Having folded Steven Casey’s 8×8 40 grid seamless chessboard and singularly failing to fold Marc Kirschembaum’s 40 grid because of crease-creep inaccuracies, I was approached by Daniel Brown and asked if I was interested in his chessboards – naturally I jumped at the chance. “Seamless” chessboards are deliciously more complicated because it required each square to be represented by an un-broken surface (as opposed to being able to be comprised of bits and pieces of layers – a much easier path):

I say CHESSBOARDS because Daniel has developed a series of coloured/white alternate seamless models of LOTS of sizes, and the skills necessary to migrate edge paper towards the centre to effect colour changes is a thing that needs some work and, often, particular “widgets” (or self-contained localised fold structures).

I started with the 4×4, rather efficiently designed on a 9×9 grid ( 0.444 efficiency). I had a piece of blue-white kami, so gave it a whirl. Even dimensions require different approaches for adjacent corners as they are different colours – the same colour corner exists on the diagonal.

Colour reverse widgets for color/white corners and some pleat migration set up an asymmetric back that lets the 4×4 happen.

Moving to the 5×5 and I started with a square of flimsy blue/white dollar store wrapping paper (I bough a roll on a whim). We started with a 13 grid (once again, so elegant and efficient, but I used a ruler to do 13th divisions) – 0.38 efficiency.

Odd board dimensions allow you to have all 4 corners similar – in this case you need a 2×2 white corner widget on each corner, a delicious (but not re-used) open square twist for the middle white square and then the introduction of what I have termed the “bastard twist” – whereby you dig out a white square from the middle of a triple pleat as the “connection” between each of the 2×2 corners. The 5×5 bastard twists were relatively easy as they were right on the edge, making the flipping out of the colour much easier than later.

The “infrastructure” necessary to efficiently mobilise colour change begins to emerge at the 5×5 – the back centre having a “tessellated” feel to create some beep pleats that let you migrate and reverse surface colour change very neatly. The only downside is that the accuracy of your grid ends up being the make or break of the fold as inaccuracy causes alignment to become an issue.

Taking another square of blue/white wrapping paper split from the roll, the 6×6 board requires an 18×18 grid = 0.333 efficiency (I folded 9ths and then halved that, but there are a few ways to get this awkward division). I was introduced to a standard starter for a 2×2 white corner, followed by 2 re-usable widgets – the “secret elephant” which is a way of stretching colour toward the centre of the board with a 3 square long trunk and parts of a “centre structure” that contains a trough you can weave the coloured elephant trunk in-between raised white squares to effect much of the board.

Because it is an even-dimensioned board, we need different strategies for adjacent corners – so we have a 2×2 white corner widget and a 3×3 secret elephant-based coloured corner. Interfacing these two widgets you need an internal “bastard twist” one layer in from the edge on all 4 edges to make up the 6×6 board (a tricksey move with such flimsy paper). Infrastructure pleats on the reverse set up “cradles” and raise white square in the interior of the board in clever, dense and re-usable ways.

The 7×7 board required a 25×25 grid (0.28 efficiency). As it is an odd dimensioned board, we can use 4 of the same widget for each of the white corners but this strategy has limitations that require a raised white square for each corner – that requires a new infrastructure structure, but this can be re-used for the next board, so is a useful skill. This raised white square straddles coloured layers, effectively allowing you to spread the edge close to the centre, but necessitates a set of bastard twists to surface connecting white squares, Interestingly the middle square is a single layer.

25 grids eat lots of paper – I used a 60cm square, the resultant squares on the finished board were just over a cm square – waaay too small to play on.

Interestingly, apparently Daniel discovered the 23-grid solution to the 7×7 chessboard while i was part way through this series – these designs are really FRESH and innovative – serious respect must go to him for working with me on this.

I decided that a change to a more substantial paper was necessary for the 8×8 board on a 28 grid (0.286 efficiency). I used a 70cm square of white/natural Kraft paper – my goto paper is Kraft, it is tough and I am quite good at coaxing it through tough processes – more importantly, it has good “fold memory” – that is it remembers the crease orientation of how it was folded – important because this dimensioned board required some pre-folding.

I was given 4 component test folds – the central infrastructure fold (folded first, creases SET, then unfolded), a white corner test fold that has a 3×3 whose point straddles a raised square (part of the infrastructure) and a coloured corner which was a secret elephant with added eyes and ears – effectively elongating the penetration of the trunk so it in combination with the coloured head gives us a 4-square penetration, weaving in through troughs in the infrastructure centre in clever ways. The 3×3 white and 4×4 coloured corners require a “connection” fold that includes a deep coloured square and bastard twists one row from each of the 4 edges to add the missing white squares.

I practiced the 4 components (each separately on 16 grids) before attempting to combine them on the one sheet folded from an astonishing 28 grid. 28 GRID!! Considering Casey and Kirschembaum’s boards require 40 grid, the efficiency here is pretty astonishing.

The central infrastucture alone is beautiful – a cursed corrugation that looks like an echidna orgy. I folded it with the intention of completing it quickly – then life got in the way so it remained folded for over a week while I travelled to Melbourne for 8OSME and Folding Australia Conferences. This had an unexpected bonus – the creases for the central infrastructure were well and truly set (making their progressive reformation MUCH easier in retrospect … all part of a plan, right?)

Once completely unfolded, there is then an intricate dance required to partially fold the coloured corner secret elephants simultaneously, untangling the pleat arguments as you go, then you partially re-collapse the outer section of part of the infrastructure, then partially fold the white corners simultaneously, sorting out the escalating pleat arguments and gently excavating the raised squares from teh infrastructure, then alternating coloured corner and white corner collapses while re-forming the remainder of the infrastructure – a sort of “everything, all at once” collapse.

You then need to perform 4 bastard twists – alternatively forwards then backwards (this did my head in as the bastard twist is complex enough, doing it mirrored was topologically challenging – to me at least). Once the bastards were in place, then the white corners need to straddle their raised squares, elephant trunks are woven through the infrastructure troughs, then the elephant ears are woven in to effect the final 4 missing coloured edge squares … easy (not!).

Well done if you have made it thus far … it has been a little TLDR;

I have really enjoyed this fold journey – it feels like I have been folding chessboards for months (in truth, it has been my July task), and through all of this I have been in awe of the design genius that is Daniel Brown. His hand-drawn progress diagrams had nearly no annotations – requiring me to actually understand what I was trying to do, using the layers I had – I think that made me a better folder, as did my practice with terrible flimsy wrapping paper – forced to be gentle and mindful BEFORE assaulting the paper.

I have no doubt Daniels exhaustive exploration of colour change patterns – work he and Jason Ku documented in a paper that was presented at 8OSME – aided him in the strategies for efficiently planning these seamless designs. The ability to visualise layer utilisation potential is a particular mind space that is not trivial – gotta love specialist knowledge and the specialists more so.

Seamless chessboards and regular colour change patterns are very interesting to the origami and mathematics community, and there is a mini competition happening in terms of efficiency chasing – you can check progress live to this – https://origamimagiro.github.io/checkerboards/