HackingTheUniverse

You are browsing the archives of group theory.

Ars Magna (The Great Art) published in 1545 by Girolamo Cardano, included techniques for solving cubic (to the third power) and quartic (to the fourth power) equations. The solution for cubic equations was developed by Scipione del Ferro, then passed on to a student who provoked Niccolo Fontana (aka Tartaglia) to also develop the solution. [...]

Topology is the study of how spaces are shaped and connected without worrying about distances. Some properties of spaces are preserved even when the space is being stretched and deformed. Often, the space that is being studied is called a manifold. A line and a circle are simple one dimensional topological manifolds. A plane and [...]

In 1872, Felix Klein, at the age of 23, having just been appointed as a professor at the University of Erlangen in Germany, published a manifesto that described a new direction for geometry. Klein was attempting to unify classical Euclidean geometry with the new non-Euclidean geometries and some elements of topology by showing that some [...]

Group theory is about groups of objects that have similar characteristics. Objects can show different kinds of symmetry when they are transformed by a standard set of operations. A basic transformation operation is turning (called rotation in the world of symmetry). When an object is rotated, depending upon the shape of the object and the [...]

Niels Henrik Abel was born in 1802 in Norway and became a mathematician. He tried and failed to find a solution for the quintic equation and then completed Ruffini’s incomplete proof that it could not always be solved using radicals. This is now known as Abel’s impossibility theorem or the Abel–Ruffini theorem. Jacobi and Abel [...]

Paolo Ruffini was born in Italy in 1765 and was a mathematician. After the cubic equation was solved in 1539 by del Ferro, Tartaglia and Cardano, and Ferrari had solved the quartic equation by 1545, mathematicians turned their attention toward attempting to solve the quintic. The previous polynomial equations (quadratic, cubic and quartic) had all [...]

In 1887, Willhelm Killing came up with a classification scheme for Lie algebras. The designation E-8 was eventually applied to an exceptional, simple Lie algegra and the associated Lie group of rank 8 and dimension 248. This is one of the most complex objects known to mankind. Mathematicians Map E8 – [aimath.org] Mathematicians have mapped [...]

Born in Erlangen, Bavaria, Germany in 1882, Emmy Noether was a mathematician who laid the groundwork for the idea of symmetry in mathematics and physics. She studied the works of Hilbert, Klein and Minkowski and was peer with Hermann Weyl. Noether’s theorem – in 1915, she proved a relationship between symmetries in physics and conservations [...]

“Mathematics is the queen of sciences and number theory the queen of mathematics. She often condescends to render service to astronomy and other natural sciences, but in all relations she is entitled to the first rank.” Johann Carl Friedrich Gauss was born in Braunschweig, Germany in 1777. He made great contributions to mathematics, but was [...]

Bertrand Arthur William Russell was born in Trellech, Monmouthshire (Wales), England in 1872 and was a mathematician, a philosopher, a historian and an author of many books. He is considered to be one of the great logicians of his time and co-wrote, “Principia Mathematica” with Alfred North Whitehead. In 1901 he discovered a contradiction in [...]