What to do · Section A of arXiv:2410.09574v2
is the selected boundary corner; are its neighbors. starts at and moves to grow the quadrilateral , as in Section A of the paper. Double-click a red dashed ray to play. Drag the progress slider to inspect the motion, then choose a ray in a remaining polygon. Hover matching objects; drag to pan and scroll to zoom. Hold A and scroll to change speed; F toggles fullscreen.
The construction. The notation follows Section A of arXiv:2410.09574v2: is the selected boundary vertex, its two neighbors, and the moving fourth corner of . Initially . The green circle moves until it reaches the first permitted candidate , shown in purple. Finishing the move adds the quadrilateral and leaves smaller polygons to fill. The ray moves with and splits into two rays when two polygons remain at that corner.
The pictures. “Before folding” shows the growing quadrilateral and construction circles. “Inverted circles” applies : circles through become lines. The green line turns while touching the fixed orange circle . “After folding” shows the origami images of the same marked points. Hover a point or circle to highlight its counterpart.
Controls. Click a red dashed ray to select it; double-click to play. The ↤ / ↦ buttons jump to the beginning / end of the selected move. Reaching the end records it automatically. “Play all” fills the remaining polygons. Space plays or pauses; Cmd/Ctrl+Z or ← plays backward; Cmd/Ctrl+Shift+Z or → plays forward. Press again during motion to finish it and start the next move in that direction. Adjust speed with the slider or hold A while scrolling. F toggles fullscreen; R restarts. “Hide controls” and “Hide details” enlarge the drawings while keeping playback visible.
Permitted tangency. Normalize . A circle of signed height has radius , counterclockwise orientation for positive height and clockwise orientation for negative height. The stopping condition is . An earlier touch on the opposite side does not stop the motion. “Show all candidate circles” includes the same-parity candidates; every candidate enters the calculation even when hidden. The table below lists their meeting parameters.
Examples and display. Choose a library polygon or generate one using an even number of vertices and a letter seed. The same number and letters reproduce the same polygon. Swapping and changes the inversion pole and rotation direction while preserving the growing quadrilateral. Drag to pan; scroll to zoom. Text size, arrow size and hover radius adjust visibility.
Candidate circles and numerical checks
| Circle | Boundary vertex | Meeting parameter | Required side |
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