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We describe an N×N optical switch for use in cross-connects. It is a free-space device, based on multiple bounces in a pair of White cells sharing a spatial light modulator at one end. In a companion paper, we described various polynomial cells, in which the number of outputs was proportional to the number of bounces raised to some power. In the binary device described here, the number of possible outputs is proportional to the number two raised to the power of the number of bounces. It allows a 1024 × 1024 switch using a single digital two-state tip/tilt micromirror array, four spherical mirrors, and a spot displacement device. It is highly scalable and insensitive to micromirror pointing accuracy.