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In mathematics, a geometric algebra (also known as a Clifford algebra) is an algebra that can represent and manipulate geometrical objects such as vectors. Geometric algebra is built out of two fundamental operations, addition and the geometric product. Multiplication of vectors results in higher-dimensional objects called multivectors ...
The Greeks created a geometric algebra where terms were represented by sides of geometric objects, [16] usually lines, that had letters associated with them, [17] and with this new form of algebra they were able to find solutions to equations by using a process that they invented, known as "the application of areas". [16] "
However, al-Khwarizmi did not use symbolic or syncopated algebra but rather "rhetorical algebra" or ancient Greek "geometric algebra" (the ancient Greeks had expressed and solved some particular instances of algebraic equations in terms of geometric properties such as length and area but they did not solve such problems in general; only ...
[1] [2] Greek mathematicians lived in cities spread over the entire region, from Anatolia to Italy and North Africa, but were united by Greek culture and the Greek language. [3] The development of mathematics as a theoretical discipline and the use of deductive reasoning in proofs is an important difference between Greek mathematics and those ...
The operations of geometric algebra have the effect of mirroring, rotating, translating, and mapping the geometric objects that are being modeled to new positions. The Clifford torus on the surface of the 3-sphere is the simplest and most symmetric flat embedding of the Cartesian product of two circles (in the same sense that the surface of a ...
A demand for conic sections existed then and exists now. The development of mathematical characterization had moved geometry in the direction of Greek geometric algebra, which visually features such algebraic fundamentals as assigning values to line
Title page of Pappus's Mathematicae Collectiones, translated into Latin by Federico Commandino (1588).. Pappus of Alexandria (/ ˈ p æ p ə s / ⓘ; Ancient Greek: Πάππος ὁ Ἀλεξανδρεύς; c. 290 – c. 350 AD) was a Greek mathematician of late antiquity known for his Synagoge (Συναγωγή) or Collection (c. 340), [1] and for Pappus's hexagon theorem in projective geometry.
Fragments of a work by Diocles entitled On burning mirrors were preserved by Eutocius in his commentary of Archimedes' On the Sphere and the Cylinder and also survived in an Arabic translation of the lost Greek original titled Kitāb Dhiyūqlīs fī l-marāyā l-muḥriqa (lit. “The book of Diocles on burning mirrors”). [3]