The Remainder

video installation, animated loop, 2014

J. Jernajczyk: Reszta, 2014

The regular pentagon is an exceptional figure – the ratio of its side to the diagonal inside the shape creates a golden proportion. That number, marked with the symbol φ (phi), is irrational – it may not be expressed in the form of a fraction. The famous Euclidean algorithm in the case of rational numbers (sections) allows us – after a finite number of steps – to find their proportion (strictly speaking their greatest common factor). However in the case of incommensurable sections (of irrational proportions), such as the side and the diagonal of a regular pentagon, algorithm has no end – there will always be a non-zero remainder left over. This animated loop shows indispensable connection of irrationality and infinity.

The animation presented on the screen shows a consistent division of the sections. At some point however, we end up at the starting point. From the cross-section of diagonals of a regular pentagon, another pentagon is created which inherits the same irrational proportions. The process of division then starts from the beginning and never ends.

The Remainder installation shows the necessary relationship of irrationality with infinity. The cognitive aspect of that fact is described in the article: J. Jernajczyk, Irrational images – the visualization of abstract mathematical terms, MATHEMATICA APPLICANDA Vol 43, No 2 (2015).

Work presented on exhibitions:

  • To, co zbylo / To, co zostało, Prague, Czech Republic, April-May 2014;
  • MathArt, Cracow, Poland, March 2015;
  • LIMITS 2015, Wrocław, Poland, December 2015;
  • Polish Scientific Networks, Wrocław, Poland, June 2016;
  • MATRIX, Leeds, Great Britain, September 2016;
  • OBRAZ PORUSZONY, Wrocław, Poland, March-April2017;
  • Dar Wrocławia, Gdańsk, Poland, May-June 2019.