Zero Intrinsic Curvature  /

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Physical Knowledge

Coupled Systems

"Physical knowledge is the meeting place of inner images and outer facts."    Wolfgang Pauli 

"If a curved surface is developed upon any other surface whatever, the measure of curvature in each point remains unchanged."        Carl Friedrich Gauss, Theorema Egregium, 1827

Zero Intrinsic Curvature 1  began with the blend of vague yet distinct notions or flavours:  the desire to work with circles, to make a window similar to those in islamic architecture;  and wanting to work with the diagonal shift in the pattern of light and shadow on the surface of the moon as it revolves around the earth and the sun through time. I knew that I needed to work with a rounded beam of wood, but the curved, continuous and uniform cylindrical / circular surface, unlike the cubic volume, offered no point of entry to begin to start measuring and applying lines. Intuitively I thus 'detached' the curved surface of this cylinder, traced it on a flat and rectangular sheet of paper onto which I was then able to draw the pattern - which was then 're'-applied to the surface of the wood. But why was there this seam / cut all along the length of the beam and the pattern which exposed the cross-section of the lines? Only much later I realised that intuitively I had understood and thus made visible an intrinsic property of the rounded beam: 

 

The curved surface of the cylinder is intrinsically flat.

The intrinsic curvature of the surface of the cylinder is zero - equivalent to that of a sheet of paper.

 

In 1827 Carl Friedrich Gauss had formulated the Theorema Egregium - the remarkable theorem- which states that the curvature of a surface can be determined entirely by measuring angles and distances on the surface itself, without any reference to the way in which the surface is embedded in the ambient space (the fact that the cylinder surface clearly is curved in the way it sits in space is not relevant for the intrinsic curvature of this surface - which is flat).The Gaussian curvature is an intrinsic invariant of a surface - a property which does not change when the surface is being bent or crumpled (but does change when stretched). It is closely related to the Euler Characteristic and the Polyhedron Formula defined by Leonard Euler (1707–1783) -   V-E+F=  - a topological invariant and intrinsic property of surface which does not change under deformation. 

 

What is remarkable to me is that by intuition and through wood carving I was able to discover for myself and able to 'understand' a mathematical truth and property of surface. It has re-affirmed my confidence in my tools. (How far can my intuition take me?)

...

The ratio between the diameter of a circle and its circumference is expressed in the mathematical symbol and value of π (pi). 

 

π  thus determines that the circle at the base of the rounded beam of a diameter of 9 cm must have a circumference of 28.27 cm - which in this 'sculpture' is consequently also the width of the flat surface onto which a circle pattern is applied. Consequently the circles at  the top and bottom 'end' in this pattern with a diameter of 28.27 cm wrap around the cylinder perfectly without cutting or overlapping. The diameter of the circle at the base of the wooden beam provides its circumference; this circumference provides the surface, the width of which determines the size of the circle inscribed.The vertical 'cut' or 'seam' which runs lengthways along the cylinder surface is not only an expression of the intrinsic flatness of the cylinder surface, but also the limit expressed by pi - the fixed ratio between circle and diameter / between cylinder base and cylinder surface.

 

...

The diameter of the circles when thought of in a horizontal manner, is a straight line in one sense and curved in another, whereas the vertical diameter is straight and un-curved / flat /straight in every sense.

Layers of meaning / alternative perspectives of one physical situation.

Zero Intrinsic Curvature 2 is in progress.

"Because a line drawn on a surface is always necessarily parallel to the surface it is drawn on, we can understand that the intrinsic curvature of a piece of rolled or crumpled up paper is zero and thus flat. 

This intrinsic curvature is invariant. This is what we can understand. This measures the quality of the lines drawn on these flat surfaces which can be be independent from this flatness and parallel-ness.

it is this separation and independence itself that I want to enlarge. The geometry and structure of the sheet of paper cannot be separated from my understanding of it. ‘I' ‘produce' the ‘paradox’ which is embodied in lines on the surfaces of bodies. Where is it’s origin exactly? To try and ‘stretch' a paradox further to see what reveals itself in this ‘gap’. To not desire to define the indefinable but to desire to hone in on it’s origin, this magnet, this desire-machine - a promise of something. There is a notion of silence."

π [pi]  / Calendar

Wolfgang Pauli, excerpts from

"The Influence of Archetypal Ideas on Kepler's Therories",

"The Interpretation of Nature and the Psyche", 

 

"[...] intuition and the direction of attention play a considerable role in the development of the concepts and ideas, generally far exceeding mere experience, that are necessary for the erection of a system of natural laws (that is, a scientific theory).

[...] What is the nature of the bridge between sense perceptions and the concepts?

Pure logic is fundamentally incapable of constructing such a link.

 

[...] It seems most satisfactory to introduce at this point the postulate of a cosmic order independent of our choice and distinct from the world of phenomena.

 

[...] 'participation of natural things in ideas' ...

 

[...] the relation between sense perception and idea remains predicated upon the fact that both, the soul of the perceiver and that which is recognised by perception, are subject to an order thought to be objective.

[...] The process of understanding nature as well as the happiness that man feels in understanding, that is, in the conscious realisation of new knowledge, seems thus to be based on a correspondence, a "matching" of inner images pre-existent in the human psyche with external objects and their behaviour.

[...] the preconscious, archaic level of cognition. On this level the place of clear concepts is taken by images with strong emotional content.

[...] these images...can be termed 'symbolical'....

 

[...] as ordering operators and image-formers in this world of symbolical images, the archetypes function as the sought-for bridge between the sense perceptions and the ides, and are, accordingly, a necessary presupposition even for evolving  a scientific theory of nature.

 

(However one must guard against transferring this a priori of knowledge into the conscious mind and relating it to definite ideas capable of rational transformation."