Solution
Several of you observed that any ray of light that enters from below will exit in the exact same direction above in the same vertical line. This makes the object an "invisible object"!
A more challenging question might have been: Find a similar setup — the same geometry — using points with integer coordinates. John Sullivan found one in the triangle A = (11, 0, 0), B = (5, 0, 12), C = (11, 0, 24). Indeed, he worked out the whole story. His formula appears below.
An animation and discussion, with a reference to a recent paper, can be found at http://www.etudes.ru/en/etudes/inviz/. Watch the first and last few seconds of the animation to see what happens when the object is rotated into position.
Nikolai Andreev, who told me about this, tells me that his institute has built a working model but, because of the need for parallel rays, it works only when viewed from a distance. It would be quite easy to make the required shape using a 3D printer. But getting perfectly reflecting faces is not so easy. In any case, this is a rather cool idea for getting an invisible objet. The web site above has more information, including a reference to a recent research paper on the subject.
The trigonometry work needed to work out the solution is easy. The diagram below shows the case with rational coordinates, but scaling up yields Sullivan's integer solution, given above.
John Sullivan's formula is this:
Start with an isoceles triangle ABC of base BC having length 2 and height t < 1 (where t is for the tangent of the base angle). Place it in the first quadrant of the xz-plane with vertical base and with A left of the base and at distance a from the z-axis.
We then want t + 2a to be the tangent of the double angle:
t + 2a = 2t/(1 − t²)
So,
a = t (1/2 + 1/(1 − t²))
Thus the vertices are (t + a, 0, 0), (a, 0, 1), (t + a, 0, 2).