Radial fold
Every other fold on this site runs in parallel strips and selects by where you stand sideways. Take the same pleat into polar coordinates and two things change, one awkward and one worth the trouble.
The awkward one first. A pleated disc cannot lie flat. Folding shortens a circle while the radial creases keep every point its full distance from the centre, and no plane holds both, so the sheet stands up as a cone. Parallel strips come off a flat sheet for free; these do not come off one at all.
The reward is a selector nothing else here has. Pleat the disc as concentric rings and every direction round it is alike, so walking past does nothing. But the ray out to the rim leans further the nearer you stand, and one family of facets goes edge-on at a radius of D·cot θ. That ring travels as you move. A clean image blooms inward from the rim as you approach and greys back into a blend as you retreat, which a parallel fold cannot do at any angle.
Creases are circles and the facets tilt in and out, so every direction round the piece is alike and moving sideways changes nothing. What changes it is distance: the ray out to the rim leans further the closer you stand, and one family of facets goes edge-on at a radius that travels as you walk.
Where you stand
Drag the piece to move round it; scroll to walk in and out. On a target, scroll slowly.
- Standing
- 0.90 m out
- Sideways
- 0°
- Resolving ring
- 328 mm
- Rim reads
- 73% one image
- Rim resolves within
- 412 mm
Past the resolving ring one image has the surface to itself; inside it the two mix, evenly at the centre. Since the ring sits at D·cot θ it shrinks as you approach, so a clean picture blooms inward from the rim and greys out again as you step back. The rim only ever resolves from inside 412 mm, which is why polar folds want to be steeper than parallel ones. At 45° this disc would need you 150 mm away.
Plates
Using the house set. Load your own to replace any of them
Plate 1
The image that survives out at the rim as you walk in.
Plate 2
Contrast with the others in colour and composition.
Parameters
It will not lie flat
- Cone half-angle
- 20°
- Circumference lost
- 65.8%
- Pleat depth at rim
- 14.7 mm
Pleating shortens a circle. At radius r the paper still holds an arc of 2πr, but folded at 70° it presents only 2πr·cos θ of circumference, while the radial creases, never bent along their length, keep that point its full r from the centre. No plane satisfies both, so the sheet becomes a cone of half-angle 90° − θ, here 20°. Only a fold angle of zero leaves it flat.
This is the real difference between polar and parallel folding. Parallel strips come out of a flat sheet for free, which is what the fold bench prints. These are not developable at all: you either accept the cone, or cut 65.8% of the circumference away as gores and join them.