video installation, animated loop, 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.