“A work based only on a line concept is scarcely more than a illustration; it fails to achieve pictorial structure. Pictorial structure is based on a plane concept. The line originates in the meeting of two planes … we can lose ourselves in a multitude of lines, if through them we lose our senses for the planes.”

— Hans Hofmann

'Terms' p. 71
Search for the Real and Other Essays (1948)

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Hans Hofmann 67

American artist 1880–1966

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“We speak of concrete and not abstract painting because nothing is more concrete, more real then a line, a colour, a surface. A woman, a tree, a cow; are these concrete elements in a painting? No. A woman, a tree and a cow are concrete only in nature; in painting they are abstract, illusionistic, vague and speculative. However, a plane is a plane, a line is a line and no more or no less than that. 'Concrete paintin.”

Theo van Doesburg (1883–1931) Dutch architect, painter, draughtsman and writer

Spirit has arrived at the age of maturity...
Quote in 'Comments on the basic of concrete painting', Paris, January 1930, in 'Art Concret', April 1930, pp. 2–4
1926 – 1931

“These formulae [in (1) and (2) above] may be shown to be valid for a circle or a triangle in the hyperbolic plane… for which K < 0. Accordingly here the perimeter and area of a circle are greater, and the sum of the three angles of a triangle are less, than the corresponding quantities in the Euclidean plane. It can also be shown that each full line is of infinite length, that through a given point outside a given line an infinity of full lines may be drawn which do not meet the given line (the two lines bounding the family are said to be "parallel" to the given line), and that two full lines which meet do so in but one point.”

Howard P. Robertson (1903–1961) American mathematician and physicist

Geometry as a Branch of Physics (1949)

“Only now [1943] I become conscious that my work in black, white and little color planes has been merely 'drawing' in oil color. In drawing, the lines are the principal means of expressions... In painting, however, the lines are absorbed by the color planes; but the limitations of the planes show themselves as lines and conserve their great value.”

Piet Mondrian (1872–1944) Peintre Néerlandais

note from his postcard, late May 1943; as quoted in Mondrian, - The Art of Destruction, Carel Blotkamp, Reaktion Books LTD. London 2001, p. 240
1940's

“[W]hereas Nature, in propriety of Speech, doth not admit more than Three (Local) Dimensions, (Length, Breadth and Thickness, in Lines, Surfaces and Solids;) it may justly seem improper to talk of a Solid (of three Dimensions) drawn into a Fourth, Fifth, Sixth, or further Dimension.
A Line drawn into a Line, shall make a Plane or Surface; this drawn into a Line, shall make a Solid. But if this Solid be drawn into a Line, or this Plane into a Plane, what shall it make? A Plano-plane? This is a Monster in Nature, and less possible than a Chimera or a Centaure. For Length, Breadth and Thickness, take up the whole of Space.”

John Wallis (1616–1703) English mathematician

Nor can our Fansie imagine how there should be a Fourth Local Dimension beyond these Three.
Treatise of Algebra (1685)

“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.”

Hans Reichenbach (1891–1953) American philosopher

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.
The Philosophy of Space and Time (1928, tr. 1957)

“A plane is not necessarily a planet. A plane may be one planet, but a plane may also exist where no planet is. One planet may have several planes. Planes may also involve various aspects of apparent time. Planes can and do intermix without the knowledge of the inhabitants. A plane may be a time... or only one iota of vitality that exists all by itself. A plane may cease to be. A plane is formed for entities as patterns for fulfillment along various lines. It is a climate conducive to the development of unique and particular capabilities and achievements... an isolation of elements.”

Jane Roberts (1929–1984) American Writer

Source: Seth, Dreams & Projections of Consciousness, (1986), p. 103-104, quoting from Seth Session 16

“Since one cannot create 'depth' by carving a hole in the picture, and since one should not attempt to create the illusion of depth by tonal gradation, depth as a plastic reality must be two dimensions in a formal sense as well in the sense of color. 'Depth' is not created on a flat surface as an illusion, but as a plastic reality. The nature of the picture plane makes it possible to achieve depth without destroying the two-dimensional essence of the picture plane.. A plane is a fragment in the architecture of space. When a number of planes are opposed one to another, a spatial effect results. A plane functions in the same manner as the walls of a building... Planes organized within a picture create the pictorial space of its composition... The old masters were plane-consciousness. This makes their pictures restful as well as vital…”

Hans Hofmann (1880–1966) American artist

'Search for the Real in the Visual Arts', p. 44
Search for the Real and Other Essays (1948)

“Always lines, never forms. Where do they find these lines in Nature? Personally I see only forms that are lit up and forms that are not, planes that advance and planes that recede, relief and depth. My eye never sees outlines or particular features or details… …My brush should not see better than I do.”

Francisco De Goya (1746–1828) Spanish painter and printmaker (1746–1828)

Goya, in a recall of an overheard conversation
conversation of c. 1808, in the earliest biography of Goya: Goya, by Laurent Matheron, Schulz et Thuillié, Paris 1858; as quoted by Robert Hughes, in: Goya. Borzoi Book - Alfred Knopf, New York, 2003, p. 176
probably not accurate word for word, but according to Robert Hughes it rings true in all essentials, of the old Goya, in exile
1800s

“It requires courage to make a frontal attack on nature through the broad planes and the large lines and it is cowardly to do it by the facets and details. It is a battle.”

Edgar Degas (1834–1917) French artist

posthumous quotes, The Shop-Talk of Edgar Degas', (1961)

“A line cannot control pictorial space absolutely. A line may flow freely in and out space, but cannot independently create the phenomenon of push and pull necessary to plastic creation. Push and pull are expanding and contracting forces which are activated by carriers in visual motion. Planes are the most important carriers, lines and points less so... the picture plane reacts automatically in the opposite direction to the stimulus received; thus action continues as long as it receives stimulus in the creative process. Push answers with pull and pull with push. … At the end of his life and the height of his capacity Cézanne understood color as a force of push and pull. In his pictures he created an enormous sense of volume, breathing, pulsating, expanding, contracting through his use of colors.”

Hans Hofmann (1880–1966) American artist

'Search for the Real in the Visual Arts', p. 44
Search for the Real and Other Essays (1948)

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Hans Hofmann 67

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