Sunday, February 26, 2017

PERSPECTIVE AND PROJECTIONS. The Entryway to Higher Dimensions

Perspective, Shapes, and individuality all coincide in the following lecture. The reader will be presented with the tools and information necessary to understand how the multitudes of understanding of what many artists and scientists thought to be the correct perspective view point, etc. has helped shape three dimensional spaces in two dimensional works. 

Platonic Solids Multiplication Chart
In three-dimensional space, a Platonic solid is a regular, convex polyhedron. It is constructed by regular polygonal faces with the same number of faces meeting at each vertex. Five solids meet those criteria: Tetrahedron. Cube. Octahedron. Dodecahedron and Icosahedron
Plato stated that as things appear, is different from as things are (Platonic view) suggesting that with the use of mathematics we can gain a sense of the ideal world. “Measuring and numbering help understand the perception of our eyes” - Plato

Kepler's view of the world and the heavens. Also geocentric, but with the added complexity, that each sphere fits with one of the Platonic solids. Resulting in a harmony of the spheres. Image originally from Kepler's Mysterium Cosmographicum (1596). This reprinting from Astronomi i billeder (1972)

No real monuments would rest on points like this, so Jamnitzer is actually displaying a kind of conceptual art.
Similar balance is seen in the following pair of monuments, where the one on the right goes even further by having the outer icosahedral shape floating freely without support

One of the oldest surviving fragments of Euclid's Elements, found at Oxyrhynchus and dated to circa AD 100 (P. Oxy. 29). The diagram accompanies Book II, Proposition 5.[15]

Euclid’s Elements consist of 13 books. It is a collection of definitions, postulates (axioms), propositions (theorems and constructions), and mathematical proofs of the propositions.
The word 'element' in the Greek language is the same as 'letter'. This suggests that theorems in the Elements should be seen as standing in the same relation to geometry as letters to language. The title may also be related to Plato’s solids as they represented 5 elements
Euclid's Elements has been referred to as the most successful and influential textbook ever written.
For more than two thousand years, the adjective "Euclidean" was unnecessary because no other sort of geometry had been conceived.
Today, however, many  non-Euclidean geometries are known, the first ones having been discovered in the early 19th century. An implication of Albert Einstein's theory of general relativity is that physical space itself is not Euclidean, and Euclidean space is a good approximation for it only where the gravitational field is weak.

Greeks and Romans understood linear perspective but it was lost overtime due to the iconoclasm in byzantine empire.
Some samples in this slide are: Orthogonal Perspective, Fishbone Perspective, Fragmentary linear perspective

Pompeiian mural of the pageant of Orestes, 2nd century AD, containing both central convergence (black lines) and ‘fishbone’ parallel convergence for the peripheral features such as the roof rafters (white lines).
No examples of Greek perspective paintings survive, but we can perhaps glean a sense of their technique from Roman copies (probably by Greek painters) from the ruins of Pompeii
• The black construction lines illustrate that the central structures adhere accurately to a single vanishing point close to the viewer’s eye level (estimated as horizontal light line).
•The light construction lines drawn from the rafters in the roof and other edges distant from the center illustrate that there was no principled adherence to a central vanishing point

Because of the varying degrees of parallelism, this painting does not obviously seem to follow the rule of thumb outlined above. Objects higher in the composition do not necessarily have higher vanishing points than objects lower in the composition.
And yet, despite all of these variances, these wandering horizons and varying levels of parallelism, the painting doesn’t look too bad. It may not be entirely consistent, it may thumb its nose at the rules of linear perspective, but the effect is not jarring.
And that, actually, is more than what can be said for the perspective drawings which art students today are taught to produce.
It takes a while, and a bit of fiddling around, but it seems no one really notices just how broken the linear perspective system is.