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Why must an integrating sphere be a sphere? Why can't it be an integrating cube? What is the difference? Could I use a cube to measure total illuminance like an integrating sphere does?
Surface coating of an integrating sphere is optimized for low losses. This white coating (barium sulfate or PTFE) acts like an ideal lambertian scatterer.
First generation stray light (blue in OP's picture) shows this light cone. Imagine this cone at the corner of a cube: some light will hit a wall again and suffers tiny losses. Detector port in cubic geometry hat a lower propability to to be hit with the ray of highest energy. With a sphere however all surface normal vectors point to its center. Remember, that these rays "carry more energy" according to Lambert's cosine law. It will have lower losses than a measurement head with a cube geometry. A spherical geometry reduces the necessary number of stray events.
In a sphere, any light emitted from the center will reflect off the sides at normal incidence come back to the center. In a cube, some rays never return to the center, so you aren't measuring all of the light emitted, which defeats the purpose of the device.
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