We are all very familiar with planar geometry – we see, for instance, a square or rectangle is a plain shape with all 4 corners being right angles (90 degrees). Curved space gets a LOT weirder:

It is possible, for instance, to construct a shape on a curved surface that has 5 (or even 6) corners, each having a right angle. Origami typically deals with sheets that start flat – a non-flat sheet affords fascinating properties.

After a conversation with Goran Konjevod (@foldsome), I wanted to try a technique he pioneered involving radially pleating such a non-Euclidean square.

I have previously attended workshops online and in person (during 8OSME) with Goran, and am particularly enamored with the technique of raising curved ridges on a pleat field. Goran at a recent online session surfaces a piece he made on a 6-sided square and I was determined to try something similar.

I cut a square from some of my stock Kraft paper, divided it into quarters through all major meridians, then split one of the horizontal meridians to the centre of the sheet and GRAFTED a new square into the split. This created a sheet that would not lie flat (it had a lump of extra paper preventing it from doing so.

I then began dividing the sheet with mountain folds radiating out from the centre, taking care to regularly divide existing creases in half. When complete, for each mountain fold I then formed a small valley (a pleat) that started at the centre and became wider the closer it was to the outside edge of the sheet. Initially I guestimated the pleat depth would need to be about 1/5 of the gap, with the aim of caching the lump of extra paper so the whole folded structure lay flat. In practice, the pleat depth was less than 1/5 the distance, and because I was estimating this there was some residual sheet concavity … nevertheless I proceeded.

Once the pleat field was established, I flipped the sheet and, using the point of my bone folder I described an oval off-centre near the middle of the sheet. Flipping back, I was able to RAISE a curved fold spanning the pleat field that stands proud. The theory behind this is that we use some of the paper cached inside each pleat to provide the material to lock in the ridges – so deliciously weird.

Once I placed the central oval, I repeated the flip – score – flip – raise process for a containing circle.

The resultant piece is sculpturally interesting, textural and appears impossible. I must explore this more.