# „The surfaces of three-dimensional space are distinguished from each other not only by their curvature but also by certain more general properties. A spherical surface, for instance, differs from a plane not only by its roundness but also by its finiteness. Finiteness is a holistic property. The sphere as a whole has a character different from that of a plane. A spherical surface made from rubber, such as a balloon, can be twisted so that its geometry changes.... but it cannot be distorted in such a way as that it will cover a plane. All surfaces obtained by distortion of the rubber sphere possess the same holistic properties; they are closed and finite. The plane as a whole has the property of being open; its straight lines are not closed. This feature is mathematically expressed as follows. Every surface can be mapped upon another one by the coordination of each point of one surface to a point of the other surface, as illustrated by the projection of a shadow picture by light rays. For surfaces with the same holistic properties it is possible to carry through this transformation uniquely and continuously in all points. Uniquely means: one and only one point of one surface corresponds to a given point of the other surface, and vice versa. Continuously means: neighborhood relations in infinitesimal domains are preserved; no tearing of the surface or shifting of relative positions of points occur at any place. For surfaces with different holistic properties, such a transformation can be carried through locally, but there is no single transformation for the whole surface.“

1891 - 1953

### „It is remarkable that this generalization of plane geometry to surface geometry is identical with that generalization of geometry which originated from the analysis of the axiom of parallels.... the construction of non-Euclidean geometries could have been equally well based upon the elimination of other axioms. It was perhaps due to an intuitive feeling for theoretical fruitfulness that the criticism always centered around the axiom of parallels. For in this way the axiomatic basis was created for that extension of geometry in which the metric appears as an independent variable. Once the significance of the metric as the characteristic feature of the plane has been recognized from the viewpoint of Gauss' plane theory, it is easy to point out, conversely, its connection with the axiom of parallels. The property of the straight line as being the shortest connection between two points can be transferred to curved surfaces, and leads to the concept of straightest line; on the surface of the sphere the great circles play the role of the shortest line of connection... analogous to that of the straight line on the plane. Yet while the great circles as "straight lines" share the most important property with those of the plane, they are distinct from the latter with respect to the axiom of the parallels: all great circles of the sphere intersect and therefore there are no parallels among these "straight lines".... If this idea is carried through, and all axioms are formulated on the understanding that by "straight lines" are meant the great circles of the sphere and by "plane" is meant the surface of the sphere, it turns out that this system of elements satisfies the system of axioms within two dimensions which is nearly identical in all of it statements with the axiomatic system of Euclidean geometry; the only exception is the formulation of the axiom of the parallels. The geometry of the spherical surface can be viewed as the realization of a two-dimensional non-Euclidean geometry: the denial of the axiom of the parallels singles out that generalization of geometry which occurs in the transition from the plane to the curve surface.“

—  Hans Reichenbach American philosopher 1891 - 1953

### „To help us to understand three-dimensional spaces, two-dimensional analogies may be very useful... A two-dimensional space of zero curvature is a plane, say a sheet of paper. The two-dimensional space of positive curvature is a convex surface, such as the shell of an egg. It is bent away from the plane towards the same side in all directions. The curvature of the egg, however, is not constant: it is strongest at the small end. The surface of constant positive curvature is the sphere... The two-dimensional space of negative curvature is a surface that is convex in some directions and concave in others, such as the surface of a saddle or the middle part of an hour glass. Of these two-dimensional surfaces we can form a mental picture because we can view them from outside... But... a being... unable to leave the surface... could only decide of which kind his surface was by studying the properties of geometrical figures drawn on it.... On the sheet of paper the sum of the three angles of a triangle is equal to two right angles, on the egg, or the sphere, it is larger, on the saddle it is smaller.... The spaces of zero and negative curvature are infinite, that of positive curvature is finite.... the inhabitant of the two-dimensional surface could determine its curvature if he were able to study very large triangles or very long straight lines. If the curvature were so minute that the sum of the angles of the largest triangle that he could measure would... differ... by an amount too small to be appreciable... then he would be unable to determine the curvature, unless he had some means of communicating with somebody living in the third dimension.... our case with reference to three-dimensional space is exactly similar.... we must study very large triangles and rays of light coming from very great distances. Thus the decision must necessarily depend on astronomical observations.“

—  Willem de Sitter Dutch cosmologist 1872 - 1934

### „... spread on a plane the surface of a sphere in such a way that the positions of all places shall correspond on all sides with each other both in so far as true direction and distance are concerned and as concerns true longitudes and latitudes.“

—  Gerardus Mercator cartographer, philosopher and mathematician 1512 - 1594
Legend on 1569 map

### „The first law is the same for both light and material bodies; they both move in a straight line, as long as they are not deflected by an outside force.The second law is also the same as that governing the reflection of an elastic ball from an impenetrable surface. Mechanics shows that such a ball is reflected from such a surface so that its angle of reflection equals its angle of incidence, as observed for light.But the third law still requires a plausible explanation. The passage of light from one medium to another exhibits behavior that is totally different from a ball moving through different media.“

—  Pierre Louis Maupertuis French mathematician, philosopher and man of letters 1698 - 1759

### „Time' has a large part to play in looking at a picture. A picture (a stupid empty surface to begin with) gets covered in the course of its creation by a rhythmically measured network of colors, lines and dots, which evokes in its final form a total of living movement. The eye jumps from a blue to red, to green (even if there is only a change of form), to a black line, suddenly comes upon a sharp white eruption, follows it, floats on to... It is impossible to take it all-in at once. Time is inseparable from surface.“

—  August Macke German painter of the expressionist group Der Blaue Reiter 1887 - 1914
Quote in Macke's letter to philosopher , in Expressionism; Praeger Publishers, New York, 1973, p. 145

### „A river is nearly the ultimate symbol for the very essence of change itself. It flows unceasing from one point of being to another, yet continuously occupies the same bed or pathway, and accommodates life’s endings with the same musical grace with which it accommodates life’s beginnings, along with all the muted and explosive moments that surface between the two extremes.“

—  Aberjhani author 1957
(Evolution of a Vision: from Songs of the Angelic Gaze to The River of Winged Dreams, p. 3).

### „The objection to the shiny, highly polished surface of albumen and gelatine papers is that, besides the fact that the surface reflects false and disturbing lights, the very polish and gloss has an artificial appearance which, from its very superfine character, irresistibly reminds us of its origin and nature.“

—  Alfred Horsley Hinton British photographer 1863 - 1908
p. 71

### „It was found [in the 1970s], unexpectedly and without anyone really having a concept for it, that the rules of perturbation theory can be changed in a way that makes relativistic quantum gravity inevitable rather than impossible. The change is made by replacing point particles by strings. Then Feynman graphs are replaced by Riemann surfaces, which are smooth - unlike the graphs, which have singularities at interaction vertices. The Riemann surfaces can degenerate to graphs in many different ways. In field theory, the interactions occur at the vertices of a Feynman graph. By contrast, in string theory, the interaction is encoded globally, in the topology of a Riemann surface, any small piece of which is like any other. This is reminiscent of how non-linearities are encoded globally in twistor theory.“

—  Edward Witten American theoretical physicist 1951
"The Past and Future of String Theory" in The Future of Theoretical Physics and Cosmology: Celebrating Stephen Hawking's Contributions to Physics (2003) ed. G.W. Gibbons, E.P.S. Shellard & S.J. Rankin

### „Evolution is no linear family tree, but change in the single multidimensional being that has grown to cover the entire surface of Earth.“

—  Lynn Margulis, What Is Life?

### „If you want to know all about Andy Warhol, just look at the surface; of my paintings and films and me, and there I am. There's nothing behind it... I see everything that way, the surface of things, a kind of mental Braille. I just pass my hands over the surface of things. [1973]“

—  Andy Warhol American artist 1928 - 1987
In: Warhol in his own words – Untitled Statements ( 1963 – 1987), selected by Neil Printz; as quoted in Andy Warhol, retrospective, Art and Bullfinch Press / Little Brown, 1989, pp. 457 – 467

### „Every measurable thing except numbers is imagined in the manner of a continuous quantity. Therefore, for the mensuration of such a thing, it is necessary that points, lines, and surfaces, or their properties, be imagined. For in them... measure or ratio is initially found... Therefore, every intensity which can be acquired successively ought to be imagined by a straight line perpendicularly erected on some point of the space or subject of the intensible thing, e. g., a quality... And since the quantity or ratio of lines is better known and is more readily conceived by us—nay the line is in the first species of continua, therefore such intensity ought to be imagined by lines... Therefore, equal intensities are designated by equal lines, a double intensity by a double line, and always in the same way if one proceeds proportionally.“

—  Nicole Oresme French philosopher

### „If the definition of simultaneity is given from a moving system, the spherical surface will result when Einstein's definition with є = 1/2 is used, since it is this definition which makes the velocity of light equal in all directions.“

—  Hans Reichenbach American philosopher 1891 - 1953

### „The centre of gravity of any hemisphere [is on the straight line which] is its axis, and divides the said straight line in such a way that the portion of it adjacent to the surface of the hemisphere has to the remaining portion the ratio which 5 has to 3.“

—  Archimedes Greek mathematician, physicist, engineer, inventor, and astronomer -287 - -212 a.C.
Proposition 6.

### „The systems approach to problems focuses on systems taken as a whole, not on their parts taken separately. Such an approach is concerned with total- system performance even when a change in only one or a few of its parts is contemplated because there are some properties of systems that can only be treated adequately from a holistic point of view. These properties derive from the relationship between parts of systems: how the parts interact and fit together“

—  Russell L. Ackoff Scientist 1919 - 2009
Cited in: Haluk Demirkan, James C. Spohrer, Vikas Krishna (2011) The Science of Service Systems. p. 274.

### „It is in our idleness, in our dreams, that the submerged truth sometimes makes its way to the surface.“

—  Virginia Woolf English writer 1882 - 1941

### „Donald Judd spoke of a 'neutral' surface, but what is meant? Neutrality must involve some relationship (to other ways of painting, thinking?) He would have to include these in his work to establish the neutrality of that surface. He also used 'non' or 'not' – expressive – this is an early problem – a negative solution or – expression of new sense – which can help one into – what one has not known. 'Neutral' expresses an intention.“

—  Jasper Johns American artist 1930
Book A (sketchbook), p 31, c 1963: as quoted in Jasper Johns, Writings, sketchbook Notes, Interviews, ed. Kirk Varnedoe, Moma New York, 1996, p. 50