“Along the River During the Qingming Festival” (清明上河圖) is a 12th-century painting by Zhang Zeduan. It is often cited as an example of isometric perspective, although the painting in fact skillfully mixes different perspective techniques to produce an overall effect that could never be seen by a single person.

The word “isometry” is made of two parts — “iso” meaning “the same” and “metry”, referring to measurement. “Isometric” in this context refers to the fact that the relative sizes of objects are preserved by the representation.

The illustration on the left shows how this works: a figure standing close to the viewer (bottom left) isrepresented as the same size as another, further away (top right). This approach is true to one aspect of reality — after all, the two figures really are about the same size.

In the Western tradition, perspective developed along different lines, and in a European painting after about 1400 it would be normal to depict the two figures at different sizes, reflecting the fact that distant objects appear smaller than close ones. That represents a different set of choices about what the image will preserve and what it will distort.

Ghirlandaio’s “Calling of the Apostles”, 1481

Zhang does in fact use this technique in some of the larger-scale aspects of his composition, meaning the whole image mixes different systems of perspective. This contrasts sharply with the systematic use of perspective that reigned in the West between approximately 1500 and 1800, and that remains the norm in many fields of illustration and computer-generated imagery.

Isometry is actually a rather general mathematical concept. The term can apply to any transformation of space that preserves some measure that interests us. Zhang’s isometric perspective transforms a 3D scene (which existed only in his imagination) into a 2D one; this will inevitably destroy or distort some measurable quantities. In this case what’s preserved are the ratios of vertical and horizontal distances.

In geometry, we’re sometimes interested only in things we can measure. If so, the isometries that preserve those measurements are known as symmetries: an object that is transformed by an isometry stays “essentially the same” from the viewpoint of that geometry. In plane Eucliean geometry, for instance, all rotations, translations and reflections are isometries.

Note that what counts as an isometry — and indeed what counts as a “measure” — depends on your choice of geometry. In fact, in 1872 Felix Klein suggested that geometry is really all about symmetries, and this view of the subject continues to be dominant today. It suggests that geometry isn’t really about objects — whether they’re triangles, spheres, or exotic 5-dimensional polytopes — but about processes. Here’s a video lecture that goes through some concepts about symmetry in 3D space from Klein’s view of things:

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In 2010 a company called Crystal CG prduced a computer-animated version of the painting, designed to be projected on a vast scale. Weirdly, the animations do not use Zhang’s original isometric perspective, producing odd effects. For example, if you watch from 3:20 to 3:59, you’ll see figures walking across the bridge and changing size as their distance from the viewer changes; this doesn’t happen in Zhang’s original painting and it therefore sits very oddly with his backdrops (this is a bit hard to see at first because the scene includes a mixture of adults and children). Initially I thought this was a crass mistake, but perhaps it’s a fitting addition to the collage of perspectives in the original:

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Our Perspective course, which focuses on the conceptual and geometric underpinnings of the most common systems of perspective in the Western tradition, continues for the next seven weeks.