Suppose a convex body wants to pass through a circular hole in a wall. Does its ability to do so depend on the thickness of
the wall? In fact in most cases it does, and in this paper we present a sufficient criterion for a polytope to allow an affirmative
answer to the question.
Some theorems from inversive and Euclidean circle geometry are extended to all affine Cayley-Klein planes. In particular,
we obtain an analogue to the first step of Clifford’s chain of theorems, a statement related to Napoleon’s theorem, extensions
of Wood’s theorem on similar-perspective triangles and of the known fact that the three radical axes of three given circles
are parallel or have a point in common. For proving these statements, we use generalized complex numbers.
Let R be a ring and define x ○ y = x + y - xy, which yields a monoid (R, ○), called the circle semigroup of R. This paper investigates the relationship between the ring and its circle semigroup. Of particular interest are the cases
where the semigroup is simple, 0-simple, cancellative, 0-cancellative, regular, inverse, or the union of groups, or where
the ring is simple, regular, or a domain. The idempotents in R coincide with the idempotents in (R, ○) and play an important role in the theory developed.
and between the two world wars, when ethnic nationalism continued to shape the Czechoslovak Republic. Rooted in the late middle ages, literary societies had been widespread in almost every town and city of Europe ever since the ﬁrst such circles