Aesthetic Morphisms


Goguen introduced the notion of semiotic morphism, in order to mathematise earlier, informal, ideas by others about how metaphor works. He didn't consider very many types of morphism: mainly products, inclusions, and simple mappings between words or meanings with related semantic structures. By which I mean, although this probably isn't one of his examples, something such as the mapping of <stallion> ([+horse,+male]) to <mare> ([+horse,+female]).

But as a cartoonist, I have become aware of many other kinds of morphism, and I want to classify them. With such a classification — an ontology — we should be able to formalise a lot more, giving rise to as yet unsuspected computer applications. I shall discuss visual morphisms, but I'm sure that analogues can be found in other media. Not least, in poetry.

This work was partly inspired by the transformations discussed in "Drawing as Translation II".

Catalogue of mappings

Without further ado, I shall list the morphisms I've found so far. At the moment, these notes are are still pretty rough: for example, I probably need to add the kind of structure discussed in "A New Kind of Aesthetics — The Mathematical Structure of the Aesthetic" (Kubota et. al. 2017). That will have to wait until I have more spare time, or money to buy proper research time.


This section is inspired by Max Ernst (Ernst):

The collage technique is the systematic exploitation of the accidentally or artificially provoked encounter of two or more foreign realities on a seemingly incongruous level – and the spark of poetry that leaps across the gap as these two realities are brought together.


A ready-made reality, whose naive destination has the air of having been fixed, once and for all (a canoe), finding itself in the presence of another and hardly less absurd reality (a vacuum cleaner), in a place where both of them must feel displaced (a forest), will, by this very fact, escape to its naive destination and to its identity; it will pass from its false absolute, through a series of relative values, into a new absolute value, true and poetic: canoe and vacuum cleaner will make love. The mechanism of collage, it seems to me, is revealed by this very simple example. The complete transmutation, followed by a pure act, as that of love, will make itself known naturally every time the conditions are rendered favorable by the given facts: the coupling of two realities, irreconcilable in appearance, upon a plane which apparently does not suit them …


[ Above the Clouds Midnight Passes by Max Ernst. From . ]

Ernst also said

I found figural elements united there that stood so far apart from each other that the absurdity of this accumulation caused a sudden intensification of my visionary facilities and brought about a hallucinating succession of contradictory images.

This is very close to Goguen’s semiotic blending.

Two-dimensional transformations

These are transformations carried out on the image alone. For example, reflections, rotations, shears. They need no knowledge of what it represents. Example:

[ Image: Mona Lisa, reflected. From , then reflected.]

Three-dimensional transformations

These are carried out by redrawing the object depicted from another point of view. Unlike the above, they do need knowledge of what it represents. Example:

[ Image: a painting of Mona Lisa's back. From . Also found on other Pinterest pages. Creator not known.]

Information-creating and -destroying morphisms

Rotating an object brings into view parts that were previously hidden, and takes out of view parts that will no longer be visible. (This might be the basis of an interesting categorical definition of rotation.) So rotation is an example of a morphism that creates or destroys information. The example above is a prime example, since the picture of Mona's back has almost no surface in common with that of her front.

Level-of-detail scaling

Depicting an object in more or less detail. Example (in particular, the woman at the left). Examples:

[ Images: Javie, however, was a gimme-pig by Jocelyn Ireson-Paine; and detail therefrom. From (Ireson-Paine 2017b). ]

Inflating significant zones

In images where a lot of detail has been lost, perhaps because of low resolution, emphasising parts which are most necessary for recognition. Example:

[ Image: Helveticality emerging from the gloom by Douglas Hofstadter. From (Ireson-Paine 2017b). Originally from Metamagical Themas by Douglas Hofstadter. ]

Another example is the previous one, where I enhanced the expression on the woman’s face.

Style transfers

Rendering one artwork in the style of another. Examples:

[ Image: "Artistic Style Transfer with Deep Neural Networks" post in (Tejani 2016). From . Adapted from (Gatys et. al. 2015). ]

Note that style can’t be completely separated from content. Every medium and every style makes it easier to depict some objects than others (Ireson-Paine 2017a). Putting it another way, style inevitably inteferes with content. Examples of this are many, and include all the representational techniques such as cross-hatching that, if the viewer didn't know them to be conventions, would mislead him. Maybe this can be expressed in terms of symmetric monoidal categories or something else that can encode a notion of entanglement (Coecke).

Looking at it in another way, style morphisms translate from one representational system to another. This is something that computer scientists are familiar with; but the vast difference is that the representational systems implemented as part of most programming languages (records, arrays, lists, etc.) can usually be exactly translated, one to another. The representational system doesn't interfere with its content. Here, it does. Looking at it in terms of category theory, this means that the relations between scenes depicted in one system won't map exactly onto the relations between the same scenes depicted in another.


Rendering an image as lots of little "facets", in Cubism. Example:

[ Image: The Portuguese by Georges Bracque. From . ]

I've seen at least two interpretations of why the Cubists painted this way. (1) To depict an object from more than one viewpoint. (2) To depict the vast number of interconnections that even a mundane scene such as a café involves. In my example, (2) is not obvious, because my image is too small. In any case, I ought to put (1) and (2) under different headings. They work very differently, even if the results look superficially similar.

Textual anchoring

There are a number of pieces of text in the above. One interpretation of this that I've come across is that it helps keep a sense of reality, despite the distortions in the image. Example: above.

Also (Hughes 1991 p. 64):

[ Image: Merz 410: "Irgendsowas" by Kurt Schwitters. From . ]


Depicting a scene such that in some sense all regions have a similar form or texture, even though they represent very different objects, perhaps at very different distances. Example:

[ Image: Mont Sainte-Victoire seen from Les Lauves by Paul Cézanne. From . ]

One interpretation of this is that painting different regions in a similar style eliminates some "inessential" properties of the component objects, enabling the viewer to concentrate more on the form as a whole.

I call this an equaliser, and I hope I can formulate it as a categorical equaliser. Or possibly a pull-back, which is closely related.

Superficially, the results of equalisation may sometimes look like facetting. But the intention is probably very different.

Optimal match-unity equalisation

Artists have discovered numerous kinds of equalisation. Compare Cézanne's View of Gardanne with Le Château de Médan.

[ Image: Le Château de Médan by Paul Cézanne . From ]

[ Image: View of Gardanne by Paul Cézanne . From . ]

[ Image: View of Gardanne by Paul Cézanne . From . Original online source unknown. ]

Le Château de Médan uses rhythmic parallel brushstrokes (Verdi 1992, pp 112-118). These give the painting a superficial unity, by giving all parts of the surface a similar texture. But they also distort individual elements. But by the mid-1880s five years later, Cézanne had a more versatile technique where his strokes followed the contours of each form, but still made all forms within the image equally salient. This style didn't distort individual elements so much, so enabled him to "attend both to the formal distinctions between things and the relations among them".

Verdi’s phrase above is crying out for a mathematical formulation. I think of it as meaning:

  1. Break the image into a set of regions, such that each has roughly the same size and shape.
  2. Render them so that each looks in some sense similar to all the others. What this would seem to mean is that the distribution of texture and colour — or certain long-range spatial-frequency components thereof — is similar. We can probably formalise this in the terms used by recent work on style transfer, e.g. (van Kleur 2018).
  3. Emphasise the formal aspects of each patch, achieving a balance between the viewer’s recognition of form and that of meaning, i.e. what the patch denotes.
  4. Tweak so that the balance achieved by (3) is roughly the same for each patch, as is the distortion introduced by (2).

The latter clause in (4) worked less well for Le Château de Médan than for View of Gardanne, which is why critics regard the latter as better. We can order equalisations by how well (4) works, and thus make a category of them. The best, or at least optimal, equalisations, are probably initial or terminal elements thereof.

Visual rhyming

Giving an image unity by making contours echo one another. Example:

[ Image: Montagne Sainte-Victoire with Large Pine by Paul Cézanne . From . ]

Here, the branch of the pine tree curls around and follows the contour of the mountain (Verdi 1992 p. 120). Another example, which I like very much, is The Bathers:

[ Image: The Bathers by Paul Cézanne. From (Verdi 1992 p.191), photographed by myself. ]

I loved this when I first saw it. In it, the contours of the trees follow those of the bodies, giving the picture a delightful waviness. I suspect this works better in black and white and at a fairly small scale, which is one reason I’ve used this rather bad photograph. (The other reason is convenience.)

There must be analogues of visual rhyming in which it's not contour that's echoed, but some other property. These, and visual rhyming, ought to be definable in terms of some measure of local-global coherence.

[ Image: Cartoon by Mike Williams. From Punch Cartoon Album. ]

Implied Visual Rhyming

[ Image: Cartoon by Fougasse. From Punch Cartoon Album. ]

Visual Balance

[ Image: Cartoon by Ffolkes. From Punch Cartoon Album. ]

[ Image: Private Eye Issue 1486. ]

[ Image: . ]


Increasing unity by, for example, making a line point at a key element in an image. Example (Verdi 1992 p. 101): Montagne Sainte-Victoire with Large Pineabove, the branch emerging from the vertical middle of the pine pointing diagonally right- and down-wards over the isolated house below.


Lessening attention to the objects denoted by an image, but increasing attention to their form. Examples (Thomas 1991 pp. 159-161):

[ Images: Spear Heads and Clubs, originally from (Williams 1858 pp. 57 and 77). These copies from . Cited in (Thomas 1991, pp. 159-161). ]

The aestheticisation techniques identified by Thomas include symmetry, balance, and repetition.

Thomas demonstrates how these, and the techniques in the following section, have been used to depersonalise "primitive" peoples with whom the British explorers and colonists came into contact.

We can subdivide aestheticisation into within-object and between-object. The latter uses techniques such as the above mentioned, which concern the relation between objects. The former manipulates the representation of individual objects, as in the next section.

Excessive ornamentation and grotesquification

This is a particular case of the above. Example:

[ Image: Specimens of New Zealand Workmanship. 1 and 2. Different views of an adze. 3. A saw. - 4. A shell. From (Cook 1770s). ]

As Thomas points out, the carving on the adze suggests precise workmanship, but it is complicated and difficult to understand, increasing our perception of the "natives" who made it as "curious".


This is closely related to aestheticisation. Example: the Cook image above, which, like its fellows, shows "incongruity in the scales of different pieces and a general sense of abstraction and remoteness" (Thomas 1991 p. 150).

Justifiable aestheticisation

Unlike in the above, images can be aestheticised for desirable purposes. Example:

[ Image: front cover of (The Garden Machine Centre 1969). Photographed by myself. ]

Appointing a representative

Making an image less "busy" by letting a few patches of texture stand-in for an entire region thereof (Ireson-Paine 2017c). Example: the bricks in the castle and the clumps of grass in the above.

Synaesthetic reduplication

Re-representing some non-visual feature visually. Example:

[ Image: Hear Me Croak by Jocelyn Ireson-Paine. From . ]

In this, some of the letters have tails alluding to the drawn-out rather fricative -k and -t sounds at the ends of the words "Dot Dot Croak".


Re-representing objects as animate, often human. Example:

[ Image: Punch Volume XXXIII, 1857 , illustrating the poem "The Two Giants of the Time". ]

This image is very striking, and so is the poem. So I've copied it to an appendix.

The least subtle forms of anthropomorphism add faces and limbs to the image. More subtle varieties recruit lines already there. Example:

[ Image: Bill Peet. ]

This was drawn by Bill Peet, one of Disney's artists. When adding faces to objects, he often uses lines already there. Of it, Peet says: "A drawing of an unhappy caboose rattling along under a cloud of train smoke was stuck on my studio wall for fifteen years before she became 'Katy' in the story 'The Caboose that got Loose'".

It may be strange to associate anthropomorphism with something as abstract as category theory, when most of us see it in the form of dancing teapots and talking Mars Bars. Or it may just sound like a bad pun on "morphism". But I can sketch a categorical interpretation. Anthropomorphising an object increases its "agency" or freedom of action, because the anthropomorphised version is free to do anything a person can. So the state space available to anthropomorphised objects is much bigger than that available to the originals. The space of possible relations has been vastly expanded.


Herschel Chipp 1968. Theories of Modern Art: A Source Book by Artists and Critics. Oakland: University of California Press.

Bob Coecke. "Introducing categories to the practicing physicist".

James Cook 1770s. A Voyage Towards the South pole and Around the World.

Amanda-Jane Doran, Miles Kington. Punch Cartoon Album: 150 Years of Classic Cartoons.

Max Ernst. Details to be provided. At least one of the quotes occurs in (Chipp), and is online at

Leon Gatys, Alexander Ecker, Matthias Bethge 2015. "A Neural Algorithm of Artistic Style". (Submitted on 26 Aug 2015 ( v1 ), last revised 2 Sep 2015 (this version, v2)).

Robert Hughes 1991. The Shock of the New: Art and the Century of Change. London: Thames and Hudson.

Jocelyn Ireson-Paine 2017a. "Drawing as Translation".

Jocelyn Ireson-Paine 2017b. "Drawing as Translation II".

Jocelyn Ireson-Paine 2017c. "Generalised Inverses, Adjunctions, Aesthetic Balance, and Too Many Cartoon Bricks".

Akihiro Kubota, Hirokazu Hori, Makoto Naruse and Fuminori Akiba 2017. "A New Kind of Aesthetics — The Mathematical Structure of the Aesthetic". Philosophies 2017, 2, 14.

Punch Volume XXXIII, 1857.

Shafeen Tejani 2016. "Artistic Style Transfer with Deep Neural Networks", posted in From Bits to Brains blog. 27 Dec 2016.

Nicholas Thomas 1991. Entangled Objects: Exchange, Material Culture, and Colonialism in the Pacific. London: Harvard University Press.

Phil van Kleur 2018. "Style transfer", posted in Chromophilia blog. 20 August 2018.

Richard Verdi 1992. Cézanne. London: Thames and Hudson.

The Garden Machine Centre. Which One 1969? Catalogue of mowers from Kingston House (Mowers) Ltd. Printed by BPC Letterpress Ltd., London, at Gale & Polden Aldershot.

Thomas Williams 1858. Fiji and the Fijians. Volume 1, The Islands and Their Inhabitants. London: Alexander Heylin.

Appendix: The Two Giants of Our Time

"WHAT can we two great Forces do?"
Said Steam to Electricity,
"To better the case of the human race,
And promote mankind’s felicity?"

Electricity said, "From far lands sped,
Through a wire, with a thought’s velocity,
What tidings I bear! — of deeds that were
ever passed yet for atrocity."

"Both land and sea," said Steam, "by me,
At the rate of a bird men fly over;
But the quicker they speed to kill and bleed,
A thought to lament and sigh over."

"The world, you see." Electricity
Remarked, "thus far is our debtor,
That it faster goes; but, goodness knows,
It doesn’t get on much better."

"Well, well," said Steam, with whistle and scream,
"Herein we help morality;
That means we make to overtake
Rebellion and rascality."

"Sure enough, that’s true, and so we do,"
Electricity responded.
"Through us have been caught, and to justice brought,
Many scoundrels who had absconded."

Said Steam, "I hope we shall get the rope
round the necks of the Sepoy savages,
In double quick time, to avenge their crime,
And arrest their murders and ravages."

"We’ve been overpraised," said both; "we raised
Too sanguine expectations:
But with all our might, we haven’t yet quite
Regenerated the nations.

"We’re afraid we shan't — we suspect we can’t
Cause people to change their courses;
Locomotive powers alone are ours:
But the world wants motive forces."