# Bongard Problem #4 (solution)

#### Left: Convex objects.

#### Right: Concave objects.

If geometric terminology puzzles you, *concave* objects
are the ones that have at least one indentation, a notch, or a
bump inwards. *Convex* objects do not have such inward
indentations.

Notice that the original BP #4 had a small square in place of
the potato-like object at second box of top-line, left. When I
presented the original problem to a friend of mine, the answer
was unexpected. This friend found as a "rule" that all
boxes at left contain simple geometric objects, treated by
elementary geometry (such as a square, an ellipse, a triangle,
etc. -- maybe the last one can be seen as a pentagon). Since this
could arguably *count* as a rule, I changed the small
square to a potato-like object, to eliminate the ambiguity.

Another interesting note: There is a simple definition in
geometry about convexity/concavity: Convex shapes are the ones
which, no matter which straight side you choose to augment
indefinitely on the plane, this side will never intersect the
shape (or, use a tangent line if the shape has curves). This is
not true for concave shapes: there is at least one straight side
(or tangent to a curved side) which will intersect the shape if
augmented in one direction or another.

This simple definition, however, seems *not* to occur
to people when they notice convex and concave shapes (unless they
are trained in geometry). If you ask people *why* is a
shape called convex, they will usually not be in a position to
formulate an answer, while for concave shapes they may say
something like "it has indentations, or bumps, pointing
inwards". It seems to me that we tend to observe bumps,
spikes, notches, etc., *as a whole*, without the need to
discover and apply the elegant geometric definition -- if we do
not already know it.

*Back to problem #4*