When Choices Outgrow the Universe — From Counting Possibilities to Inevitable Structure
Begin with arranging short and long syllables, then move through lattice paths, integer partitions, colorings, and networks. Follow binomial expansions in Baghdad, prosodic recurrences in Patan, triangular arrays in Hangzhou, systems of combination in Paris and Leipzig, graphs and generating functions in Königsberg and Berlin, inevitability and probabilistic existence in Cambridge and Budapest, symmetry in Zürich, computer-assisted proof in Urbana, and internet collaboration. The journey asks how a subject that counts every possibility learned to prove structure without seeing every case.
Scene 1 of 15
c. 1000
Baghdad
Calculating Power-Expansion Coefficients in a Triangular Array — al-Karaji
In algebraic works written in Baghdad, al-Karaji treated rules for expanding powers of a binomial and a triangular arrangement of coefficients. Parts of his own text are lost, but al-Samaw’al’s later work preserves the method. This is not labeled the lone invention of the modern ‘Pascal triangle’; binomial knowledge developed through several authors and transmission networks in the Islamic world.
Replay on the map from scene 1Scene index