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The Gauss-Bonnet formula has several formulations. The simplest one expresses the total Gaussian curvature of an embedded triangle in terms of the total geodesic curvature of ...

The Kakeya needle problems asks for the plane figure of least area in which a line segment of width 1 can be freely rotated (where translation of the segment is also ...

Find the plane lamina of least area A which is capable of covering any plane figure of unit generalized diameter. A unit circle is too small, but a hexagon circumscribed on ...

Roughly speaking, a tangent vector is an infinitesimal displacement at a specific point on a manifold. The set of tangent vectors at a point P forms a vector space called the ...

An obtuse triangle is a triangle in which one of the angles is an obtuse angle. (Obviously, only a single angle in a triangle can be obtuse or it wouldn't be a triangle.) A ...

Circumscribe a triangle about a circle, another circle around the triangle, a square outside the circle, another circle outside the square, and so on. The circumradius and ...

Consider a game, first proposed by Nicolaus Bernoulli, in which a player bets on how many tosses of a coin will be needed before it first turns up heads. The player pays a ...

If two similar figures lie in the plane but do not have parallel sides (i.e., they are similar but not homothetic), there exists a center of similitude, also called a ...

Given a simplex of unit content in Euclidean d-space, pick n>=d+1 points uniformly and independently at random, and denote the expected content of their convex hull by ...

Square line picking is the selection of pairs of points (corresponding to endpoints of a line segment) randomly placed inside a square. n random line segments can be picked ...