting! And inspirative, too
 I like it 
  February 22, 2015
Mesmerising and rather beautiful.
 I like it 
  February 21, 2015
Fascinating puzzle art! One becomes completely Dizzy.
 I like it 
  February 21, 2015
Positively mind boggling! Awesome!
 I made it 
  February 19, 2015
Quoting David FNJ Wow, this is really awesome looking! Love how so many different shapes and patterns can be made with this! Phenomenal!
Thanks, David. The pattern-forming power in these simple little tiles really is amazing -- especially considering that all 64 are exactly alike.
 I made it 
  February 19, 2015
Quoting Turbo Charger Brilliant MOC! Very thorough presentation of your work, really shows how much dedication goes in your creations.
Thanks, Turbo Charger. These are all labors of love, but my pages really should be a =lot= shorter. To quote Ben Franklin, "I'd have written a shorter letter if I'd had more time."
 I made it 
  February 19, 2015
Quoting Matt Bace I have to completely agree with Giorgio, although the word that came to me is "mesmerizing". The connected-patch pictures are particularly interesting. This tiler reminds me of a card game from my youth in which players had to place cards with water pipes oriented in a somewhat similar manner (but I can't recall the name).
Thanks, Matt. I know exactly the card game you're referring to but can't remember the name, either. (We're dating ourselves here.) Agree, there are some strong similarities.
 I made it 
  February 19, 2015
Quoting Giorgio Ferrannini I'd say the best word to describe this MOC is HYPNOTIC. Too bad I have not the pieces to build and play with it :(
Thanks, Giorgio. The most expensive parts were the baseplate and the 8x16 tiles used to floor the frame, and there are good alternatives for both. Got the rest fairly cheaply, but it took a lot of shopping around on BrickLink.
 I made it 
  February 19, 2015
Quoting Deus "Big D." Otiosus Amazing! I thought the die tile puzzle-thing was a non-LEGO comparison, haha! As for the video, I really loved the subtly rising frustration in your movements, it made the video very climactic. Beautiful work.
Deus, I see what you mean. It does get a little fiddly at times.
Jeremy McCreary
Matt Bace
  February 18, 2015
I have to completely agree with Giorgio, although the word that came to me is "mesmerizing". The connected-patch pictures are particularly interesting. This tiler reminds me of a card game from my youth in which players had to place cards with water pipes oriented in a somewhat similar manner (but I can't recall the name).
 I like it 
  February 18, 2015
Wow! Incredible work here!
 I like it 
  February 18, 2015
Quite impressive!
  February 18, 2015
Amazing! I thought the die tile puzzle-thing was a non-LEGO comparison, haha! As for the video, I really loved the subtly rising frustration in your movements, it made the video very climactic. Beautiful work.
 I like it 
  February 18, 2015
Wow, this is really awesome looking! Love how so many different shapes and patterns can be made with this! Phenomenal!
 I like it 
  February 18, 2015
Brilliant MOC! Very thorough presentation of your work, really shows how much dedication goes in your creations.
 I like it 
  February 18, 2015
I'd say the best word to describe this MOC is HYPNOTIC. Too bad I have not the pieces to build and play with it :(
 I made it 
  February 18, 2015
Quoting Topsy Creatori Elegant! :)
Thanks, Topsy. Elegant made my day.
 I made it 
  February 18, 2015
Quoting Andrew JN This is quite cool (as well as very confusing), but hey, looks really neat!
Thanks, Andrew.
 I made it 
  February 18, 2015
Quoting matt rowntRee Mesmerizing at it's quantum level. Just simply beautiful and fun at a more fundamental state. And absolutely cool everywhere else. Awesome!
Thanks, Matt. Quantum. Hmmm. You know, I wouldn't be a bit surprised if the whole universe turned out to be a quantum Truchet tiling embedded in a subspace tachyon field.
 I like it 
  February 17, 2015
Mesmerizing at it's quantum level. Just simply beautiful and fun at a more fundamental state. And absolutely cool everywhere else. Awesome!
 I like it 
  February 17, 2015
Elegant! :)
 I like it 
  February 17, 2015
This is quite cool (as well as very confusing), but hey, looks really neat!
 I made it 
  February 17, 2015
Quoting Damon Corso Wicked cool!
Thanks, Damon!
 I like it 
  February 17, 2015
Wicked cool!
 I made it 
  February 17, 2015
Quoting Sam the first Pure genius. And I love the way you present your MOCs, especially the data at the bottom. I do like some data, I do.
Very kind, Sam. Pretty sure most visitors take one look at my write-ups and run the other way as fast as they can, and I don't blame them. The pages are way too long with way too many words. Glad somebody sees some value in them, though, as I don't seem to know how to do a MOCpage any other way. This is my brain on LEGO. And yes, data is good.
 I made it 
  February 17, 2015
Quoting Yann (XY EZ) Such a fantastic creation! Well done!
Thanks, Yann. I always appreciate your support.
 I like it 
  February 17, 2015
Pure genius. And I love the way you present your MOCs, especially the data at the bottom. I do like some data, I do.
 I like it 
  February 17, 2015
Such a fantastic creation! Well done!
 
By Jeremy McCreary
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