Cluster Puzzles‎ > ‎

M. C. Escher

Maurits Cornelis Escher (17 June 1898–27 June 1972), the life-life tessellation pioneer who scarcely needs a further introduction, shows three cluster puzzle instances (although of course not known by the title of the day), with Sun and Moon, with 28 (bird) motifs, of 1948, Plane Filling I, with 36 animal motifs, of 1951 and Plane Filling II, with 40 animal motifs, with one exception, of 1957. As suggested by the different names, these differ in style, and from two broad subgroups. Plane Filling I is also shown as a periodic drawing, No. 83. The Plane Filling works are also synonymously titled Mosaic I and Mosaic II. Given that ‘Plane Filling’ is generally preferred, I thus use this title below.

Regarding the images below, all Escher's works are copyright of the Escher Foundation. Despite two previous requests to use, they did not respond, and so this time I did not pursue them again. However, there is a workaround to showing these. All three images have been licensed for use as actual jigsaw puzzles, and so instead I show these. Although not ideal, it still more than admirably serves to illustrate the works. 

Unlike other cluster puzzle artists who refrained from comment on their works, Escher at least discussed all these (among some of his other works), in The Graphic Work of M. C. Escher (all three), but not particularly in-depth, and in lecture notes (excluding Sun and Moon) for a proposed lecture tour of the USA in 1964, unrealised due to ill-health. However, there are various difficulties in quoting the passages in this book. It has been through many editions, expanded revisions and languages. For instance, in the original Dutch 1960 edition only Plane Filling II appears, whilst in an English 1970 edition both appear. Further, the commentary appears to be notably different. Given that both works appear only in the English edition, I will defer to this edition below, save for specific instances of the Dutch edition. 

Of interest is to how the genre was described by Escher, and by extension, others. Escher described the cluster puzzle nature in four different ways: ‘Free plane-filling’ (Periodic Drawing 83), ‘irregular plane filling’ (letter to George Paulus, 1964), ‘unequal figures’ (Grafiek en Tekeningen), and ‘no repeating figures’ (Lecture notes, 1964). By others, Bruno Ernst (Magic Mirror) called these ‘free surface-fillings’ and Marianne Teuber (M.C. Escher: Art and Science) ‘irregular space-filling patterns’. All are broadly descriptive. Of Escher's terminology, I prefer ‘free plane filling’ and ‘irregular plane filling’, in no particular order. Of course, my favoured description, ‘cluster puzzles’, is derived from Alex Palmer’s usage, of which I have chosen as the preferred term, for reasons as I give in the main page, despite the term not being particularly descriptive! Indeed, quite how ‘best’ to define the genre is unclear. Nonetheless, although Escher’s terminology is more exact than Palmer’s, I have decided to retain his terminology for the genre.

In The Graphic Work, these are shown as Plates 34-35 with commentary on p. 17. However, the latter was somewhat brief, both of just five lines, of which I transcribe below. Such relative brief commentary was not an outlier; it was typical throughout the book. The unrealised text is shown in Escher on Escher: Exploring the Infinite, p. 32. Again, this is relatively brief and lightweight. Even so, his comments are interesting. He also discussed this again, briefly, in private correspondence with George J. Paulus (who was also active in the field), in 1964 (and enclosed his Plane Filling II). Latterly, in 2017, I corresponded with Escher’s son, George, on the relation between Simplex cluster puzzles in his native Netherlands, and of which he commented on related matters of Plane Filling I and Plane Filling II, which were insightful, and added to the story. Subsequently, these prints have been discussed by others, including the Escher authority Jeffrey Price, in M.C. Escher Amazing Images, c. p. 21 on Mosaic II and Escher Het Paleis (Escher in the Palace) by Erik Kristen (all three) and Bruno Ernst (Sun and Moon). All this I discuss in more detail below. The page is in three parts. Part 1 gives Escher's own commentary on all three works. Part 2 gives commentary by others, including Erik Kersten, Jeffrey Price, and Bruno Ernst. Part 3 gives a commentary of my own.

Part 1

Escher’s Commentary on Sun and Moon, Periodic Drawing 83, Plane Fillings I and II

Here I repeat the commentary by Escher with additional essentially clarifying comments of my own immediately below the comments.

Sun and Moon

Sun and Moon Puzzle

Sun and Moon,  Woodcut, 9⅞” x 10⅝” (251mm x 270mm) 1948, April

In Graphic Work.

The subject of this colored woodcut is once again the contrast between day and night. But in this instance  the two notions are not, as in plate 11 [Day and Night] pictured as next to each other but in the same place - though not simultaneous, being separated by a leap of the mind. It is day-time when there is a sun shining in the centre, where the sun is shooting out yellow and red rays. Against this background stand out fourteen dark blue birds. As soon as one divests them of their function as objects and regards them as background against a night sky, with a crescent moon in the centre and with stars, planets and a comet.

Escher's commentary essentially discusses the intricacies of the print itself, focussing on matters of figure-ground rather than the cluster puzzle aspect. No mention is made of what he later called ‘free plane-filling’ (1951) and ‘irregular plane filling’ (1964) of the cluster puzzle aspect. 

Plane Filling I

Periodic Drawing 83 for Plane Filling I, 1951, March. 

In Visions of Symmetry

Escher notes on the lower drawing margin, translated:

Free plane-filling, based on a rectangular system, with 36 different motifs (design for mezzotint).

A brief comment, understandably, in the lower margin. However, the term ‘rectangles’ is not to be taken literally, rather than rectangles per se quadrilaterals are used, presumably within a ‘rectangle system’. This was used immediately for the print. As an aside, oddly, there is no periodic drawing for the later 1957 instance. 

Plane Filling I, Mezzontint, 5¾” x 7¾” (14.5 x 20 cm) 1951, March

In Graphic Work.

Regularity of construction can be recognised in this rectangular mosaic in that, both as regards height and breadth, three light and dark alternate like the squares on a chessboard. With the exception of the shapes around the edge, every white one is surrounded by four black ones and every black by four white. The sum total can immediately be ascertained: 36 pieces, 18 white and 18 black.

Escher's commentary essentially discusses peripheral matters of the print itself, focussing on the shape, and aspects of black and white and number of motifs, rather than the ‘unusual’, and far more obvious point of interest, the cluster puzzle aspect. No mention is made of what he later called ‘free plane-filling’ (1951) and ‘irregular plane filling’ (1964) of the cluster puzzle aspect. 

Plane Filling II

Plane Filling II Puzzle

Plane Filling II, Lithograph 12⅜” x 14⅝” (32 x 37 cm)

In Grafiek en Tekeningen (A Google translation of the Dutch text). 

The series of images shown so far, in which the plane extension plays an important role, are covered with a print consisting of a number of unequal figures.

However, each of them is in the form of something, whether a living being or an object, which the viewer "recognizes". Compiling such a surface fill is a tiring activity and at the same time a mindless game. It is the draftsman, as if he were not the game leader himself, but as if, without will, he allows his creatures the freedom to determine their own shape and character.

As such, this does not discuss the print directly, but rather background matters. Strictly there is nothing of real interest here.

Although of the same basic premise, this lithograph differs from the mezzotint, in that the motifs are far less structured.

On Plane Filling II

In this case the only regularity to be noted is the rectangularity of the complete surface. There are but few of the inner figures bordered by four adjacent ones. The direct environment of the frog consists of two figures; the guitar is hemmed in by three, the cock by five and the ostrich (if that is what it really is) by six. The sum total can be arrived at by careful counting.

Letter to George J. Paulus

Some of my other American acquaintances showed me their efforts in making regular plane fillings, but you are the first who shares my (secondary) hobby for irregular fillings. Please find enclosed another example which I made (it’s a copper engraving, a “mezzotint”) in 1951, not reproduced in my book. Though all the animals are irregularly bordering each other, they form together a rectangle with 18 black and 18 white figures.

In 1963, an American architect, George Paulus, in a letter dated 20 March 1963, sent Escher examples of his cluster puzzle work, of a single instance. Escher replied, the next year, 29 March 1964. Again, there is nothing of a revelatory nature here. Escher's commentary essentially discusses peripheral matters of the print itself, with once again talk of rectangles and numbers, reminiscent in style from his later 1970 book.

On Both Plane Filling I and II 

In Escher on Escher. Lecture notes made for a cancelled tour, p. 32:

I should like to round off this introduction to my periodic patterns with these two tessellations [Plane Filling I and II], which contain no repeating figures whatsoever. So they do not really belong to the subject of this talk, but I show them all the same because they clearly illustrate my two main requisites recognisability and colour contrast. If one couldn’t recognise them as living beings or a well known object (the guitar, for instance), it would have been a senseless game to put them together; and without a shade contrast between two adjacent figures they would simply be invisible!

Composing such a jigsaw puzzle is done more or less unconsciously. While drawing, I feel as if I were a spiritualist medium, controlled by the creatures that I am conjuring up, and it is as if they themselves decide on the shape in which they like to appear. 

A contour line between two interlocking figures has a double function, and the act of tracing such a line therefore presents a special difficulty. On either side of it, a figure takes shape simultaneously. But, as the human mind can’t be busy with two things at the same moment, there must be a quick and continuous jumping from one side to the other. The desire to overcome this fascinating difficulty is perhaps the very reason for my continuing activity in this field.

As such, this does not discuss the cluster puzzle nature of the print directly, but rather background matters once more. Strictly there is nothing of real interest here as regards cluster puzzle interest.

Plane Fillings I and II 

In Graphic Work

f. Irregular filling of plane surfaces.


The next two prints consist of figures that do not in any way repeat themselves in similar form. So they do not really belong to group II; nevertheless they were added to it because they do in fact have their surfaces filled up, with no spaces left empty. What is more, they could never have been produced without years of training in regular surface-filling. The recognizability of their components as natural objects plays a more important role. The only reason for their existence is one’s enjoyment of this difficult game, without any ulterior motive.

Here, as well as an introductory, both of the Plane Fillings are discussed, albeit of peripheral matters.

George Escher email 2017

In regards to cluster puzzles I contacted Escher’s son, George. He told me:

I can tell you that although we were familiar with some Simplex puzzles, I cannot remember ever discussing their designs with father, or him showing any interest in them. When talking about Plane Fillings I and II he mentioned that he created them just out of curiosity, and that Plane Filling II allowed him to use "real" objects with "reality" constraints on their contour (e.g. the guitar), and also that he had the freedom to choose how many designs of one colour he wished to use to delineate a design of another colour. In general, the lack of constraints killed his interest in this kind of tessellation.

I must admit, the comments here, of relative extensive extent, in the round, are somewhat disappointing, in that the cluster puzzle premise is not detailed, but is rather passed over. Instead, generalities are discussed, with simple calculation and observational comments thereof.

Part 2

Commentaries by Others

As alluded to above, these prints have been discussed by others of some degree of extent, including the Escher authority Jeffrey Price in M.C. Escher Amazing Images, p. 21* on Plane Filling II, Erik Kristen in Escher Het Paleis (Escher in the Palace) and Bruno Ernst in Magic Mirror. Although there are indeed others, these are so lightweight, of a brief mention in passing, as not deemed worth mentioning. There may be other instances, but of an initial survey of the literature, this is what I found. 

* The book is not paginated, the page number is from my own count.

Jeffrey Price, 2011

In M.C. Escher Amazing Images, ‘p. 21’

Price discusses the print in relative depth, although I don't always agree with his points. For instance,  he relates the white bird to that in Day and Night. I don't agree; the birds are notably different. He also added some other speculation of his own as to the reasons why Escher included specific objects, such as a guitar and snail, but left open-ended. Again, this does not discuss the cluster puzzle intricacies of the print.  

Erik Kristen (1), 12 July 2017

In ‘Escher Het Paleis’ (Escher in the Palace)

Kirsten discusses both prints on two separate occasions, oddly out of sequence, on 12 July 2017 (Plane Filling II) and 9 March 2019 (Plane Filling I), albeit in differing degrees of depth; Plane Filing I much more than Plane Filling II. Plane Filling I is much more expansive and can be considered the first detailed treatment. However, again, I have reservations here, but even so, this is a more than welcome contribution. For instance, he compiles a near-complete inventory.

This is Plane Filling II, a lithograph from July 1957 without an underlying system. The shapes extend out in all directions. It may well be the weirdest print from Escher’s oeuvre.


There is no other work that features this many different animals, people, satyrs and other bizarre creatures: 21 black ones and 19 white ones. Six fish, four birds, a walrus, an elephant, a monkey, a frog, a camel, a kangaroo, seven humanoids (among them an E.T.-like creature), a meditating buddha, two devils, a lobster, a kind of platypus, a dragon, a unicorn, a snake, several dogs, a braying donkey, a turtle and a snail. And a guitar.

Erik Kristen (2), 9 March 2019

Plane Filling I

In March 1951 Escher produced a print with the deceptively simple name Plane Filling I. I say ‘deceptively simple’ because at that point in time he had been a graphic artist for 30 years and had already produced countless tessellations. The principle of the regular division of the plane formed the core of his artistry, the subject to which he always kept returning. Why, then, did he suddenly produce a work that seems to suggest it is the first time he is tackling such a subject?

Plane Filling I is an anomaly, a tessellation that does something diametrically opposed to the typically Escherian ones featuring birds, fish, insects and other recognisable figures. Sometimes he combined two or three of those figures in a print, but all these tessellations are characterised by the principles of symmetry, repetition and reflection. In Plane Filling I none of these principles are respected—every figure is different and nothing is mirrored or repeated. It is an experiment in which Escher shows that it is possible to have a tessellation with forms that are all dissimilar. Plane Filling I does contain two other principles that he used in his regular divisions of the plane: recognisability and colour contrast. Escher always strove to use identifiable forms in his tessellations. Although sometimes difficult to define, and sometimes featuring animals that are more in the realm of fantasy than reality, all these forms are immediately recognisable as ‘animals’. Or, in a number, of cases as ‘people’. The second characteristic is colour contrast—as in a chessboard, every white figure alternates with a black figure, both horizontally and vertically. Each row contains six figures and, because there are six of those rows, the tessellation is populated by 36 figures, 18 white and 18 black. Incidentally, this alternation between white and black is entirely logical. If everything were to be the same colour without any contrast, then all the figures would be invisible.

In 1957, Escher would once again venture to produce such a print, filled with all kinds of special creatures that are all different. Number two goes one step further: whereas the creatures in Plane Filling I are arranged roughly in a checkerboard pattern, in Plane Filling II he sets them free completely. In a lecture that Escher would have given in Canada in 1964 (it was cancelled due to health problems), he would have said of the two pieces:

‘Yet each of them has the form of something, be this either a living being or an object, which the viewer “recognises”. Putting such a tessellation together is a tiring activity and at the same time a thoughtless game. It tires the draftsman, as if he were not the ringleader himself, but as though, stripped of his will, is allowing his creatures the freedom to determine their own shape and character.’

By far, this is the most detailed discussion of all, Escher included. However, it is still lacking. This is addressed in Part 3, with comments of my own.

Bruno Ernst. In Magic Mirror, p. 77

… On closer inspection we discover that all the birds are different; so we are dealing with one of the very few entirely free surface-fillings that Escher has made. (Mosaic: I, 1951, and Mosaic: II, 1957).

Of a general discussion on Sun and Moon, concentrating on the figure and ground aspects, Ernst notes the cluster puzzle aspect, but again, is of a lightweight nature.

Part 3

David Bailey Commentary 

Of note is that despite the three works all being of a cluster puzzle nature, they all differ subtly in their respective approaches. Of note is that all instances are of the higher standards, of whole-bodied motifs, with generally good articulations, and furthermore all upright, or mostly so, and not of the lower quality ‘amputations’ and as used by others above. They are all rendered to a high quality, and do indeed have contrast, and so are readily identifiable, something which is generally lacking in others.

Of note is that all three works consist of a variety of animal and human-like forms, some instantly recognisable and some of an extremely fanciful nature. However, there is one exception, where a non-animal motif is included, namely a guitar, in Plane Filling II.

Further distinctions of the animal theme can be made. Sun and Moon is solely of birds, whilst Plane Filling I and II are of animals, of various kinds. 

It can be seen that Escher does not merely repeat himself here each time; in effect, he sets new challenges. This is easily shown by drawing the underlying tiles. Sun and Moon is based on equilateral triangles, Plane Filling I quadrilaterals, all of the same broad proportions, and Plane Filling II without any real structure at all, of triangles, quadrilaterals and pentagons.

The three works of a motif count consist of 24 (Sun and Moon), 36 (Plane Filling I) and 40 (Plane Filling II) motifs. Whether by design or not, the motif count increased with each successive work. The number is relatively high in the context of cluster puzzles, but by far from being exceptional, or even unduly high. 

For Plane Filling I and II below, I give an inventory of the motifs for analysis purposes. Sun and Moon is excluded, consisting entirely of generic birds. However, although it may be thought that this would be a simple task, with one-to-one correspondence, of a bird, fish etc., in practice this is not so. Around half of the motifs, due to their fantasy aspect, are ambiguous as to their description. For instance, there are creatures consisting of parts of two animals. Other creatures are wholly fantasies. 

Therefore, the listing below should be read with all this in mind. In short, it can be regarded as an indication, and nothing more. I now discuss each work in turn, in detail. 

Sun and Moon

As a bald, simple statement, under the overarching title, the print consists of a single subject matter, of birds, underpinned by a grid of equilateral triangles and, of the extremities, a ‘near equilateral’ triangle. 28 birds are shown, of alternate colours. The birds are all broadly generic, in typical Escher portrayal. 

Interestingly, rather than a new design, Escher in effect reuses periodic drawing 71 (as pointed out by Schattschneider) with the central region of a 2 x 6 configuration based on those birds. From this base, he then continues in a free-form manner, i.e, as a cluster puzzle, without any repeat nature as previously. Of course, the birds at the extremities, having no need of a double contour line, can be made more exact as to bird anatomy.

An analysis of viewpoints:

Under View 16

Over Views  4

Above 4

Both Wings Upright 3

Unclear 1

Plane Filling I

As a broad statement, a life-like themed print, of 36 animals and human figures, mostly of a fantastical nature. 

An inventory gives:

Fish (7); Bird (6); Quadruped (4); Fantasy (3); Bat (2); Human (2); Bug, Jellyfish, Reptile, Snake, Tortoise (1)

All are based on a broad ‘average’ quadrilateral, nowhere excessively broad in height and width. Quite how Escher began here is unclear. Possibly, rather than immediately with quadrilaterals, he started with a rectangular grid, possibly in the centre, before the demands of the creatures dictated a strict loosening. However, all remain clearly identifiable in silhouette as animals and are not merely (inferior) surface design added to arbitrary shapes. On the rectangle premise, Erik Kersten outlines a possibility, but if so I doubt very much if Escher began in this manner, or at least as strictly implied by Kersten. However, he does make some interesting points, pointing out, in effect, the cluster puzzle nature:

...but all these [periodic] tessellations are characterised by the principles of symmetry, repetition and reflection. In Plane Filling I none of these principles are respected—every figure is different and nothing is mirrored or repeated. It is an experiment in which Escher shows that it is possible to have a tessellation with forms that are all dissimilar. 

The print is coloured as according to Escher’s self-imposed rule of contrast, effectively of black and white. Given that each motif is based on a quadrilateral, each motif is surrounded by four others. This contrasts with Plane Filling II, where the surrounds differ, as pointed out by Escher.

Plane Filling II

As a broad statement, a life-like themed print, of 40 animals and human figures, some of a fantastical nature, with one exception, a guitar. 

An inventory gives:

Fantasy (8); Fish (6); Humanesque (4); Unidentified (2), Quadruped (2); Bird, Buddha, Camel, Cockerel, Dog-like,  Dragon, Elephant, Donkey, Frog, Guitar, Human, Kangaroo, Lobster, Rabbit-like, Reptile-like, Simurgh, Snake, Turtle (1) 

Here, the underlying structure is decidedly less so than in his other two cluster puzzle works. Indeed, save from saying that these show various polygons, this can be considered unstructured in its underpinning. Escher can only have designed this by beginning with an animal, likely in the central region, and then successively continued adding other motifs, stopping within a broad rectangle frame.  All remain clearly identifiable in silhouette as animals and are not merely (inferior) surface design added to arbitrary shapes.

The print is coloured as according to Escher’s self-imposed rule of contrast, effectively of black and white.

Of note here, as regards the motifs, is the appearance of a guitar, the only manufactured object in the print (and indeed, of all three works). Its appearance is obviously somewhat out of kilter. Although Escher mentions the motif in his writings, this is in passing, there is no explanation as to its appearance.

Of interest here are what can be called constraints. Strictly, a guitar is restricted by its shape, essentially of a single outline; there can be no leeway as with other animals, of which they can twist and move in various ways and so can be portrayed in different ways and yet still remain recognizable.

Each motif is surrounded by others in different ways, of three, four or five motifs. This contrasts with Plane Filling I, where due to the quadrilaterals underpinning each tiles is surrounded by four others, as pointed out by Escher


1948, April, Sun and Moon, Woodcut

1951, March, Periodic Drawing 83, used for Plane Filling I

1951, March, Plane Filling I, Mezzotint 

1957, July, Plane Filling II, Lithograph

1960, Grafiek en Tekeningen M. C. Escher published. Shows Plane Filling II only

1963, George J. Pulaus sends letter to Escher, including his Cluster Puzzle work, Animals

1964, Replies to George J. Paulus, enclosing Plane Filling I.

1970, The Graphic Work of M. C. Escher, in English, published. Shows all three works, albeit of different sections.


Escher, M. C. The Graphic Work of M. C. Escher. Oldbourne, London 1970.

Grafiek en Tekeningen M. C. Escher 1960  Koninklijke Uitgeven Van de Erven J. J. Tijl N.V., Zwolle 

Wilson, Janet, editor. Ford, Karin (translator). Escher on Escher. Exploring the Infinite. Harry N. Abrams, Inc. 1989.

Price, Jeffrey. M.C. Escher Amazing Images. Self Published. 2011. 

Ernst, Bruno. The Magic Mirror of M. C. Escher. Tarquin Publications, 1985.

Web (Sun and Moon) (Plane Filling I) (Plane Filling II)

Private Correspondence

Escher, George. Email of 5 March 2017.

Page Created 30 April 2020