How does making "pyramid hologram projectors using smartphones" work? 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,
 
 A: 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. 
A: 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
A: It's just pure reflection. 
The "hologram" is made by tricking the eyes with a 2 dimensional image. 
First of all,  look at the pyramid dimensions used to make the hologram. 

As it it is clear from the diagram,  the pyramid is built in such a way that its every sides makes an angle of about $50^0$ with the smartphone screen.
This is well above the critical angle of most plastics (highest refractive index for a polymer till date is  1.76, according to Google.) 
So,  the pyramid plays a double role here. 


*

*Projects the image on the screen upright 

*Allows the background objects to be seen through it. 


The combined effect gives us an illusion of the 2D jellyfish floating around in your familiar surroundings. 
