Cube Folding — Fundamental Concepts

Fundamental Concepts

Choose a square as the front or base and fold its neighbours through right angles. Keep a six-position table so that a square cannot silently occupy two positions. If following different paths through the layout produces conflicting orientations, or two squares occupy the same side, the proposed net is invalid. Six squares provide the correct area but do not alone prove foldability.

In the worked net A–B–C–D, E sits above B and F below it. Holding B as front puts A left, C right, D back, E top and F bottom. Opposite pairs are therefore A–C, B–D and E–F. A and D, although far apart in the flat row, meet at the closing seam. This is a useful example of why page distance cannot determine opposition.

A three-face corner cannot contain an opposite pair. Thus A, C and E cannot appear together around one corner, whereas A, B and E can. When a question specifies top, front and right positions, also check their order: a mirror arrangement can preserve the same opposite pairs while reversing the cycle around a face.

Rigid rotation preserves both geometry and handedness. Use the opposite-pair test as a quick rejection tool, then complete the orientation check when the option supplies ordered positions.