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One of the most classical models in mathematics is the Clebsch Diagonal Surface.
As any smooth cubic surface, it contains 27 lines, but for this special surface, all 27 lines are real, and very symmetric. In addition, the object shows two planes which cut the surface in one line and three lines, respectively.
Some mathematics details:
The model illustrates part of the proof of the fact that any cubic surface contains 27 lines:
Let us assume that we know that any such surface contains at least one line. One may consider all planes through this line. Any such plane cuts the given cubic surface in a cubic plane curve which factors into a line and a conic. It turns out that it happens exactly 5 times that the conic is not smooth, but factors itself into 2 lines. With some other arguments, this can be used to prove that there are exactly 27 lines.