DONALD RICHARDSON BA, Dip.Art, T.Dip.Art, RSASAART AND DESIGN History, Theory, Education, Practice21 Druids Avenue, Mount Barker, South Australia 5251 08/83982185

donaldar@ozemail.com.au

CRACKING THE NUT OF CURVED PERSPECTIVE

Donald Richardson

December, 2008

Abstract

Erwin Panofsky, in his essay, Perspective as Symbolic Form

(p.31ff),i asserts the view that all modern schemes of

perspective representation, because they are drawn with

straight lines, do not represent how humans really see

the world. This is an elaboration of an observation of

the seventeenth-century German polymath and artist,

Wilhelm Schickhardt (1592-1635), that 'all lines, even

the straightest, which do not stand directly in front of

the eyes….necessarily appear somewhat bent….' but that

'….nevertheless, no painter believes this; this is why

they paint the straight sides of building with straight

lines, even though according to the true art of

perspective this is incorrect….' [and] 'contrary to

nature.'ii

Schickhardt concludes his statement with the

challenge: 'Crack that nut, you artists!'iii

This paper will show that – contrary to

Schickhardt's assumption – European artists at least

since the Proto-Renaissance have been aware of the

conflicting nature of the raw information that is

presented to our visual perception and have come to terms

with it in their own individual ways. They have done

this just as our perception itself does – by a

synthesising of multifarious individual perceptual fixes

into a stable representation of the Euclidian world.

Further, the artists have often been able to manipulate

the disjunctions to suit their own pictorial ends (see

Figure 2).

But, Schickhardt's challenge is a pseudo-problem, a

mathematical abstraction that has no bearing on how we

perceive the world or how artists represent it.

________________________

The academic discussion

Panofsky illustrates Schickhardt's thesis with the

following diagram (ibid, p. 83, figure 10).

2

and quotes Schickhardt as continuing: 'For, as in the

figure above, the median lines CL and FH are the nearest

to the eye….they must appear larger; whereas the sides

BD, DM, MK, KB are further from it an so must appear

smaller. Thus the sides become narrower and necessarily

curved….'

Panofsky agrees with Schickhardt's position but

acknowledges that '….only a very few of us have perceived

these curvatures.' He lists (ibid, p.82, note 9) 'the

great psychologists and physicists at the end of the last

century' who have noted them – specifically Johannes

Kepler, Hermann von Helmholtz and Guido Hauck. And he

attributes the dearth of awareness of the phenomenon to

our 'habituation to linear perspective construction'….'a

quite specific, indeed specifically modern, sense of

space' which has become endemic in the western world

since the Renaissance. Nevertheless, he is clear that

linear perspective is factually incorrect in that it

calculates, and draws, linear lengths rather than angles

of visioniv and he adduces Euclid's eighth theorem as

proof.v Both visual and tactile space are qualitatively

different from Euclidian space, he avers; however, he is

forced to conclude that rendering space other than

through linear perspective is 'an impossible task, for a

sphere cannot be unrolled on a surface.'vi

There clearly is an issue that begs for discussion,

yet – apparently – it did not excite much interest among

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art historians and theorists in the period between 1925

and 1947-1948, when Panofsky re-stated the problem,

almost verbatim, in his Charles Eliot Norton Lectures,

given at Harvard University (1971, p.12).

Notwithstanding this, neither of the two major art

historians who did give the issue some attention – John

White and E H Gombrich – understood its principles.

White's statement (1957/1972, p.208) that Panofsky's

position 'possesses some of the qualities of Einstein's

finite infinity'vii is patent nonsense.viii And Gombrich's

(single, grudging) concession – that vapour-trails in the

sky appear curved, when they are actually straight – is

not a function of anything but the curvature of the

Earth, just as the curved appearance of the horizon, when

seen from a great height, is.ix

Moreover, Panofsky himself seems not to have fully

understood his own position: for instance, he uses it to

'explain' entasis and optical corrections in Doric

architecture (ibid pp.34-35, 89-90).x And his opinion that

the phenomenon is related to the curvature of the retina

(ibid, p.81) and that this is 'the entirely analogous

operation of the camera' (ibid, p. 31) are both patently

unjustifiable. Gombrich, too, is inclined to attribute

the phenomenon to the fact that our retinas are curved (op.

cit., p.258). In both cases, this is outmoded

neurophysiology and psychology.

Damisch (1995, p.6) condemns Panofsky's position as

'absurd'xi: 'One can only regret that even so prodigious

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an intelligence as Panofsky's could be sufficiently

misled by it to devote extended preliminary remarks to

so-called curvilinear perspective'. And Pirenne (1952)

claims to demonstrate that Renaissance perspective, far

from being 'artificial' (as White styled it – 1957/72,

p.113 ff), is the only natural system and 'corresponds to

the way we see the world' (op. cit., p.171).xii

The issue stems from a deficiency in Albertian

perspective that causes distortion at the periphery of a

projection. Panofsky illustrates this with the following

diagramxiii – a plan representation of three identical

cylindrical columns – that demonstrates that, whereas all

three columns are actually identical in size, the

representation in Albertian perspective (represented by

line A-F) shows A-B and E-F larger than C-D, whereas – in

perception – the reverse would be true. The fact that

the arcs alpha, beta and gamma are of equal length is

taken by some to indicate that our perception and,

therefore, space and perspective are somehow curved in a

concave way. That it is the reverse that is true is

demonstrated below.xiv

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The problem is demonstrated in such drawings as the

illustration from Viator (Figure 1) in which the left-

hand column should be represented smaller than its

neighbour because it is further from the viewer's eye,

but Albertian theory fails to provide the means to make

it so. A corollary effect is that this column thus

appears to be larger than the other one when it is clear

that the drawing is intended to represent two

dimensionally-identical columns. The degree of this

discrepancy is due to the vanishing-point being placed at

the extreme right of the picture. Such peripheral

distortions are rarely obvious in the works of the

masters because they usually veiled such discrepancies

with justly-placed shadows, drapery or foliage – one of

the ways they 'cracked the nut.' This applies in

particular to the interiors of seventeenth-century Dutch

masters like Jan Vermeer and Pieter de Hooch.

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Figure 1 A drawing by Viator from his De Artificiali Perspectiva(1509) (from Ivins, 1973 (unpaginated)).

The oil-painting by Canaletto (Figure 2) is a

curious picture with a (possibly) unique indication of

how an artist can use perspective in a creative way. It

was painted about a century after Schickhardt threw down

the gauntlet. While Canaletto uses the – by then – well-

established single-point perspective convention of

drawing the vertical and horizontal lines of the front

face of the palazzo, and of the monumental column,

parallel to the lines of the frame, his representation of

their orthogonals is not in accord with the practice.

Given that the spectator's viewing point is at the centre

of the picture, receding planes and lines to the right of

centre normally show left faces (which is also the way

they appear to us in the real world). However, in this

picture, the left face of the building has simply been

7

omitted and, instead, the right face of the column

capital and the right faces of various details of the

building are shown. It is not obvious why Canaletto made

this deviation from practice,xv but it seems not to have

been noticed by the curators of the National Gallery of

Victoria, for it is not remarked upon in the description

of the work on the gallery's website.

Figure 2 Canaletto (Italian 1697-1768. Worked in England) Bacino di S. Marco from the Piazzetta (c.1735-45). National Gallery of Victoria,

Melbourne. Felton Bequest, 1986.

However, the construction under discussion is a

mathematical fiction. It has no reference to either

nature or art – or how we perceive both. And the

theoretical position that, under ideal viewing

8

conditions, the peripheral objects that have been

rendered larger than they should be will appear correctly

diminished is just that – a purely theoretical one – for

it only applies to a one-eyed viewer standing immobile

directly before the centre of the picture and not moving

his/her eye at all. Such a scenario is extremely rare,

to say the least. Panofsky's (1991, pp.31-32)

attribution of the problem to the discrepancy that exists

between Albertian perspective construction and how an

image actually 'paints itself upon' the concave surface

of our retina is little more than a statement of the

obvious.xvi

But, common logic and knowledge of optics and

geometry are not adequate to rationalise what is really a

psychological phenomenon.

The psychology of visual perception

Many of the academic reflections quoted above suffer from

ignorance of the physiology and psychology of human

visual perception. This is surprising given that

Gibson's earliest book on the subject was published as

long ago as 1952 and even Gombrich's collaboration with

Gregory dates from the 1970s. All these predate many of

the articles quoted above.

We can state quite briefly that it is now well

established that the sensation of vision does not occur

in the eye itself. It is a mental operation in the

brain, and at no stage in the process does an image –

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either rectilinear or curvilinear – form on the retina.xvii

Further, all theories of perspective that rely upon the

assumption that we view – or can view – the world with a

single, fixed eye are erroneous because, even when we

close one eye, the open one is never still: "The eye is

continually moving –even when a person tries to fixate a

i Perspective as Symbolic Form was first published in the Vortrage der Bibliotek Warburg, 1924-25, and later translated into English by Christopher S Wood (1991). It is a commentary on one of the symbolic forms of human culture which Ernst Cassirer discussed in his Philosophie der Symbolischen Formen (1923).ii The last phrase taken from Frangenberg, 1992, p.2.iii Or 'Crack this little nut, oh artists!' in the translation in Frangenberg (1992).iv In taking this position, Panofsky is following Cassirer (who, in turn, follows Ernst Mach) - both of whom he quotes in this passage. Of course the problem is considerably older: Frangenberg (1992, p.4ff) reports Johannes Kepler's response to Schickhardt and then goes on to discuss Piero della Francesca's contribution. That Leonardo considered it an import issue is documented by Pedretti (1963), among others. v 'And perhaps it is more than mere accident that in Renaissance paraphrases of Euclid….this eighth theorem was either entirely suppressed of 'emended' until it lost its original meaning', he adds.(Ibid, p.35.) For a discussion of the possibility that Euclid may have considered that perspective, or space, is curved, see Knorr (1991). vi Ibid, p.36. By this expression Panofsky clearly means a flat surface. This, of course, is the dilemma of the map-maker, which isrecogised by Gombrich (1972, p.134). However, the map-maker's problem is caused by the spherical shape of the Earth, not by any peculiarity in human perception.vii Gombrich (c.1972, p. 135) also seems to make this error, but he admits that he has 'still failed to grasp what this is supposed to mean.' viii White does, however, make a brief and tentative reference to the phenomenon in his discussion of the work of Maso di Banco (op. cit., p.87).ix Other misunderstandings are expressed in his 'The "What" and the "How"' (c.1972).x Pirenne notes the error of this (1952, p.182, n.2).xi Even though he speaks from the perspective of an additional fifty years of neuropsychological research,

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well-marked point as steadily as he can which cause[s]

irregular oscillations of the retinal image across the

retina….' (Gregory, 1988, p.682). These oscillationsxviii

are, of course, unconscious and extremely rapid and are

integrated automatically by the brain into what we 'see.'

And applying the facts of visual perception to

viewing a representation of the visual world (a realistic

picture), reveals the irrelevance and falseness of the

concepts of viewing point and viewing distance. By this

point in time we will all, no doubt, have had the

experience of viewing a movie film from an extreme angle

from a seat in the front stalls – and having little

difficulty in perceiving the represented depth of the

projected images.

xii Pirenne adheres to this stance in (1970).xiii Ibid, p.79, figure 9. White (1957/1972) uses a similar diagram (p. 209, figure 9) as does Gombrich (1960, p.216). As Panofsky acknowledges in his caption, all such illustrations are derived from Leonardo's E16v (1513-14).xiv Incidentally, this does not explain how the columns further from the eye actually appear - and should be represented as - smaller thanthe one directly in front of the eye.xv One possible explanation would be if the picture was the left-handsection of a diptych or triptych, which would place the viewing pointto the right of the present canvas. Less likely it could be the left-hand section of a very wide panorama that had been cut down. The Royal Collection has a similar view of this subject but, here, the column is seen more frontally.xvi Apart from Panofsky's own fiction of an image being 'painted' on the retina.xvii See, for example, Zeki (1999), in particular Chapter 3 - 'The Myth of the "Seeing Eye"', Pirenne (1970), Kaufman (1974) and Gibson (1960).xviii Also known as saccades, saccadic flicks, saccadic movement or physiological nystagmus (the latter, e.g., in Turner, 1992).? The fish-eye lens, of course, does not have this correction, but printing from such a photograph usually eliminates the peripheral distortion.

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Artists' practice

The Italian architect and theorist, Leon Battista Alberti

is usually considered to be the originator of the theory

of perspective rendition – or, at least, the one who

revived and refreshed the knowledge of the ancient Greeks

and Romans. However, the extent of Alberti's

understanding is questioned by manyxix due to the fact

that his De Picturaxx of 1435 fails to give a complete and

unquestionable account of linear perspective and also

because it is clear that both the painter, Masaccio, and

the sculptors, Ghiberti and Donatello (as well as other

early Renaissance masters) made totally convincing

perspective representationsxxi which are innocent of the

deficiencies outlined above at least ten years before

1435. It is certain that these artists 'cracked the nut'

in their own ways. In view of this, and considering that

– in Early Netherlandish Painting (p.278) – Panofsky recognises

that such artists' scientific approach to visual

representation is a major aspect of the Renaissance in

both southern and northern Europe, one wonders at his

pursuing the issue at all. In fact, the painters of the

early Italian Renaissance were even ahead of twentieth-

century perceptual science in that they were aware that

the rendition of a receding plane cannot be achieved

xix And see 'Alberti on Perspective: Calling his Bluff' by the present writer in Online Journal of Art and Design, April, 2015.xx De Pictura was written in Latin in 1435 and translated into Italian as Della pittura c.1436.xxi The sculptors, of course, in reliefs.

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without gradation of tone: thus, they anticipated

Gibson's discovery of the 'perceptual gradient' by

centuries (see endnotes xxiv and xxv).

The interest of Leonardo da Vinci in problems

arising from Albertian perspective construction –

especially peripheral distortion – have been examined by

Pedretti (1963), Elkins (1988), Frangenberg (1992) and

Farago (1994). With the exception of Pedretti, all agree

that – although he may have contemplated the idea of

curved perspective – Leonardo preferred to 'operate on

the empirical judgment of the eye'xxii as, no doubt,

Masaccio, Ghiberti and Donatello did before him.

However, that he was aware of the problem which the

present writer attempts to elucidate below, is shown in

is Ms E 4r, the lower drawing of which is in principle

like Figure 4(b).xxiii

In a more general way, we should observe that the

most common and obvious principle that western artists

almost universally have used is to adopt the convention

of rendering all the vertical lines of buildings both

straight and vertical. And the horizontals of buildings

which run parallel to the picture plane are similarly

rendered straight and horizontal, while those that run

into space from the picture plane are drawn as

orthogonals according to various systems of perspective

representation. This, of course, is the convention that

Schickhardt criticised, but artists (who were, no doubt, xxii Farago, op.cit.xxiii Pedretti (1963) discusses this at length.

13

oblivious to his comments anyway) have adhered to it –

and it is accepted without question by viewers – because

it is a reasonable compromise between the instability of

perception and the need to create a picture that has

stability. And, as Gombrich observed (1982, p.164): 'If

we did not recognise a straight line as straight….we

would soon come to grief.'

Secondly, we must acknowledge something about which

all theories of linear perspective are silent. This is a

fact that the earliest Renaissance painters learned: the

fact that a uniformly-coloured flat surface (such as a

wall or a floor) that recedes in space cannot

successfully be rendered in a picture with a uniformly-

flat coloured paint. Although most viewers are

blissfully unaware of this, the illusion in a painting of

a flat surface receding in space can only be successfully

rendered with a gradation of the tonal value of the

colour of the surface being rendered. The difference may

be minutely subtle, but – unless the image has strong

linear orthogonals – it will be there. Thus, pictorial

depth can be suggested in some circumstances without

recourse at all to the geometry of linear perspective.xxiv

Perhaps the most easily accessible examples of this

principle are some of floors in pictures of the theatre

by Edgar Degas. This is, in fact, the principle of

perceptual gradient which Gibson 'discovered' and documented

in 1952xxv and is illustrated by Figure 9.2 in his The

xxiv See Gibson (1960), p.218.

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Senses Considered as Perceptual Systems (1968). 'Nut-cracking' by

another means!

'Curvilinear' perceptual space

The linear curvature that Schickhardt and Panofsky refer

to is not in any way the same as – or, even, related to –

the curved macro space of Einstein's Theory of General

Relativity, the shape of the retina, monocular or

binocular vision, the curvature the Earth's surface, or

the optical corrections of the Doric architects. To

avoid confusing this phenomenon with Einstein, the term

'curvilinear' perceptual space will be used in what follows, the

inverted commas being justified because the impression of

curvilinearity is more conceptual, or intuited, than

perceptual. It is Schickhardt's diagram in direct

perceptual experience.

'Curvilinear' perceptual space is a general

phenomenon of visual perception which can appear

convincingly under certain – and repeatable – viewing

conditions. But it is important to recognise – as

demonstrated below – that the apparent curvilearity is

convex, not concave. That this fact was known to Leonardo

is evidenced in the drawing from his Ms E 4r referred to

above. It may appear to anyone while seated directly in

front of – and quite close, and perpendicular, to – a

long, neatly-laid brick wall, uniformly lit (as

illustrated in Figure 3). Because the impression of

xxv And see Arnheim (1974), p.276.

15

convex curvilinearity may take a few moments to generate,

in what seems to be a synthesis of percept and concept,

the phenomenon is rather of time than space – or a

combination of both.xxvi

The following diagrams are an attempt to represent

the experience of 'curvilinear' perceptual space. In

reading the representation of the wall in Figure 3 we

should remember that the lens of a single lens reflex

camera is ground to eliminate peripheral distortion.xxvii

Figure 3 A photograph of a flat brick wall taken ateye-level with a single lens reflex camera

Figure 4(a) is a diagrammatic representation of a

photographic image like that in Figure 3. Figure 4(b) is

a schematic drawing showing each brick, apart from the

one in the centre, represented with its own individual

pair of orthogonals as the wall recedes further to the

left and right of the centre of vision. It is a

rationalisation of the perceptual experience of convex

curvilinearity which results from many individual fixes

that are synthesised in the brain. However, even given

xxvi Of course we should recognise the prior condition of light, as Gibson (1960), p.220, shows.xxvii The fish-eye lens, of course, does not have this correction, butin prints of an image from such a lens, the peripheral distortions are usually cropped off.

16

its radical simplification, Figure 4(b) gives a clear

impression of convex curvature.

Here we should note that these diagrams ignore that

a similar process would be operating with the vertical

lines as with the horizontal (as indicated in the first

diagram in this paper). The verticals have been left out

here the better to demonstrate the principle. This

course has been chosen because the foreshortening of

vertical lines is less noticeable in both our perception

of the world and the way artists represent buildings. No

doubt this is because, in our primitive origins, as well

as – in principle – in modern life, we use the horizontal

rotation of our heads more than vertical rotation to

orientate ourselves in space, to forage for food and to

defend ourselves from predators and enemies.

Figure 4(a) (top) Diagrammatic representation of a flat brick wall.

(b) (bottom) Diagrammatic representation of a flat brick wall drawn

with a separate vanishing-point for each brick.

17

Figures 5 is a composite photograph of a car-park

comprised of five separate exposures taken with the same

SLR camera from the same stand-point within seconds of

each other. In this gross simulation of the effect of

saccadic eye movement, scanning of the ends of the

'vertical' white lines clearly illustrates the convex

'curvilinearity' of perceptual space.

Figure 5 A composite photograph of a car-park comprised of five separate exposures taken with the same SLR camerafrom the same stand-point within seconds of each other.

In Figure 6, the top photograph of a car-park was

taken with a panoramic camera, thus the effect of

'curvilinearity' has been cancelled, the lens having been

ground specifically to do this. The bottom image is a

composite of the same scene collaged from three separate

shots taken with a SLR camera from the same stand-point

within a few seconds of each other and of the photograph

above. The convex 'curvilinear' effect of the three

successive 'fixes' is evidenced in the 'horizontal' white

line at about the middle of the picture and the distant

perimeter of the rectangular pavement.

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Figure 6 The top photograph of a car-park was taken witha panoramic camera. T he bottom is a collage of three separate shots of he same view taken with a SLR camera.

A related phenomenon can be experienced when undertaking

a journey in a vehicle along a straight stretch: looking

out a side window, the passing scene yields successive

parallaxes and central vanishing-points which resemble

the spokes of a wheel with the hub at infinity and the

vehicle running along its perimeter – a peculiar

sensation of convex curvilinearity which one knows cannot

be.

Other examples from works of art

Perhaps the most obvious reference to curvilinear

perceptual space is that by the nineteenth-century Dutch

painter, Vincent van Gogh, in all three versions of his

representation of his bedroom in the yellow house at

Arles (Figure 7). His rendering of the floor-tiles is a

recognition of convex 'curvilinear' perceptual space (as

indicated by the dotted lines superimposed by the author

19

on the illustration). Similarly, perhaps, in the

placement of both the chairs and the table. For an

instinctive artist who was dedicated to expressing

reality, this approach must have resulted from naïve

perception of the convex curvilinearity of perceptual

space. No doubt out of respect for conventional

perspective, van Gogh has not represented any other lines

as curved.

Figure 7 Vincent van Gogh, Bedroom in Arles

(1889), Art Institute, Chicago.

In his painting, The Construction Fence (1976) (Figure

8), Australian painter, Jeffrey Smart (1921-2013),

20

delineates the base of the fence with a convex

curvilinear line (here emphasised with a superimposed

black line) – plus Albertian perspective construction in

the concertina folds – to indicate that the fence is

receding markedly to both left and right. In this way

Smart has documented the convex curvilinearity of his

perception (and also made the space in which the girl

runs more believable). This would not have been

necessary if the picture had had a more square format,

but then it would have been a different picture. The

objects placed in front of the fence mask this stratagem.

Figure 8 Jeffrey Smart, The Construction Fence (1976)

Another artist who 'cracked the nut' in his own

individual way is the British painter David Hockney – in

his well-known1970s-1980s series of composite photograph

assemblages. But, as Tyler and Ione (www reference)

state, these works are 'entirely synthetic, rather than

natural': each individual shot is in the traditional

perspective of the photograph and the effect of curvature

is 'a property of the synthetic perspective of the moving

eye' – just as is our perception of real space. 'There

is no way to combine the directional snapshots into a

coherent image with this curved property…' they observe,

correctly.

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An academic pseudo-problem

In view of the foregoing, one must ask just which

artists Schickhardt was referring to. No doubt he

assumed that all artists were applying Albertian or

similar theories in rendering space whereas the most

talented were probably relying on their superior

perceptual sensitivity. Perspective theorists like

Viator? Architectural draughtsmen and renderers? He

cannot have been referring to the master artists.

The only nut that there is to crack is that of

Schickhardt's mathematical abstraction itself. Actually

the entire controversy is due to a misconception, a

disjunction between mathematical thought and optical

theoryxxviii, on the one hand, and the way artists have been

perceiving and representing the world since the late

Middle Ages, on the other. For, it is not that the world

actually appears to us as Schickhardt maintained but that

logically (or, rather, theoretically) – given the

observable fact that parallel lines receding in space

appear to converge – it should do. As Gombrich wrote

(1982, p.167): 'If I am right, the curvature does not

represent what we really perceive, but what we really do

not perceive.'xxix Schickhardt's 'nut' is no more than an

xxviii A reference in Kline (1979, pp.124-125) to Riemannian geometry adds little to the discussion.xxix Onians (2007) traces the progress of Gombrich's understanding of psychology.

22

academic pseudo-problem that has no bearing on how we

perceive the world or on how artists represent it, as is

proven by the following representation of a checkerboard

taken with a pin-hole camera.

23

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