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An illusion due to Franz Joseph Delboeuf, a Belgian philosopher and expert on hypnosis. If the same circle is placed inside two different concentric circles, its size will ...

A curve which can be turned continuously inside an equilateral triangle. There are an infinite number of delta curves, but the simplest are the circle and lens-shaped ...

If, in an interval of x, sum_(r=1)^(n)a_r(x) is uniformly bounded with respect to n and x, and {v_r} is a sequence of positive non-increasing quantities tending to zero, then ...

sum_(k=-n)^n(-1)^k(n+b; n+k)(n+c; c+k)(b+c; b+k)=(Gamma(b+c+n+1))/(n!Gamma(b+1)Gamma(c+1)), where (n; k) is a binomial coefficient and Gamma(x) is a gamma function.

A dodecagon is a 12-sided polygon. Several special types of dodecagons are illustrated above. In particular, a dodecagon with vertices equally spaced around a circle and with ...

The ending of a game in which neither of two players wins, sometimes also called a "tie." A game in which no draw is possible is called a categorical game.

The intersection of an ellipse centered at the origin and semiaxes of lengths a and b oriented along the Cartesian axes with a line passing through the origin and point ...

A 1-cusped epicycloid has b=a, so n=1. The radius measured from the center of the large circle for a 1-cusped epicycloid is given by epicycloid equation (◇) with n=1 so r^2 = ...

The parametric equations of the evolute of an epitrochoid specified by circle radii a and b with offset h are x = ...

Line segment ranges and pencils which have equal cross ratios are said to be equicross.