In 1974, the brilliant Hungarian professor of architecture, Ernö Rubik, invented the puzzle that has made his name familiar all over the world. When it was mass-produced, from 1980 onwards, the cube became an international craze; vast numbers were made and they brought both great fun and great frustration to millions of children of all … Continue reading Rubik’s Cube Solvable in 20 Moves
Tag: Group Theory
The Potency of Pattern: Mind the Gap
In his book A Mathematician’s Apology, leading British mathematician G H Hardy wrote “A mathematician, like a painter or poet, is a maker of patterns.” He observed that the mathematician's patterns, like the painter's or the poet's, must be beautiful; beauty is the acid test [TM245 or search for “thatsmaths” at irishtimes.com]. Everyone is familiar with the concept of … Continue reading The Potency of Pattern: Mind the Gap
A Grand Unification of Mathematics
There are numerous branches of mathematics, from arithmetic, geometry and algebra at an elementary level to more advanced fields like number theory, topology and complex analysis. Each branch has its own distinct set of axioms, or fundamental assumptions, from which theorems are derived by logical processes. While each branch has its own flavour, character and … Continue reading A Grand Unification of Mathematics
John Horton Conway: a Charismatic Genius
John Horton Conway was a charismatic character, something of a performer, always entertaining his fellow-mathematicians with clever magic tricks, memory feats and brilliant mathematics. A Liverpudlian, interested from early childhood in mathematics, he studied at Gonville & Caius College in Cambridge, earning a BA in 1959. He obtained his PhD five years later, after which … Continue reading John Horton Conway: a Charismatic Genius
The Brief and Tragic Life of Évariste Galois
On the morning of 30 May 1832 a young man stood twenty-five paces from his friend. Both men fired, but only one pistol was loaded. Évariste Galois, a twenty year old mathematical genius, fell to the ground. The cause of Galois's death is veiled in mystery and speculation. Whether both men loved the same woman … Continue reading The Brief and Tragic Life of Évariste Galois
Motifs: Molecules of Music
Motif: A short musical unit, usually just few notes, used again and again. A recurrent short phrase that is developed in the course of a composition. A motif in music is a small group of notes encapsulating an idea or theme. It often contains the essence of the composition. For example, the opening four notes … Continue reading Motifs: Molecules of Music
Sophus Lie
“It is difficult to imagine modern mathematics without the concept of a Lie group.” (Ioan James, 2002). Sophus Lie grew up in the town of Moss, south of Oslo. He was a powerful man, tall and strong with a booming voice and imposing presence. He was an accomplished sportsman, most notably in gymnastics. It was no … Continue reading Sophus Lie
The Klein 4-Group
What is the common factor linking book-flips, solitaire, twelve-tone music and the solution of quartic equations? Answer: $latex {K_4}&fg=000000$. Symmetries of a Book --- or a Brick Take a book, place it on the table and draw a rectangle around it. How many ways can the book fit into the rectangle? Clearly, once any … Continue reading The Klein 4-Group
The Langlands Program
An ambitious programme to unify disparate areas of mathematics was set out some fifty years ago by Robert Langlands of the Institute for Advanced Study in Princeton. The “Langlands Program” (LP) is a set of deep conjectures that attempt to build bridges between certain algebraic and analytical objects. Substantial strides have been made in the … Continue reading The Langlands Program
Speed Cubing & Group Theory
The article in this week’s That’s Maths column in the Irish Times ( TM038 ) is about Rubik's Cube and the Group Theory that underlies its solution. The Rubik's cube craze ran through the world like wildfire in the 1980s. This simple mechanical puzzle is made from small pieces, called “cubies”, in a 3x3x3 structure … Continue reading Speed Cubing & Group Theory
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