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The Volumetric Eye

Originally published 12 Oct 2010

The 3D modeller is an expert in volumetric analysis. They can look at complex objects in the real world such as trees, vehicles and architecture, and see volumetric primitives (cubes, spheres, tubes etc.) underlying the complexity.

3D model - wireframe view - by Gordon Tarpley @

Without this ability, a would-be 3D modeller would flounder. Industry software like 3D Studio Max quickly trains modellers to think of objects in terms of primitives, polygons, planes and extrusions.

Perspective Study of a Chalice, Paolo Uccello, circa. 1450, Uffizi Gallery, Florence

Some of the perspective drawings from the Italian Renaissance, including Paolo Uccello’s Perspective Study of a Chalice, look spookily like a 3D software wireframe view. His drawing represents the vase in terms of volumetric primitives in a stack.

This presents a striking bond between contemporary 3D artists and Renaissance painters. Here the bond is not only one of aspiration to represent reality, but also one of conceptual thinking. How can I represent volumetric shapes convincingly and efficiently? The answer is geometry: understanding basic shapes and how they fit together as building blocks of reality.

This geometric education is a prerequisite of 3D software mastery, but I start to wonder why Italian Renaissance artists were moved to develop this deep understanding. Italian artists as a group (particularly the Florentines) are largely credited for the development of scientific perspective in 15th century art. We are generally aware (if even vaguely) that artists such as Leonardo da Vinci and Michelangelo were multi-talented geniuses who not only painted masterpieces but also created scientific inventions and architectural plans. But delving deeper, one discovers that this connection between artists and math wasn’t confined to one-off superstars such a Leonardo.  Piero della Francesca, a master painter of the early Italian Renaissance, also wrote pure math books on subjects such as Regular Solids, in addition to Perspective in Painting.  Leonardo was the culmination of a tradition of art and math being connected disciplines.

But what was it about Italy that created this interconnection?

I found the fascinating answer to this question is this excellent book by Michael Baxandall, Painting and Experience in Fifteenth-Century Italy: A Primer in the Social History of Pictorial Style.  It turns out that Italians considered artists to be part of the merchant class and the merchant class needed super advanced skills in volumetric gauging (amongst other skills) in order to do business. (ref 1) Basically, there were no standardised containers in Italy at this time, and in order to calculate the cost of a barrel of wine or a sack of onions, merchants were able to look at the size and shape of any bespoke container, and accurately compute the volume of it, despite its irregularities. And in order to begin to contemplate ‘gauging’ you needed a strong foundation in complex algebra and geometry. You can get an idea from this extract:

extract from Painting and Experience in Fifteenth-Century Italy by Michael Baxandall

The result of this is that the majority of artists and their clients in Renaissance Italy had a strong mathematical education, particularly in the area of volumetric geometry. It therefore begins to make sense why the interest in accurate volumetric depiction in art was so strong in this region at this time. Perspective, the sister treatise placed these volumetric objects convincingly in space.

Contemporary audiences, whilst not expert gaugers, are now used to seeing perfect 3D rendering. We are unforgiving of things that look ‘wrong’, but still able to be delighted by inventive and beautiful 3D art. In creating for spatially sensitised and demanding audiences, 3D artists again find something in common with their 15th century equivalents.

References 1. Baxandall, Michael, Painting and Experience in Fifteenth-Century Italy: A Primer in the Social History of Pictorial Style, Oxford Paperbacks; 2 edition (1988), pp. 86-91.



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