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Mathematics is the study of representing and reasoning about abstract objects (such as numbers, points, spaces, sets, structures, and games). Mathematics is used throughout the world as an essential tool in many fields, including natural science, engineering, medicine, and the social sciences. Applied mathematics, the branch of mathematics concerned with application of mathematical knowledge to other fields, inspires and makes use of new mathematical discoveries and sometimes leads to the development of entirely new mathematical disciplines, such as statistics and game theory. Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, and practical applications for what began as pure mathematics are often discovered. ( Full article...)

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animation of the construction of a fourth-degree Bézier curve
animation of the construction of a fourth-degree Bézier curve
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 P0 through P4. 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.

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The frontispiece of Sir Henry Billingsley's first English version of Euclid's Elements, 1570
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Euclid's Elements ( Greek: Στοιχεῖα) is a mathematical and geometric treatise, consisting of 13 books, written by the Hellenistic mathematician Euclid in Egypt during the early 3rd century BC. It comprises a collection of definitions, postulates ( axioms), propositions ( theorems) and proofs thereof. Euclid's books are in the fields of Euclidean geometry, as well as the ancient Greek version of number theory. The Elements is one of the oldest extant axiomatic deductive treatments of geometry, and has proven instrumental in the development of logic and modern science.

It is considered one of the most successful textbooks ever written: the Elements was one of the very first books to go to press, and is second only to the Bible in number of editions published (well over 1000). For centuries, when the quadrivium was included in the curriculum of all university students, knowledge of at least part of Euclid's Elements was required of all students. Not until the 20th century did it cease to be considered something all educated people had read. It is still (though rarely) used as a basic introduction to geometry today. ( Full article...)

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General Foundations Number theory Discrete mathematics

Algebra Analysis Geometry and topology Applied mathematics

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