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Treat nature in terms of the cylinder, the sphere, and the cone, the whole put into perspective so that each side of an object, or of a plane, leads towards a central point. Lines parallel to the horizon give breadth, whether a sections of nature, or, if you prefer, of the spectacle which Pater omnipotens aeterne Deus unfolds before your eyes. Lines perpendicular to this horizon give depth.. ..Everything I am telling you (Joachim Gasquet, fh) about - the sphere, the cone, cylinder, concave shadow – on mornings when I’m tired these notions of mine get me going, they stimulate me, I soon forget them once I start using my eyes.
--
What he told me – The motif, Joachim Gasquet, as quoted in Joachim Gasquet’s Cézanne, - a Memoir with Conversations, Thames and Hudson, London 1991 pp. 163-164

 

Paul Cezanne

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...the stereographic projection of the spherical surface. From the north pole P we draw radial lines to project every point of the surface of the sphere upon the horizontal plane [below, perpendicular to a line joining it to P and the sphere's center]. In general this transformation is unique and continuous , although the metrical relations are distorted; for the point P, however, it shows a singularity. Point P is mapped upon the infinite; i.e., no finitely located point of the plane corresponds to it. It can be shown that every transformation possesses a singularity in at least one point. The surface of the sphere is therefore called topologically different from the plane. Only a "sphere without a north pole" [point] would be topologically equivalent to a plane. ...such a sphere has a point-shaped hole without a boundary and is no longer a closed surface.

 

Hans Reichenbach

 

We are now in the middle of a long process of transition in the nature of the image which man has of himself and his environment. Primitive men, and to a large extent also men of the early civilizations, imagined themselves to be living on a virtually illimitable plane. There was almost always somewhere beyond the known limits of human habitation, and over a very large part of the time that man has been on earth, there has been something like a frontier...
Gradually, however, man has been accustoming himself to the notion of the spherical earth and a closed sphere of human activity. A few unusual spirits among the ancient Greeks perceived that the earth was a sphere. It was only with the circumnavigations and the geographical explorations of the fifteenth and sixteenth centuries, however, that the fact that the earth was a sphere became at all widely known and accepted. Even in the thirteenth century, the commonest map was Mercator's projection, which visualizes the earth as an illimitable cylinder, essentially a plane wrapped around the globe, and it was not until the Second World War and the development of the air age that the global nature of tile planet really entered the popular imagination. Even now we are very far from having made the moral, political, and psychological adjustments which are implied in this transition from the illimitable plane to the closed sphere.

 

Kenneth Boulding

 

Cézanne made a cylinder out of a bottle. I start from the cylinder to create a special kind of individual object. I make a bottle — a particular bottle — out of a cylinder.

 

Juan Gris

 

It is not given to a cylinder to move everywhere by its own motion, nor yet to water nor to fire nor to anything else which is governed by nature or an irrational soul, for the things which check them and stand in the way are many. But intelligence and reason are able to go through everything that opposes them, and in such manner as they are formed by nature and as they choose. Place before thy eyes this facility with which the reason will be carried through all things, as fire upwards, as a stone downwards, as a cylinder down an inclined surface, and seek for nothing further. For all other obstacles either affect the body only, which is a dead thing; or, except for opinion and the yielding of reason itself, they do not crush nor do any harm of any kind; for if they did, he who felt it would immediately become bad.

 

Marcus Aurelius

 

The floorboards point in parallel lines to a vanishing point that does not concern us -- somewhere beyond the opera house, across the streets, across the houses of the suburbs, all the way to a hypothetical single dot... on the sea's horizon. Far from this sour drama.

 

Peter Greenaway