Leonardo

Projections from ActualityAuthor(s): Quentin WilliamsSource: Leonardo, Vol. 28, No. 4 (1995), pp. 334-335Published by: The MIT PressStable URL: http://www.jstor.org/stable/1576205 .

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fractions. In particular, I let h and k as- sume values 1 through 15. (The program code provides additional details.) Even if we place infinitely many Ford circles, none will overlap, and each will be tan- gential to the x axis. We can confirm this visually by magnifying the froth.

When I showed colleagues my com-

puter graphics of Ford circle froth, they were at first confused by the occasional instances of circles within circles. (In fact, the only other clear diagrams I had seen of Ford circles were in Rademacher's mathematics text [1], and in it the figures were hand-drawn diagrams containing only about 13 circles.) These anomalous interior circles, which I call "Chrysler circles," arise from the fact that my computer programs produce various h/k values that are equivalent, for example 1/2, 2/4, 4/8.... This, in turn, produces circles with different radii but having the same x coordinate. In addition, I also plot circles at (h/k, -1/2k2) to in- crease the symmetry of the representa- tion and heighten its aesthetic appeal.

Note that two fractions are called

"adjacent" if their Ford circles are tan- gential. Any fraction has, in this sense, an infinitude of adjacents [2]. Any circle can have an infinitude of tangen- tial circles.

Consider a godlike archer who launches an arrow at the Ford froth.

Depending on where the archer aims, the outcome is different. To under- stand this, place the virtual archer high up-that is, select a position above the Ford froth with an appropriately large y value. To simulate the shooting of the arrow, next draw a vertical line from the location of the archer (e.g. at x = a). Trace the arrow's straight-line trajec- tory as gravity pulls it down to the x axis. It turns out that if the archer's po- sition is at a rational point (a is ratio- nal), the line must pierce some Ford circle and hit the x axis exactly at the circle's point of tangency. However, when the archer's position is at an irra- tional number (a non-repeating, endless decimal value such as TC = 3.1415 .. .), it cannot pass directly to the x axis from a Ford circle. In other words, the arrow must leave every circle it enters. How- ever, as I mentioned previously, every circle it leaves is completely sur- rounded by a chain of adjacents. There- fore the archer's arrow travelling along x = a must enter another circle. This is

fractions. In particular, I let h and k as- sume values 1 through 15. (The program code provides additional details.) Even if we place infinitely many Ford circles, none will overlap, and each will be tan- gential to the x axis. We can confirm this visually by magnifying the froth.

When I showed colleagues my com-

puter graphics of Ford circle froth, they were at first confused by the occasional instances of circles within circles. (In fact, the only other clear diagrams I had seen of Ford circles were in Rademacher's mathematics text [1], and in it the figures were hand-drawn diagrams containing only about 13 circles.) These anomalous interior circles, which I call "Chrysler circles," arise from the fact that my computer programs produce various h/k values that are equivalent, for example 1/2, 2/4, 4/8.... This, in turn, produces circles with different radii but having the same x coordinate. In addition, I also plot circles at (h/k, -1/2k2) to in- crease the symmetry of the representa- tion and heighten its aesthetic appeal.

Note that two fractions are called

"adjacent" if their Ford circles are tan- gential. Any fraction has, in this sense, an infinitude of adjacents [2]. Any circle can have an infinitude of tangen- tial circles.

Consider a godlike archer who launches an arrow at the Ford froth.

Depending on where the archer aims, the outcome is different. To under- stand this, place the virtual archer high up-that is, select a position above the Ford froth with an appropriately large y value. To simulate the shooting of the arrow, next draw a vertical line from the location of the archer (e.g. at x = a). Trace the arrow's straight-line trajec- tory as gravity pulls it down to the x axis. It turns out that if the archer's po- sition is at a rational point (a is ratio- nal), the line must pierce some Ford circle and hit the x axis exactly at the circle's point of tangency. However, when the archer's position is at an irra- tional number (a non-repeating, endless decimal value such as TC = 3.1415 .. .), it cannot pass directly to the x axis from a Ford circle. In other words, the arrow must leave every circle it enters. How- ever, as I mentioned previously, every circle it leaves is completely sur- rounded by a chain of adjacents. There- fore the archer's arrow travelling along x = a must enter another circle. This is

located at an irrational point, the archer's arrow must pass through an in- finity of circles!

This all relates to the fact that even

though there are an infinite number of rational and irrational numbers, the in- finite number of irrationals is, in some sense, greater than the infinite number of rationals. Three-dimensional froth renditions using spheres instead of circles, although not shown in this ab- stract, also have considerable aesthetic appeal.

References

1. H. Rademacher, Higher Mathematics from an El- ementary Point of View (Boston, MA: Birkhauser, 1983).

2. L.R. Ford, "Fractions," American Mathematics Monthly 45 (1938) pp. 586-601.

PROJECTIONS FROM ACTUALITY Quentin Williams, 10 Nevil Road, Bishopston, Bristol BS7 9EQ, England.

Received 18 May 1995. Acceptedfor publication by Roger Malina.

Paintings are often wrongly described as

"photographically realistic." Indeed, they more usually impose on the viewer quite different conditions of accommodation

located at an irrational point, the archer's arrow must pass through an in- finity of circles!

This all relates to the fact that even

though there are an infinite number of rational and irrational numbers, the in- finite number of irrationals is, in some sense, greater than the infinite number of rationals. Three-dimensional froth renditions using spheres instead of circles, although not shown in this ab- stract, also have considerable aesthetic appeal.

References

1. H. Rademacher, Higher Mathematics from an El- ementary Point of View (Boston, MA: Birkhauser, 1983).

2. L.R. Ford, "Fractions," American Mathematics Monthly 45 (1938) pp. 586-601.

PROJECTIONS FROM ACTUALITY Quentin Williams, 10 Nevil Road, Bishopston, Bristol BS7 9EQ, England.

Received 18 May 1995. Acceptedfor publication by Roger Malina.

Paintings are often wrongly described as

"photographically realistic." Indeed, they more usually impose on the viewer quite different conditions of accommodation

and assimilation than do photographs. Those copied from fixed photo-

graphs usually look inert and embar- rassingly inaccurate, and few that are

copied from any source show genuine photographic qualities. Perhaps a paint- ing can appear photographically realis- tic only when the aesthetic values of painting are absent. In any event, a

photographically realistic appearance has best been achieved when a painting is copied from a real image projected by a camera obscura or camera lucida. An extremely small number of master painters, using this method, achieved a

unique form of painting that produced the equivalent of color photographs, 300 years before the advent of color

photography. Jan Vermeer became adept, for prac-

tical purposes, in using the camera obscura. Apart from the now famous light spots he felt obliged to copy verba- tim from the ground-glass screen, and his frequent typically photographic dis-

position (rather than composition) of content [1], he captured a very particu- lar, soft atmospheric variegation found hardly anywhere else in painting and, perhaps consequently, nowhere in scholarly discussion. Contrast his Alle- gory of Paintingwith the Van Eyck Arnolfini wedding portrait; the Van

and assimilation than do photographs. Those copied from fixed photo-

graphs usually look inert and embar- rassingly inaccurate, and few that are

copied from any source show genuine photographic qualities. Perhaps a paint- ing can appear photographically realis- tic only when the aesthetic values of painting are absent. In any event, a

photographically realistic appearance has best been achieved when a painting is copied from a real image projected by a camera obscura or camera lucida. An extremely small number of master painters, using this method, achieved a

unique form of painting that produced the equivalent of color photographs, 300 years before the advent of color

photography. Jan Vermeer became adept, for prac-

tical purposes, in using the camera obscura. Apart from the now famous light spots he felt obliged to copy verba- tim from the ground-glass screen, and his frequent typically photographic dis-

position (rather than composition) of content [1], he captured a very particu- lar, soft atmospheric variegation found hardly anywhere else in painting and, perhaps consequently, nowhere in scholarly discussion. Contrast his Alle- gory of Paintingwith the Van Eyck Arnolfini wedding portrait; the Van

Fig. 2. A View of Delft, with a Musical Instrument Vendor's Stall, sometimes called The Music Dealer, Carel Fabritius, 1652. Detail. (National Gallery, London) The atmospheric color, degree of obscurity and absence of paint value in this detail demonstrate the photographic resemblance achieved by the use of the camera obscura and camera lucida. (See Artist's Statement by Quentin Williams.)

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Fig. 2. A View of Delft, with a Musical Instrument Vendor's Stall, sometimes called The Music Dealer, Carel Fabritius, 1652. Detail. (National Gallery, London) The atmospheric color, degree of obscurity and absence of paint value in this detail demonstrate the photographic resemblance achieved by the use of the camera obscura and camera lucida. (See Artist's Statement by Quentin Williams.)

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true for all the Ford circles it pierces. Therefore, when the godlike archer is true for all the Ford circles it pierces. Therefore, when the godlike archer is

334 Artists' Statements 334 Artists' Statements

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Eyck is painterly. In the Vermeer, how- ever, paint quality is not significant and makes no contribution to the art of

painting [2]. Vermeer's The Milkmaid mixes pho-

tography with painting. The distilled

light spots in the still-life passage (see, also, the shadowy barge in his A View of Delft) are in direct disparity with the

impasto in parts of the figure [3]. In The Lacemaker, however, Vermeer paints totally according to a projected image. It is, in effect, a photograph fixed and

printed by painting [4]. Passages of the same quality are evident throughout Vermeer's later work, notably including The Girl in the Red Hat and The Girl with a Pearl Earring [5], both of which imme-

diately reveal the genuinely timeless ap- pearance of actuality.

The tiny, distant townscape left of the church in The Music Dealer by Vermeer's

contemporary, Carel Fabritius, is

painted in the same way. Its atmo-

spheric color, degree of obscurity and absence of paint value offer one of the most tantalizing frustrations of the

genre (heightened by its banishment to

the far distance in the deliberate distor- tion effected by Fabritius's camera box) [6] (Fig. 2).

Canaletto's camera painting, Whitehall and the Privy Garden, however, frustrates not at all. The magnificent townscape is laid out for inspection. The more we look, the more we probe its sunlit intimacies and the more we re- ceive a weird, time-travelled sense of

eyewitness privilege blossoming directly out of the light-projected actuality, a source much more vital than fixed pho- tography [7]. The figures in this paint- ing display the same loose technique evident in Vermeer's highlit draperies, and this generic difference between Canaletto's figures (often posed at long range) and background is present be- cause their incidence is transient, af-

fecting the pace of work. The stunning evocations of Gilman

and Hopper are, we know, this side of

Impressionism and Niepce and consti- tute, therefore, painting intellectually apart from photography. Conversely, the breathtaking realities of Terborgh and Chardin come from an alliance be-

tween painting and photography that stems from some recognition of a pro- jected image. But the evidence suggests that absorbed consultation of that im-

age, as in Vermeer and Canaletto, pro- duces a unique, rare and heretofore

unsung quality of painting.

References

1. See Officer and Laughing Girl, Jan Vermeer, c. 1658. (The Frick Collection, New York)

2. The Allegory of Painting, Jan Vermeer, 1666-1667 (Kunsthistorisches Museum, Vienna); The Betrothal of the Arnolfini, Jan Van Eyck, 1434 (National Gal- lery, London).

3. The Milkmaid, Jan Vermeer, 1658-1660. (Rijksmuseum, Amsterdam)

4. The Lacemaker, Jan Vermeer, 1669-1670. (The Louvre, Paris)

5. The Girl with a Red Hat, Jan Vermeer, 1666-1667 (National Gallery of Art, Washington, D.C., Mellon Collection); The Girl with a Pearl Earring, Jan Vermeer, 1665 (Maurithuis, The Hague).

6. A View of Delft, with a Musical Instrument Vendor's Stall, sometimes called The Music Dealer, Carel Fabritius, 1652. (National Gallery, London).

7. See Whitehall and the Privy Garden from Richmond House, Antonio Canaletto, 1746. (Goodwood House, Chichester, West Sussex, England).

Artists' Statements 335

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