“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

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

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Hans Hofmann67
American artist 1880–1966

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“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)

“To a given right line to apply a parallelogram equal to a given triangle in an angle which is equal to a given right lined angle.
According to the Familiars of Eudemus, the inventions respecting the application, excess, and defect of spaces, is ancient and belongs to the Pythagoric muse. But junior mathematicians receiving names from these, transferred them to the lines which are called conic, because one of these they denominate a parabola, but the other an hyperbola, and the third an ellipsis; since, indeed these ancient and divine men, in the plane description of spaces on a terminated right line, regarded the things indicated by these appellations. For when a right line being proposed, you adapt a given space to the whole right line, then that space is said to be applied, but when you make the longitude of the space greater than that of the right line, then the space is said to exceed; but when less, so that some part of the right line is external to the described space, then the space is said to be deficient.”

Proclus (412–485) Greek philosopher

And after this manner, Euclid, in the sixth book, mentions both excess and defect. But in the present problem he requires application...
The Philosophical and Mathematical Commentaries of Proclus on the First Book of Euclid's Elements Vol. 2 (1789)

“Depth, in a pictorial, plastic sense, is not created by the arrangement of objects one after another toward a vanishing point, in the sense of the Renaissance perspective, but on the contrary (and in absolute denial of this doctrine) by the creation of forces in the sense of push and pull. Nor is depth created by tonal gradation (another doctrine of the academician which, at its culmination, degraded the use of color to a mere function of expressing dark and light).”

Hans Hofmann (1880–1966) American artist

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

“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 (1880–1966) American artist

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

“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)

John Wallis photo

“[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)

George Pólya photo

“The differential equation of the first order
\frac {dy}{dx} = f(x, y)
… prescribes the slope \frac {dy}{dx} at each point of the plane (or at each point of a certain region of the plane we call the field")…. a differential equation of the first order… can be conceived intuitively as a problem about the steady flow of a river: Being given the direction of the flow at each point, find the streamlines…. It leaves open the choice between the two possible directions in the line of a given slope. Thus… we should say specifically "direction of an unoriented straight line" and not merely "direction."”

George Pólya (1887–1985) Hungarian mathematician

Mathematical Methods in Science (1977)

Alexander Calder photo

“How can art be realized? Out of volumes, motion, spaces bounded by the great space, the universe. Out of different masses, tight, heavy, middling - indicated by variations of size or color - directional line - vectors which represent speeds, velocities, accelerations, forces, etc...”

Alexander Calder (1898–1976) American artist

these directions making between them meaningful angles, and senses, together defining one big conclusion or many. Spaces, volumes, suggested by the smallest means in contrast to their mass, or even including them, juxtaposed, pierced by vectors, crossed by speeds. Nothing at all of this is fixed. Each element able to move, to stir, to oscillate, to come and go in its relationships with the other elements in its universe. It must not be just a fleeting moment but a physical bond between the varying events in life. Not extractions, but abstractions. Abstractions that are like nothing in life except in their manner of reacting.
1930s, How Can Art Be Realized? (1932)

Carl Friedrich Gauss photo

“In researches in which an infinity of directions of straight lines in space is concerned, it is advantageous to represent these directions by means of those points upon a fixed sphere”

Carl Friedrich Gauss (1777–1855) German mathematician and physical scientist

&quot;Gauss&#x27;s Abstract of the Disquisitiones Generales circa Superficies Curvas presented to the Royal Society of Gottingen&quot; (1827) Tr. James Caddall Morehead &amp; Adam Miller Hiltebeitel in General Investigations of Curved Surfaces of 1827 and 1825 http://books.google.com/books?id=SYJsAAAAMAAJ&amp; (1902) <br class="br">Context: In researches in which an infinity of directions of straight lines in space is concerned, it is advantageous to represent these directions by means of those points upon a fixed sphere, which are the end points of the radii drawn parallel to the lines. The centre and the radius of this auxiliary sphere are here quite arbitrary. The radius may be taken equal to unity. This procedure agrees fundamentally with that which is constantly employed in astronomy, where all directions are referred to a fictitious celestial sphere of infinite radius. Spherical trigonometry and certain other theorems, to which the author has added a new one of frequent application, then serve for the solution of the problems which the comparison of the various directions involved can present.

James Jeans photo

“Minkowski… supposed that this fourth dimension of time was not detached from and independent of the three dimensions of space. He introduced a new four-dimensional space to which ordinary space contributed three dimensions, and time one; we may call it 'space-time'…. The succession of positions which a particle occupied in ordinary space at a succession of instants of time would be represented by a line in space-time; this he called the 'world-line' of the particle…. Newton's absolute space and absolute time fell out of science, and they carried much with them in their fall. First to go was the concept of simultaneity…. It now became necessary to find a way of treating gravitation which should not involve simultaneity. Einstein found through the medium of his 'Principle of Equivalence'.”

James Jeans (1877–1946) British mathematician and astronomer

The Growth of Physical Science (1947)

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