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The basis of this question is this video, where a 3D hologram like effect is achieved using a 4 sided hollow pyramid. I don't know if this is holography in the usual sense. Can some one explain this phenomenon in a detailed, mathematical way using principles of optics ?

There is a partial explanation here, but I did not find it rigorous or self contained

Here is a picture of the final effect, enter image description here

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Please explain more precisely in the question what the effect is like. I should not have to watch a video to understand a question, and "a 3D hologram-like effect" isn't really very specific. – ACuriousMind Feb 24 at 11:58

it is just 2D image available in 4 direction. As in some spectacle and illusions, using glass or mirror gives you the illusion of a floating image (glass is better because you see through instead of seeing the reflection of the dark areas of the phone screen).

note that when the camera turn around, the display is very well chosen so that there is content mostly when you face one of the 4 directions.

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Simple reflection of a symmetrical image.

A more sophisticated version would be Pepper's Ghost which uses a partially silvered mirror to create the illusion of 3D. Prsumably also the origin of the phrase "all done with mirrors". The best modern example I know of are the concerts of Hatsune Miku

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