Existence is an interesting concept. Even in the most rigorous of sciences, it is not free of dispute.
Above is an image of the Horgan surface, a minimal surface that does not exist, at least not with all the glorious property it appears to have. There are less convincing attempts to convey the existence of a non-existing object, like the catenoid with a handle:
In Mathematics, pictures provide evidence and enjoyment. Existence or non-existence requires proof, and provides certainty. Still, the membrane between to be and not to be can be very thin, as in the example at the top.
The pattern made by the catenoidal necks in the top image – two up and two down, at the corners of a square – does actually occur. One of the first examples I have seen is the Callahan-Hoffman-Meeks surface with an added handle as above. Its existence is not so easy to prove – does it therefore exist a little less?
The triply periodic surface above is the simplest version of the Catenoid-Square pattern I know of, and much more recent. So, in a sense, all this research is to an unconfessed extent concerned with probing the border of existence, from either side.