class: center, middle, inverse, title-slide # Tessellations from Eppalock to Heidelberg ## the tessellating of an original algorithmic piano work composed for the opening ceremony of the 2019<br/>Heidelberg Laureate Forum ### Charles T. Gray ### Heidelberg, Germany ### Sonntag, September 22, 2019 --- <!-- class: center, middle --> # "You type so fast it's scary." --- class: right, middle # _Music prepared me for mathematics._ --- class: left, middle # Listening to music is doing algebra. --- class: middle, center, inverse ## Less than or equal to # `\(\leqslant\)` --- ## `\(\leqslant\)` in music  The first note is a *perfect fifth* **below** the second note. --- # Define `\(\leqslant\)` Let `\(P\)` be a set. An *order* on `\(P\)` is a binary relation `\(\leqslant\)` on `\(P\)` such that, *** for all `\(x, y, z\)` in `\(P\)`, we have - `\(x \leqslant x\)`; - `\(x \leqslant y\)` and `\(y \leqslant x\)` imply `\(x = y\)`; - `\(x \leqslant y\)` and `\(y \leqslant z\)` imply `\(x \leqslant z\)`. *** We then say `\(\leqslant\)` is *reflexive*, *antisymmetric*, and *transitive*, for each of these properties, respectively. --- class: inverse, middle, center # Ostinati and Homomorphisms --- ## Homomorphism A *homomorphism* is a structure-preserving map between two algebraic structures of the same type. *** For a map `\(f: A \to B\)` between sets `\(A\)` and `\(B\)`, we say `\(f\)` preserves an `\(n\)`-ary operation `\(\mu_A\)`, if, for all `\(a_1, \dots, a_n\)` in `\(A\)`, we have $$ f(\mu_A(a_1, \dots, a_n)) = \mu_B(f(a_1), \dots, f(a_n)). $$ --- class: right ## Order-preserving homomorphism  --- class: right ## Order-preserving homomorphism <center> <img src="figs/motif.png" style="max-width:80%;"> <img src="figs/sequence.png" style="max-width:80%;"> </center> --- ## Order-reversing homomorphism  --- class: right ## Order-reversing homomorphism <center> <img src="figs/motif.png" style="max-width:80%;"> <img src="figs/inversion.png" style="max-width:80%;"> </center> --- class: left ## Ostinati ### An *ostinato* is a repeated musical pattern. #### Rhythmic ostinati #### Harmonic ostinati #### Melodic ostinati --- ## Tessellating homomorphisms and ostinati <center> <img src="figs/penrose-tiling-wikipedi.png" style="max-width:60%;"> </center> Image source: A penrose tiling from wikipedia. --- class: inverse # Let's tessellate some ostinati ## Let's do musical algebra together *Tessellations from Eppalock to Heidelberg* by Charles T. Gray An algorithmic piano work of musical homomorphisms tessellated for the opening of the 2019 Heidelberg Laureate Forum.