The continuum hypothesis is a hypothesis, advanced by Georg Cantor, about the possible sizes of infinite sets. Cantor introduced the concept of cardinality to compare the sizes of infinite sets, and he showed that the set of integers is strictly smaller than the set of real numbers. The continuum hypothesis states the following:
- There is no set whose size is strictly between that of the integers and that of the real numbers.
Or mathematically speaking, noting that the cardinality for the integers $|\mathbb {Z} |$ is $\aleph _{0}$ ("aleph-null") and the cardinality of the real numbers $|\mathbb {R} |$ is $2^{\aleph _{0}}$, the continuum hypothesis says
- $\nexists \mathbb {A} :\aleph _{0}<|\mathbb {A} |<2^{\aleph _{0}}.$
This is equivalent to:
- $2^{\aleph _{0}}=\aleph _{1}$
The real numbers have also been called the continuum, hence the name. (Full article...)
A
Bézier curve is a
parametric curve important in
computer graphics and related fields.
Widely publicized in 1962 by the
French engineer
Pierre Bézier, who used them to design
automobile bodies, the curves were first developed in 1959 by
Paul de Casteljau using
de Casteljau's algorithm. In this animation, a
quartic Bézier curve is constructed using
control points P
_{0} through P
_{4}. The green line segments join points moving at a constant rate from one control point to the next; the
parameter t shows the progress over time. Meanwhile, the blue line segments join points moving in a similar manner along the green segments, and the
magenta line segment points along the blue segments. Finally, the black point moves at a constant rate along the magenta line segment, tracing out the final curve in red. The curve is a
fourth-degree function of its parameter.
Quadratic and
cubic Bézier curves are most common since higher-
degree curves are more computationally costly to evaluate. When more complex shapes are needed, lower-order Bézier curves are patched together. For example, modern
computer fonts use
Bézier splines composed of quadratic or cubic Bézier curves to create scalable
typefaces. The curves are also used in
computer animation and
video games to plot smooth paths of motion. Approximate Bézier curves can be generated in the "real world" using
string art.