Can manholes be made in other shapes than circles, that prevent the cover from being able to fall down its own hole? Circular manholes are great because the cover can not fall down the hole. If the hole were square, the heavy metal cover could fall down the hole and kill some man working down there.
Circular manhole: 
Can manholes be made in other shapes than circles, that prevent the cover from being able to fall down its own hole?
Semi rigid math formulation:
Let us say that we have an infinite matematical 2D plane in 3D space. In this plane is a hole of some shape. Furthermore we have a flat rigid 2D figure positioned on one side of the plane. This figure has the same shape as the hole in the plane, but infinitesimal larger.
Is it possible to find a shape, where there is no path twisting and turning the figure that brings the figure through the hole?
Here is one such shape (only the black is the the shape):
But if one put the restriction on the shape, that it needs to be without holes (topological equivalent to a circle in 2D), then I can not answer the question!?
Edit:
Because of the huge amount of comments and answers not about math, I fell the need to specify that:
I am not interested in designing manholes. I am interested in the math inspired by the manhole problem.
 A: Q Are non-circular manholes possible?
A Yes

Q A better question is why are non-circular manholes less practical?
A Corners are weakest part of a lid and consume more material cost.
Round means no corners.
A: Here are two examples of shapes, that does not fit through their own holes (I think). But it does not help to get any closer to the general answer.


Edit: Ok - second shape is not good. But maybe the basic idea is still good - I will think some more.
A: A manhole cover can't fall into the hole if the minimum width of the cover is greater than the maximum width of the hole.
For example, consider a one-meter square cover over a square hole slightly smaller than $1\over\sqrt 2$ meter on a side. The diagonal of the hole is slightly less than 1 meter, so the cover won't fit into it.
The point is that manhole covers aren't the same size as the manholes they cover; they have flanged edges.
EDIT :
Oops, I missed this sentence in the question:

This figure has the same shape as the hole in the plane, but infinitesimal larger.

so my answer, though it does have real-world applications, doesn't really answer the question as stated.
A: This question was frequently asked on technical interviews for software engineering positions, up until developers started using counterfactual reasoning. 
There is an excellent article "If Richard Feynman applied for a job at Microsoft" showing that there is actually very little practical link between manhole shape and it's conventional representation as a circle.
If I may, I would like to throw a few quotes:

Interviewer: Why are manhole covers round?
Feynman: They're not. Some manhole covers are square. It's true that there are SOME round ones, but I've seen square ones, and rectangular ones.
  
Interviewer: I mean, why are there round ones at all? Is there some
  particular value to having round ones? 
Feynman: Yes. Round covers are used when the hole they are covering up
  is also round. It's simplest to cover a round hole with a round cover.
  
Interviewer: Do you believe there is a safety issue? I mean, couldn't
  square covers fall into the hole and hurt someone?
Feynman: Not likely. Square covers are sometimes used on prefabricated
  vaults where the access passage is also square. The cover is larger
  than the passage, and sits on a ledge that supports it along the
  entire perimeter. The covers are usually made of solid metal and are
  very heavy. Let's assume a two-foot square opening and a ledge width
  of 1-1/2 inches. In order to get it to fall in, you would have to lift
  one side of the cover, then rotate it 30 degrees so that the cover
  would clear the ledge, and then tilt the cover up nearly 45 degrees
  from horizontal before the center of gravity would shift enough for it
  to fall in. Yes, it's possible, but very unlikely. The people
  authorized to open manhole covers could easily be trained to do it
  safely. Applying common engineering sense, the shape of a manhole
  cover is entirely determined by the shape of the opening it is
  intended to cover.

A: A solution nobody has mentioned yet is to make the cover in the shape of a cone.  The hole can be any shape at all as long as the cover is an appropriately-shaped cone; if the hole is square, for example, then the cone is actually a square pyramid.  Such a cover can fall into the hole, but not all the way in, unless the hole is sufficiently  large that the base of the cone fits through, in which case the results could be spectacular.
A: Any manhole cover bounded by a curve of constant width will not fall through.  The circle is the simplest such curve.
A: Sorry, maths is hard so I posted a picture. These are the opposite of what was suggested in that they always fall down the hole.  Triangular man holes.  These are proper heavy duty ones that go in main carriageways.  The semi clever point of design is the placement of the lifting eyes.  They are at the mass centroid of each triangle so that you can lift them vertically.  They still fall down the hole if you try it though...

A: I hope manhole of equilateral Triangle Shape will also work.
