# Exercise to the 9^{th} Lecture

## Shape — System

# 5 Themes of Composition About Structure

**»Our universe appears to be constructed according to the principle of hierarchy: small things are part of bigger things and those are part of yet bigger things. The molecules of aluminium silicate make up clay that can be shaped into bricks, and bricks — in certain combinations with other materials — make up houses, garden walls and workshops, that in turn make up a city. It is true that these things are manmade, but the same principle governs nature — the human body for example. Brick cannot be used for building modern skyscrapers, which demand steel construction, and if we use bamboo and paper as building materials we are faced with different limitations altogether. These limitations are the reason for many ›laws‹, but in reality these limitations are laws. This hierarchical order of the universe, however, facilitates an encoding system that can be used for the formulation of efficient general guidelines that allow us to detect complex data.«**

Milan Butina, Elementi likovne prakse (Elements of Art Practice)

# Theme IV

√2

Structure → Elaboration

I divide the format of 70 x 50 units into 7 x 5 squares with 10—unit sides, or a very similar format of 70 x 49 units into 10 x 7 squares with 7—unit sides. I continue to divide each square into smaller squares with 1—unit sides. The result is a grid of thick and fine lines which serves as a structure (framework) for my graphics. Within each square, I develop a theme through elaboration: I join squares with one—unit sides into 3 x 2, 7 x 5, 17 x 12 and 41 x 29 planes. I can combine two longitudinal planes because the ratio does not change. If I am not happy with four different pairs of numbers, I can multiply the divisor (the smaller number in the pair) by a factor of two (the arithmetically expressed previous graphic law). Then I arrange the compositions of squares so that their inner composition logically continues horizontally and vertically. I respect the immanent linear (discursive) logic of √2.

**It is all about the play of ratios or fractions that are arithmetic approximates of the √2/1 ratio. In the proposed procedure, I play with the law: (√2) ^{-1} = √2 / 2. [The reverse value of √2 equals its half value, which is (1.414)^{-1} = 0.707; the law also says that (1+√2) = 1/(√2-1) or (1+√2) = (√2-1)^{-1}.]**

# Theme V

Φ (Golden Section)

Structure → Elaboration

I divide the format of 72×45 units into 8×5 squares with 9—unit sides, or a similar format of 65 x 40 units into 13 x 8 squares with 5—unit sides. I continue to divide each square into smaller squares with 1—unit sides. The result is a grid of thick and fine lines which serves as a structure (framework) for my graphics. Within each square, I develop a theme through elaboration: I join squares with 1—unit sides into 3 x 2, 5 x 3, 8 x 5, 13 x 8, 21 x 13, 34 x 21 and 55 x 34 planes. Each plane can be enlarged by a square along the longer side, its edge coinciding with the side, or reduced by a square located within the plane along the shorter side, its edge coinciding with the side. If you take a closer look at the fractions, you will see that they feature the numbers 2, 3, 5, 8, 13, 21, 34 and 55 (a_{n} = a_{n-1} + a_{n-2}). Then I arrange the compositions of squares so that their inner composition logically continues horizontally and vertically. I respect the immanent plane logic of φ.

**It is all about a play of ratios or fractions that are arithmetic approximates of the φ ratio (φ = (√5+1) / 2). In the proposed operation, I play with the law: φ ^{-1} = φ-1. [The reverse value of φ equals its value minus 1. (1.618^{-1} = 0.618)].**