Linear Perspective and Mathematics

 

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Linear perspective is a branch of geometry that has been developed and is useful to both mathematicians and artists/architects/engineers. It was actually developed by people interested in both - mathematics and architecture and engineering were not so much divided once upon a time. People in the past used to be less 'specialised' than they are now. These days you can use computer graphics with much less trouble, but people still enjoy perspective because it gives them a great opportunity for meditating on space through trying to present it in a clear way.

Roots of the word 'perspective' are Latin - the word 'perspicere' means 'to see clearly' and indeed this is the main purpose of perspective - to present the world in such a way that the observer of a perspective picture has the feeling of space as it would appear in nature.  

Brunelleschi, Italian architect and artist (1377-1446)

Linear perspective was invented by an Italian architect called Filippo Brunelleschi. He became very famous even in his lifetime and was buried in a church Santa Maria del Fiore - the picture of his statue is placed just outside of the church, and Brunelleschi appears as if he is looking at the dome of the church. It is the same church that he drew in a first ever 'scientific' perspective drawing, and the process of which he described in a letter written in a letter in 1413.

Santa Maria del Fiore in Florence, Italy, famous for Brunelleshi's perspective drawing

Linear perspective is based on a few very simple principles: that the parallel lines, which we know don't really meet (unless in some other type of geometry) do seem to meet to us when we look at them going away from us. For example, if you stand (but just imagine it don't do it!) at the train tracks and look at them going away from you, they will appear as if they meet at a distant point. This point is called the 'vanishing point'. The line where all the vanishing points (of these horizontals) lie is called 'horizon'.

Linear perspective was further developed by Italian artists and mathematicians, such as Alberti and Piero della Francesca, and was later one of the most admired subjects of study of a famous English mathematician Brook Taylor (1685-1731), who is better known for his work on Taylor Series.

Taylor series is an approximation of a function calculated as a sum from the values of its derivatives at a single point.

Some of the famous paintings linked to the development of perspective are to be seen to the side of this text. Click on the images to enlarge them.

 

   

Download the worksheet on perspective

Famous paintings all having linear perspective as an underlyining structure

Masaccio's Trinity painted around 1426, Florence, Sta. Maria Novella. Pay particular attention to the slabs on the ceiling and how they give the sense of space - which is done by linear perspective.

Paolo Uccello's The Hunt in the Forest , painted around 1460, now in the Ashmolean Museum, Oxford, England.

Piero della Francesca's Flagellation of Christ, around 1460, now in Urbino, Galleria Nazionale della Marche.

Leonardo da Vinci's Annunciation , around 1472, now in Florence, Uffizi.

 

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