FBE05 – Mathematics and Statistics Lecture 6 - Proportion, scale and golden section

Introduction The English author Henry Wotton translated ‘de Architectura’, which was written by Roman architect Vitruvius in 1624, and interpreted Vitruvius’ three conditions of a good building to be commodity (usefulness), firmness (solidity or strength) and delight (beauty). Proportion plays an important role in all three, as it provides guidelines for laying out useful spaces, for designing structural systems, and for creating an aesthetically pleasing environment. The proportions of a space can dramatically change how visitors feel in it, and the proportions of a facade design can affect whether a building appears welcoming threatening, or impressive.

Introduction

1 How would you describe the two spaces above 1 How would you describe the two spaces above? 2 What kind of activities will you suggest to be held at the two places? 3 How does proportion affect the sense of place?

1 How do the spaces below satisfy/dissatisfy the conditions of a good building? 2 Can scale and proportion in art and architecture be perceived in similar ways?

The Golden Ratio in Art and Architecture The Ancient Greeks were one of the first to discover a way to harness the beautiful asymmetry found in plants, animals, insects and other natural structures. They expressed this mathematical phenomenon with the Greek letter phi, but today, we call it the golden ratio—also known as the divine proportion, the golden mean, and the golden section. Much like the rule of thirds, this mathematical concept can be applied to your graphic designs to make them more visually appealing to the viewer.

Spiral sea shell in nature

The Golden Ratio in Art and Architecture The golden ratio (also called the golden rectangle) is a proportioning system that governs the relationship of smaller parts to the whole. It has long been believed to produce some of the most aesthetically pleasing shapes in nature, and as such has been used in many works of art and architecture. The ratio is AB:BC=BC:AC=1:1.618 as shown in figure 5.4.

The Golden Ratio in Art and Architecture Figure 5.4 The Golden Rectangle diagram

The Golden Ratio in Art and Architecture Mona Lisa’ by Leonardo da Vinci

The Golden Ratio in Art and Architecture Stamp design by Stephan Bundi

Creating a golden rectangle Here’s a more advanced technique for using those numbers in a more visual way. A “golden rectangle” is one that fits the parameters of the golden ratio—but the more times you divide a golden rectangle according to the golden ratio, the more useful it becomes. Let’s start by creating a rectangle with golden proportions. To make this simple, we’ll start with a width of 1000 pixels and divide it by 1.618 to get a height of about 618 pixels.

Creating a golden rectangle

Creating a golden rectangle Now add a 618 x 618 square on the right side of the canvas, leaving behind a 382 x 618 rectangle on the left side—another golden rectangle!

Creating a golden rectangle If you take that new rectangle and create another square within it, you’ll end up with another golden rectangle in the leftover space, which you can then divide up again, and so on and so forth.

Creating a golden rectangle Notice how each time you divide your golden rectangle, the largest dividing line kind of spirals in onto itself? That’s no accident—it forms the shape of a “golden spiral,” one of the more ubiquitous shapes that you’ll deal with when working with the golden ratio.

Creating a golden rectangle