M. C. Escher - Research

An ad hoc assembly of some of my Escher researches, with latest entries at the top of the page

‘Huize Ekeby’ (House Ekeby, Escher’s one-time residence in Baarn) 10 August 2021.

Upon recent research in the round (August 2021) of an article on Escher by Elly Witesenburg-Horsman, in Baarnse, ‘Baarnse Personalities, ‘Huize Ekeby’ was mentioned and illustrated. As such, I did not recall having seen this before, but upon research I saw that I was indeed familiar with it, but only in passing. First, I looked in two other publications: Paleis soestdijk escher wandelingen in de tijd van Italië naar Baarn and then Ontmoeting met Escher, below, of which of the latter I noted the new finding in my bibliography notes. There my active research has ended; there does not appear to be any other publication with this pencil drawing and/or commentary. That below is my initial research on it.

‘Huize Ekeby’ is also shown/discussed in:

Witsenburg-Horsman, Elly. In Baarnse, ‘Baarnse Persoonlijkheden VIII: M. C. Escher’, 1982, pp. 4-8 

Translated: Baarnse Personalities VIII: M. C. Escher

‘Huize Ekeby’ Text, p. 4, illustration p. 5

After Rome, Château d'Oex in Switzerland and Brussels, our village [Baarn] may have seemed like a quiet place. It was not that quiet either, because during the war he was evicted from his house by the Germans. The Eschers found shelter in Ekeby, the spacious housing with extension of the Patijn family on Van Heemstralaan. When this villa was demolished in the fifties, Escher built his own house there, devoted as he was to this beautiful spot. Was it any wonder that he was mainly concerned with the beautiful studio, which soon became very dear to him? The red house encompassed the whole family; his wife found peace there, the “children” came to stay, and MaurIts himself worked there to his heart's content.

Author/s unclear. Ontmoeting met Escher. Exhibition catalogue, Kasteel Groeneveld, Baarn, 1982

= Meeting with Escher

Entirely in Dutch, small format, of 48 pp. Of a 15 April to 11 June 1984 exhibit at the Stedelijk museum, Sint-Niklaas. With contributions by J. W. Vermeulen, W. F. Vermeulen, and D. Anthuenis.

See p. 34, of a picture of Huize Ekeby, without any obvious commentary in the main text. The caption merely states: ‘Huize Ekeby’

Toonen, Ellen. Paleis soestdijk escher wandelingen in de tijd van Italië naar Baarn. In Dutch. Translated: Soestdijk Palace ESCHER walks in time from Italy to Baarn. Amsterdam. 2012. 80 pp. soft cover. In color and b/w. White. Catalog for exhibition MC Escher 17 August - 18 November 2012 in Paleis Soestdijk, Baarn. See p. 79 for Huize Ekeby. (5 August 2021)

Available online:






Pencil, drawing

In 1943 Escher is forced to evacuate his house by order of the Wehrmacht. His friend Patijn comes to the rescue and makes part of his home Ekeby, at 56 van Heemstralaan, available.

The rent for occupancy and additional heating costs is only twenty guilders. Patijn himself lives in the front house, but he is seriously ill and will soon die.

The back house of villa Ekeby is occupied by baronet A. E. Rutgers van Rosenberg with his wife. The studio is only accessible through the Secret Annex. In exchange for the nuisance, he gives the baron, among other things, this drawing of villa Ekeby. That villa has since been demolished.

Original Dutch:


Potlood, tekening

In 1943 wordt Escher, op bevel van de Wehrmacht gedwongen zijn huis te ontruimen. Zijn vriend Patijn komt te hulp en stelt een deel van zijn woning Ekeby, aan de van Heemstralaan 56, beschikbaar.

De huur voor bewoning en extra stookkosten bedraagt maar twintig gulden. In het voorhuis woont Patijn zelf, maar hij is ernstig ziek en compt spoedig te overlijden.

Het achterhuis van villa Ekeby wordt bewoond door jonkheer A. E. Rutgers van Rosenberg met zijn vrouw. Het atelier is alleen bereikbaar via het achterhuis. In ruil voor de overlast schenk hij de jonkheer onder andere deze tekening van villa Ekeby. Da villa is inmiddels afgebroken.


1. Escher’s Missing Periodic Drawing 135 (23 July 2021)

Doris Schattschneider, in Visions of Symmetry, p. 318, simply notes in regards to Escher's numbered drawings, of 1–137, there is no number 135. I’ve long known about this missing drawing and previously I thought it was just simply a clerical error on Escher’s part, i.e. no such drawing existed. However, after correspondence with Nicholas Walker, an Escher enthusiast and collector, who asked me if I knew anything on this, thinking on this led in short time to an amazing discovery, the missing Escher periodic drawing! The missing drawing was found, (largely) serendipitously, in an article by Sjoerd Hekking, ‘Zuilen van Escher Gered!’ (In Dutch), in TEGEL, a Dutch tile annual journal, of 2013. Translated: Columns of Escher saved! This I relate below with an edited email to Walker, which contains the story, and again with Doris Schattschneider, who has confirmed the validity of my conjecture. In truth, the finding, in terms of importance of a new drawing, is a little disappointing, in that the drawing is simply essentially the same as the next one in the sequence, 136, which corrects an error in 135. Even so, it's not every day a new Escher turns up, and in any case it at least clears up the mystery! As such, this is an interim report; the investigation is still ongoing to an extent. There are still aspects to investigate. The reproduction is not ideal, in that I believe the drawing is cropped. The author has been contacted and I am awaiting a reply.

The article can be found here:


At the end of a 20 July email by Nicholas Walker on Escher matters he added the throwaway comment (he has told me this was so):

Do you recall which periodic drawing is missing? Michael Sachs told me once. Anyway, one is. The running theory is Escher must have loaned it to someone and not got it back. The reason this is assumed is that some of his periodic drawings have address stamps on them as to indicate that is where they should be returned.

I’ve long known about this missing drawing (from Visions) and previously I thought it was just simply a clerical error on Escher’s part. But now I believe I have found it! In short, it is a precursor of 136, containing an error, of which 136 thus corrects. What follows, slightly revised and expanded, is my train of thought in replying to Nick:

This has got me thinking, especially on the address stamp aspect. I have once more examined the drawings in Visions in this regard. Only three of these have his address stamped, namely 126, 134, and 136, and notably all relating to the Baarn columns. (As an aside, he frequently used the same address stamp in his latter-day correspondences.) Given that (as a number per se) 135 is between 134 and 136, a distinct possibility is that the missing drawing is of the same type. Anyway, for my general reading up on the columns, I then turned to an in-depth article I have (although seemingly little known):

Hekking, Sjoerd. ‘Zuilen van Escher Gered!’. (In Dutch) TEGEL, 41, 2013, pp. 36-42.

Translated: Columns of Escher saved!

Of note is picture 4, p. 38. On a casual glance, this could be mistaken for 136, as it shares many characteristics of that periodic drawing (as I must have thought, I believe, when I first read the article (in 2014) and thought nothing more of it at the time or since). However, viewed more circumspectly today, it can be seen that this is not so, as there are subtle differences! For reasons as I explain below, I can now only believe that it must be the missing 135! That said, this is not definitive, as the drawing does not explicitly state 135, but this can be explained. I would be amazed if it was otherwise; there are no less than three different aspects of evidence to support my conjecture:

1. The general style of the drawings, which is essentially the same.

2. The fishes’ eyes, of which Escher comments on 126 are in error. Therefore, I surmise that he must then have realised the discrepancy late in the day on this drawing, and then redrew, which thus is 136.

3. The address stamp, in line with the other three periodic drawings listed above, but no others.

That said, a slight drawback to my conjecture is that 126, 134 and 136 bear the comment ‘after using, return to’, whilst picture 4 (my putative 135) does not. This seems a little odd, in that being in the middle of 134 and 136 one would expect this comment to appear. Also, the all-important number is missing. However, a plausible explanation for this is that it has simply been cropped for the demands of the article. However, to me at least, these are minor objections. The minor quibbles aside, to me, this is proof of it being 135 beyond all reasonable doubt! 

The picture credit is interesting in that it cites the Rijksdienst voor kunsthistorische documentatie (National Office for Art Historical Documentation). How it ended up there, separated from the three others, if my conjecture is correct, is a mystery! Of course, I then investigated. However, upon searching for Escher on their site, picture 4 is not there! But given such a national institution, there must be a record of it somewhere in their archive:


I have now read (Google translated) Hekking’s entire article, without gain, beyond establishing there is nothing of direct interest to the investigation. Picture 4 is only discussed briefly in the text:

In his design, Escher named the columns. Column A became 'Birds and Fish', column B 'Flowers' (fig. 4). To allow the pattern of the motifs to continue, two different tiles had to be made for both columns. Tile A1 got a white fish and tile A2 got a black bird with white fish. Two types of tiles were also required for column B. The image is the same for both, but the rounding of the column had to be taken into account. To do this, tile B2 had to be turned a quarter turn with respect to B1.

It would be interesting to hear what Hekking and also the institute has to say on the drawing. However, I have not been able to find much on Hekking in the round. Here is the translated text on him:


Sjoerd Hekking (1951), trained at the Academy of Fine Arts in Amersfoort as a crafts teacher, has been a teacher of Art Visual Education at the Baarnsch Lyceum since 1977. The school building (1969) was only eight years old. When, after almost forty years, it was decided to build a new building, he saw it as his task to save Escher's columns. He takes a great interest in Escher's work and is a collector of ceramics from the 'new art' period.

2. Square Limit (updated June/July 2020)

In short, Square Limit is a mixture of inconsistencies and anomalies. A few points:

1. The fish motif is based on periodic drawing 119. Note here the four-fold symmetry: four heads meet at a vertex as do four tails and a right lower fin. Of note is that there is no reflection symmetry. However in the print, although this is indeed the arrangement, with suitable minor adaptation, the fishes are mirrored!  Why is this? Is it a necessity, caused by the adaptation of the plane tiling to the underlying square limit framework? No. This is not necessarily so.  For instance, Branko Grünbaum in ‘Mathematical Challenges in Escher’s Geometry’, p. 69 (in M.C. Escher: Art and Science), gives a Square Limit of his own devising, of indented tiles, that are entirely consistent throughout, truly self-similar and without distortion or reflection. Therefore I query the aesthetics of Escher’s print, in that it can be described as strictly ‘lacking’ in comparison. Ideally of course, it would exemplify Grünbaum’s ‘perfect’ instance. Of interest is how Grünbaum devised this, with each of the two different side lengths consisting of three straight line elements. Unfortunately, no detail is given. Simply deforming an isosceles right-angled triangle tile with appropriate symmetry indentations is not enough. In the course of my own investigations (1994) I showed three instances, vaguely bird-like, again with side lengths consisting of three elements, with one exception. Only No. 3 is perfect, as by the ‘Grünbaum model’. No. 1 and No. 2 can be seen to have distortions to greater or lesser degrees. No. 1 can be described as aesthetically wholly unacceptable; the tiles are distorted beyond all reasonable recognition. No. 2  can be seen to retain more of the tile, and is more recognisable of its source, although still a little distorted. What explains these discrepancies? I don’t know! Why a line with indentations will tile ‘perfectly’ in some instances but not others is unclear.


Bool 443

Woodcut in red and black, printed from two blocks

April 1964, drawing 119, February 1964

Graphic Work, page 15; Magic Mirror, page 103-105, Escher on Escher, page 41; Visions of Symmetry 252-253 (letter to H.S.M. Coxeter, 1964, page 253); Magic (letter to Gerd Arnzt, April 1964), page 182.

Preparatory drawings in: World, page 256 and Magic, page 183.

After a relatively brief period of four years since the Circle Limit prints, Escher then returned to this theme of reduction outwards, with a square border rather than a circle as previously. For this, he composed a suitable framework of his own making, based upon a series of right-angled isosceles triangles that systematically reduce in size as the (square) boundary is approached. As such, this framework can be seen in earlier prints, of a differing format, namely with Smaller and Smaller (1956) and Regular Division of the Plane VI (1957). Somewhat surprisingly, for such a simple diagram that essentially anybody could compose, the first such occurrence was indeed of Escher's own devising, of which credit must duly be made.

  As to the print itself, fish-like motifs pertaining to the framework thus reduce in size towards the edge, of which again, as with the Circle Limits I, III and IV these are viewed as seen from above. As such, the reason for such a relatively poor fish-like motif is that the demands of the framework require a line that is of a certain sinuous nature. As such, it may be quite reasonably thought that any tessellation based upon a right-angled isosceles triangle, possessing the correct symmetry (of which a 180° rotation along the hypotenuse, with 90° at the apex) would suffice. However, this is not so, as upon trying to do so with one of my own tessellation, the distortion was too great to be regarded as acceptable, resulting in a malformed creature. Therefore, a certain amount of ambiguity is ideal, as in Escher's fish motifs here.

Books and Articles

Anon. M. C. Escher Universe of Mind Play, pp. 132, 153-163, 166, 170

Print p. 132; numerous sketches but no discussion, pp. 153-163, 166. Cancelled wood blocks, p. 170

Bell, Marc. The Magical World of M.C. Escher, p. 164.

Just the print and details, no apparent discussion.

Coxeter, H. S. M. et al. M.C. Escher: Art and Science, pp. 57-60.

Branko Grunbaum ‘Mathematical Challenges in Escher's Geometry’, pp. 53-67, Square Limit discussion pp. 57-60

Ernst, Bruno. The Magic Mirror of M. C. Escher, pp. 103-105.

An in-depth discussion by Ernst, the best of what little there is.

Escher, M. C.  Regelmatige vlakverdeling

See Wilson, which reprints this.

Escher, M. C. The Graphic Work of M. C. Escher, p. 15. 

Brief commentary of a few lines of just basic detail, print, No. 25

Locher, J. L. The World of M. C. Escher, pp. 256-257.

Preparatory drawings, p. 256 and print, p. 257, but no discussion.

Locher, J. L. The Infinite World of M. C. Escher, p. 148.

Print, p. 148, no discussion.

Locher, J. L. et al. Escher The Complete Graphic Work, pp. 118, 325, 343.

Bool catalogue 443. p. 325, p. 343 (no other apparent references) Square Limit type division digam in p. 169, in reprint of Regelmatige vlakverdeling, published 1958

Schattschneider, Doris. Visions of Symmetry, pp. 252-253.

Letter to H. S. M. Coxeter, 1964, p. 253. A relatively in-depth discussion.

Schattschneider, Doris. ‘Coxeter and the Artists - Two-way Inspiration’. In The Coxeter Legacy: Reflections and Projections pp. 262-280. 2006. Edited by Harold Scott Macdonald Coxeter, Chandler Davis, Erich W. Eller. Fields Inst. Comm. ser. no. 46, Amer. Math. Soc., 2005

Primarily of a detailed discussion on Coxeter and Circle Limits, with a brief, in passing references to Square Limit p. 266 with an  illustration and brief discussion, possibly continued, but curtailed in Google Books! However, it seems unlikely to be extensive.

Teuber, Marianne L. 'Sources of Ambiguity in the Prints of Maurits C. Escher.' Scientific American 231 No. 1 (July 1974): 90-104. See pp. 89.

Thé, Erik (designer). The Magic of M. C. Escher, p. 182.

Large scale print, p. 182, and accompanying letter to Gerd Arnzt, 14 April 1964 (quoted by Ouyang), and detail, p. 183, but no discussion as such.

Wilson, Janet, (editor). Escher on Escher, p. 41.

Brief reference to Square Limit, p. 41, in prepared notes of Escher’s proposed 1964 US lecture. A minor discussion in passing, in context of infinity, with Development II, Path of Life I, Whirlpools, Sphere Surface with Fish, Smaller and Smaller, Square Limit, Circle Limit IV, and Circle Limit III. In short, inconsequential.

Not in (or believed to be):

Jeffrey Price Amazing Images

Letters to Canada

M.C. Escher’s Legacy

List of such types:

Smaller and Smaller I, October 1956, Bool 413

Plate IV Regelmatige, June 1957, Bool 421

Regelmatige, 1958

Circle Limit I, November 1958, Bool 429

Circle Limit II, March 1959, Bool 432

Circle Limit III, December 1958, Bool 434

Circle Limit IV, July 1960, Bool 436

Square Limit, April 1964, Bool 443

An open question is to when Escher devised the ‘square limit’ process. This can be dated as of least, and likely, of October 1956, where Smaller and Smaller I shows the process later used in a different form in Square Limit.


Square Limit




3. A Listing and Examination of Escher’s Spiral Themed Works

Following recent 2020 correspondence with Peichang Ouyang, in which for a joint paper, including myself and others, on ‘Generation of Advanced Escher-like Spiral Tessellations’, as part of the study a short paragraph on Escher’s spiral works was given, of a complete, or intended, listing. In the course of this compilation, given that there is apparently no such listing (perhaps surprisingly), I consulted the main reference of his prints, namely Escher The Complete Graphic Work by J. L. Locher, which includes an all-inclusive print listing with a Bool catalogue number, and of which I refer to below. However, although one can extract from this listing a one-to-one correspondence, there are various nuances and problems which should be considered for numeration purposes. For instance, there are different versions of the same work (Rind).

Do I count these as distinct works or two? Some works possess (or are said to) a spiral underlying grid that in the finished work is not obvious (Butterflies, Division). Again, how to count? Further, the works can be differentiated into tessellation and non-tessellation. Therefore, to say categorically how many Escher-like spiral tessellations there are is not a straightforward task as it may otherwise appear. To this end, I broadly consulted the main books, albeit not from cover to cover; I simply do not have the time to study all books for what are mostly limited references. Indeed, some books have only been skimmed over.

What struck me about this research was how little in-depth discussion on Escher's spiral works there are. Indeed, none address this issue in its entirety, or indeed the larger majority of the works, with largely ad hoc discussions of individual works of at best a few lines. Certainly, there is no dedicated discussion.

Therefore, for a variety of reasons this research is thus subject to considerable revision in presentation and format.

As even, an open invitation, can anyone add to what is here, or care to comment?

Below, I give a listing in two ways:

(i) Commonly accepted spiral works taken from Escher The Complete Graphic Work, along with the year and Bool catalogue number in brackets.

(ii) References as discussions in books and articles.


This is of three orderings: ‘all’, and then filtered as to ‘tessellation’, and ‘non-tessellation’. 

* Included with considerable reservation

1. Development II, 1939 (310)

2. Butterflies*, 1950 (369)

3. Spirals, 1953 (390)

4. [Study for Rind], 1954 (396)

5. Rind, 1955 (401)

6. Bond of Union, April 1956 (409)

7. Division*, July 1956 (411)

8. Smaller and Smaller, 1956 (413)

9. Whirlpools, 1957 (423)

10. Path of Life I, 1958 (424)

11. Path of Life II, 1958 (425)

12. Sphere Surface with Fish, July 1958 (427)

13. Aula, 1958

14. Sphere Spirals, October 1958 (428)

15. [Path of Life III], 1966 (445)


1. Development II, 1939 (310)

2. Butterflies*, 1950 (369)

3. Division*, July 1956 (411)

4. Smaller and Smaller, October 1956 (413)

5. Whirlpools, 1957 (423)

6. Path of Life I, 1958 (424)

7. Path of Life II, 1958 (425)

8. Sphere Surface with Fish, July 1958 (427)

9. Aula, 1958

10. [Path of Life III], 1966 (445)


1. Spirals, 1953 (390)

2. [Study for Rind], 1954 (396)

3. Rind, 1955 (401)

4. Bond of Union, 1956 (409)

5. Sphere Spirals, October 1958 (428)

Alternate Listing



1. Development II, 1939 (310)

2. Spirals, 1953 (390)

3. [Study for Rind], 1954 (396)

4. Rind, 1955 (401)

5. Bond of Union, April 1956 (409)

6. Whirlpools, 1957 (423)

7. Path of Life I, 1958 (424)

8. Path of Life II, 1958 (425)

9. Sphere Surface with Fish, July 1958 (427)

10. Aula, 1958

11. Sphere Spirals, October 1958 (428)

12. [Path of Life III], 1966 (445)


1. Development II, 1939 (310)

2. Whirlpools, 1957 (423)

3. Path of Life I, 1958 (424)

4. Path of Life II, 1958 (425)

5. Sphere Surface with Fish, July 1958 (427)

6. Aula, 1958

7. [Path of Life III], 1966 (445)


1. Spirals, 1953 (390)

2. [Study for Rind], 1954 (396)

3. Rind, 1955 (401)

4. Bond of Union, 1956 (409)

5. Sphere Spirals, October 1958 (428)

References, Works Consulted




Coxeter, H. S. M., M. Emmer, R. Penrose, and M. L. Teuber, eds. M.C. Escher: Art and Science. Amsterdam: North-Holland, 1986.

Branko Grünbaum p. 58. Path of Life I and III, but not discussed as a spiral per se. 


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

Escher gives a general discussion on a broad range of his works, as according to ten categories, but not by a spiral grouping as such. Of the commonly accepted (or said to be) spiral prints, these include, filtered:

Path of Life II, Sphere Surface with Fish, and Whirlpools mention spirals.

Smaller and Smaller I, Butterflies, no mention is made of spirals. 


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

In Chapter 15, ‘An Artist's Approach to Infinity’, pp. 102-111, Ernst discusses some of Escher’s spiral prints (better than most), pp. 103, 106-107. P. 103 gives a brief discussion on Development II, Path of Life I-III, Butterflies and Whirlpools, albeit he includes the last with reservation! In contrast, Butterflies is stated to be ‘spiral’, the only such reference I have seen. However, it is not overt by far, and of which I refrain from categorically considering as such, although I leave this possibility open-ended. Somewhat alarmingly, Ernst only considers Whirlpools, which is overt, as an afterthought! A subheading of ‘Birth, Life, and Death’ on pp. 106-107 mostly discusses the three Path of Life prints (in considerable depth), with occasional reference to Development, Butterflies, and Whirlpools. Here he asserts the underlying Butterfly spiral framework, acknowledging the lack of the network in the drawing shown, of which I query above.


Forty, Sandra. M C Escher. Taj Books 2003

Smaller and Smaller I, frontispiece, Rind, p. 68, Sphere Spirals, p. 78, Path of Life III, p. 83. No commentary. 


Locher, J. L. (general editor). Escher The Complete Graphic Work. Thames and Hudson, 1992.

Preliminary pencil studies for Spiral, p. 71, Butterflies, p. 76, Bond of Union, p. 83, Whirlpools p. 95,  Sphere Surface with Fish, p. 96. There is too much material to survey on this, but there are no obvious discussions.


————. The World of M. C. Escher. Abradale Press, 1988 

Prints without commentary: Rind, p. 68; Spirals pp. 206-207; Bond of Union, p. 221; Division, p. 224; Whirlpools, pp. 228-229; Path of Life I, p. 230; Sphere Surface with Fish, p. 234; Sphere Spirals, p. 235; Path of Life II, p. 237; Path of Life III, p. 260.

Coxeter alludes to spiral matters with a discussion on Smaller and Smaller I, p. 53.


Price, Jeffrey. M. C. Escher Amazing Images. (privately published book/catalogue).

Spirals, pp. 75, 81. P. 81 is a discussion.


Schattschneider, Doris. Visions of Symmetry, W. H. Freeman, 1990

Discussion on spirals, on Path of Life I-III, p. 248, P. 249, Whirlpools, ‘a double spiral’, Wall cemetery mural mention, p. 249. Whirlpools, Sphere Surface with Fish, discussions, p. 250. Development II, p. 291, mentions spiral. Path of Life III, p. 316-317, mentions spirals. Aula aspects, pp. 321-322, Sphere Surface with Fish, p. 322, discussion.


Schattschneider, Doris, and Michele Emmer, eds. M. C. Escher's Legacy: A Centennial Celebration (with CD Rom). Berlin and Heidelberg: Springer-Verlag, 2003.

Bruno Ernst, pp 14-15, Spirals, 1953 and discussion, Sphere Spirals, 1958. 


Thé, Erik, designer. The Magic of M. C. Escher. Introduction by J. L. Locher and foreword by W. F. Veldhuysen. New York: Harry N. Abrams, 2000. Joost Elffers Books Harry N. Abrams 2000. Foreword by W.F. Veldhuysen. Introduction by J. L. Locher.

Development I, p.67, 70 (detail), Spirals, p. 164, a minor mention mention in passing in a leet to George Escher

and Corrie. Eight concept sketches for Spirals, pp. 166-167, but no discussion. Whirlpools, p. 176, with minor commentary by escher on this,  and prints in various states p. 177. Sphere Spirals and Sphere Surface with Fish, p. 178, and prints in various states p. 179.


Wegman, William, designer. M. C. Escher: 29 Master Prints. New York: Harry N. Abrams, 1983.

Smaller and Smaller, p. 10. Escher comments on this, but no mention of any spiral features.


Wilson, Janet, editor. Ford, Karin (translator). Escher on Escher. Exploring the Infinite. Harry N. Abrams, Inc. 1989. With a contribution by J. W. Vermeulen. Compiled by W. J. van Hoorn and F. Wierda. Originally published under the title Het oneindige




Bell, Marc. Marc Bell Presents the Magical World of M. C. Escher. Boca Raton Museum of Art January 20–April 11, 2010

Occasional spiral prints, but not described as such. Development II, p. 78, with spiral mention, the only one!,  Banknote, p. 87 (in error?), Spirals, p. 127, Rind, p. 135, Bond of Union, p. 144, Whirlpools, p. 151, Sphere Surface with Fish, p. 153, Path of Life III, p. 166. No analysis of any kind.


M. C. Escher’s Universe of Mind Play. Tokyo: Odakyu Department Store,  Edited by Fuji Television Gallery, 1983.

Has spiral themed prints amid the usual collection of his works, but no discussion of any kind.


M.C. ESCHER, Magic art EXHIBITION ESCHER. Catalogue 1981. 1-12 May Isetan Museum of Art, Tokyo; 28 May-9 June Marui Imai Art Gallery, Sapporo; 17-22 September Daimaru Art Galley, Kyoto. 


Super Escher M.C. Escher, tracing the creative path of a unique print artist. Exhibition Catalogue 2006


M.C. Escher. The World of M.C. Escher. Exhibition Catalogue 1987. In English and Japanese

Has spiral themed prints amid the usual collection of his works, with commentary by ? at the end of the book

Development II, p. 68 (135),  Rind,  p. 92 (141), Bond of Union, p. 94 (142, with spiral made explicit),

Whirlpools, p. 97 (143, with spiral made explicit) Smaller and Smaller, p. 98 (143)


Various. The Collection of Huis Ten Bosch. M.C. Escher. 1994. Publisher: Huis Ten Bosch / Nagasaki 199 pages Language: English / Japanese 1st Edition, Illustrated.

Has spiral themed prints amid the usual collection of his works, along with ‘explanations of works’ section pp. 165-192, but in Japanese, unfortunately without a translation.


M.C. ESCHER, Exhibition 100th Anniversary of his birth, Kohga Collection /1996. In English and Japanese

Has spiral themed prints amid the usual collection of his works. A picture book, with effectively no discussion. Development II, p. 72, spiral mention in essay ‘The Art of Escher, (1940~1972)’ preceding p. 77; Spirals, p. 117; Rhind, p. 122; Bond of Union, p. 126; Smaller and Smaller, p. 128; Whirlpools, p. 133; Path of Life I, p. 134; Path of Life II, p. 135; Sphere Surface with Fish, p. 136; Sphere Spirals, p. 139; Path of Life III, p. 153. 



Chris den Engelsman. On the Aula. In Dutch


Another work for possible inclusion is ‘Division’. Although not overtly spiral, the underlying grid can at least seem to be. See pp. 9-11 specially of R. A. Berger's thesis:


Spiral Articles, not necessarily on Escher

Burgiel, H. and M. Salomone. ‘Logarithmic spirals and projective geometry in M.C. Escher’s Path of Life III’. Journal of Humanistic Mathematics, Volume 2 Number 1, January 2012, pp. 22-35.

Some advanced mathematics of a substantial article. Oddly, Escher’s Path of Life III is not shown, which seems strange given the premise, and the frequent mentions throughout.

Gailunas, Paul. ‘Spiral Tilings’. In Bridges 2000, 133-140

Nice treatment indeed. Comments on Grunbaum and Shephard comment on little literature on the subject. Building on their work, Gailunas shows a ‘Zig-zag spiral tiling’. It is not entirely clear the extent of originality here. I suspect  it may be based on others, with prominent use made of the versatile. Profusely illustrated. 

No Escher-like tilings.

Gardner, Martin. ‘Extraordinary nonperiodic tiling that enriches the theory of tiles’. Scientific American. January 1977 110-121.

On Penrose tiling. Minor mention and illustration of Voderberg spiral, p. 111. Reprinted in Penrose Tiles to Trapdoor Ciphers, pp. 2-4. 

Goldberg, M. ‘Central Tessellations’, Scripta Math. 21, 1955,  253-260.


Grunbaum B. and Shephard, G. C. ‘Spiral Tilings and Versatiles’, Mathematics Teaching, no.88, Sept. 1979, pp. 50‑51.


Grunbaum B. and Shephard, G. C. ‘Some Problems on Plane Tilings’. 167-196.  In The Mathematical Gardner

by David A. Klarner, 1981.

Voderburg tile as a plane tiling and spiral tiling pp. 189-192, 196, Versatile 192-194 as a spiral tiling

Grunbaum, B. and Shephard, G. C. Tilings and Patterns.  W. H. Freeman, 1987. 512-518  related p. 123

Chapter 9.5 on spiral tilings, pp. 512-516, 518. Begins with mention of Voderburg. Note and references  pp. 517-518, where they lament the lack of literature, but not here the definition [CHECK]

Grunbaum, B. ‘Patch determined tilings’. The Mathematical Gazette. 31-38.

In the context of ‘patch determined tilings’, a five-arm spiral tiling is shown, Fig. 10. 

No Escher-like tilings.

Hatch G. ‘Tessellations with Equilateral Reflex Polygons’, Mathematics Teaching, no.84, September 1978, p.32.


Kanon, Joseph. ‘The Saturday Review December’ 16 1972 ** 

Sphere Spirals

Klaassen, Bernhard. ‘How to Define a Spiral Tiling?’ Mathematics Magazine December 2017, pp. 26-38

On the difficulties of defining a spiral tiling, implicit building on Grunbaum and Shephard’s conjecture.

Shows Voderburg spiral. Largely popular, with occasional advanced maths.

Lalvani H. US Patent 4,620,998, 1986. Cited in Meta Architecture, in Architecture and Science (ed. Di Cristina G.), Wiley Academy, 2001.

Occasional spiral-like tilings, but not stated as such.

Mann, Casey. ‘A Tile with Surround Number 2’. The American Mathematical Monthly. Vol. 109 No. 4 April 2002 pp. 383-388

On coronas, something of which I am not particularly interested in. In the course of the ‘surround’ study, there is a Voderberg tile discussion, but not in the context of spirals.

Marcotte, James and Matthew Salomone. ‘Loxodromic Spirals in M. C. Escher's Sphere Surface’. Journal of Humanistic Mathematics Volume 4 Issue 2 July 2014

Palmer, Chris K. ‘Spiral Tilings with C-curves Using Combinatorics to Augment Tradition’. In Bridges Renaissance Banff 2005, pp. 37-46

The use of the word spiral in the title is somewhat overblown; it is then not mentioned again until the references page! Of no real interest as to spirals as such.

Pickover, Clifford. ‘Mathematics and Beauty: A Sampling of Spirals and Strange Spirals in Science, Nature and Art’. Leonardo Vol. 21, 1988, No. 2, pp.173-181

A good general, popular guide as to all things spiral applications as to the real world, and more, with much interest. No tiling or Escher-like as such.

Rice M. and Schattschneider D. ‘The Incredible Pentagonal Versatile’, Mathematics Teaching, no.93, Dec. 1980, pp. 52‑53.

Sharp, J. ‘Golden Section Spirals’. Mathematics in School. November 1997pp.  8-12

Of general interest in spirals. No tiling. Notable authors such as Keith Devlin and Ian Stewart are taken to task for misattributing the Nautilus shell cross section as a Golden Section Spiral.

Simonds, D. R. ‘Central Tesselations (sic) with an Equilateral Pentagon’. Mathematics Teaching No. 81, December (1977), pp. 36-37

————. Untitled note Mathematics Teaching 84 (1978), p. 33

Stock, Daniel L. and Brian A. Wichmann. ‘Odd Spiral Tilings’ Mathematics Magazine Vol. 73, No. 5 (Dec., 2000), pp. 339-346

Seemingly borrowing from Grunbaum and Shephard, Stock and Wichmann comment on the little literature on the subject. Building on their work, with a regular decagon on odd numbers of arm spirals, they show any number of odd numbered spiral tilings. A versatile is also shown.

No Escher-like tilings.

Tóth, Fejes L. Regular Figures. Pergamon Press 1964 (12 December 2010), partial copy, of Chapter 1 up to p. 43...

Regulare Figuren. Akademi kiado, Budapest , 1965. English translation

Largely theoretical. Mostly concerning group theory, which is out of my remit. Occasional tiling. Escher mention p. 39. Tilings Plates 1-3. As such, of what I have seen (Chapter 1 Plane Ornaments only), of no consequence (likely, the book is even more obscure in succeeding chapters).

Voderburg, H. ‘Zur Zerlegung der Umgebung eines ebenen Bereiches in kongruente’.

Jahresbericht der Deutschen Mathematiker-Vereinigung 46 pp. 229-231, 1936 

The first of two articles by Voderburg, in German. Four figures of Voderberg spiral tile, with one figure of the resultant spiral tiling.


 ————. ‘Zur Zerlegung der Ebene eines in kongruente Bereiche in Form einer Spirale’.

Jahresbericht der Deutschen Mathematiker-Vereinigung 47 pp. 159-160, 1937

In German. Two figures of Voderburg spiral tile, but not shown as an actual tiling.

Waldman, Cye H. ‘Voderberg Deconstructed & Triangle Substitution Tiling’ 2014. No article 

Much of interest; spiral tilings. Both popular and academic.


4. Flatworms

Upon (more or less) stumbling across a piece on Escher’s Flatworms print on the US ‘Indiana Illustrators and Hoosier Cartoonists’ blog (about the lives of Indiana's artists) some new detail has come to light. Incidentally, for those not in the know (as I was), Hoosier is the official demonym for a resident of the U.S. state of Indiana. This was a comment (at the end of the piece) ostensibly by Sherry Buchsbaum (although actually written by Monte S. Buchsbaum, her husband), curious in itself, and this opened up a whole new investigation. Specifically, this concerned a nuance on the print as to the flatworms themselves:


For convenience, I repeat the text by Sherry Buchsbaum/Monte S. Buchsbaum below:

Escher was definitely influenced by Elizabeth Buchsbaum's drawing of planaria. This can be seen in the chapter heading drawing for Chapter 10 and 12 and following drawings in Animals Without Backbones. The Buchsbaum originals were published in 1939. My father, Ralph Buchsbaum, visited Escher in the Netherlands and Escher showed him his edition of Animals Without Backbones. The Escher Flatworms (Catalogue 431) is dated 1959.

In short, this asserts that Escher was influenced in the portrayal of the Flatworms by Ralph Buchsbaum’s book, Animals Without Backbones (which went through different editions), and more specifically of the drawings, in black and white. Previously I was unaware of this detail. Anyway, I then followed it up with the people involved, of which the family chain has expanded, initially with Sherry, and then Vicki, below. To clarify matters, Sherry is the husband of Monte, and Vicki (also an invertebrate biologist) is Ralph’s daughter. I must say that these folks are most friendly and helpful! A later email gives more detail than the initial account on the Indiana blog:

From Vicki Pearse, 12 June

As I reconstruct it, Ralph's visit to Escher very likely took place in fall 1971, shortly after my parents had moved from Pittsburgh to Pacific Grove, California. I recall my father returning to the house in Pacific Grove after seeing Escher and our excitement at his bringing the prints. Therefore, if that memory is accurate, the visit had to have been somewhere in that window of a few months before Escher's death in March 1972. It's unlikely that the two men visited when Escher was very close to death, and also that my father would have traveled there in winter. Hence my estimate of the timing.

I don't know the circumstances of how they connected and arranged to meet. My father's passport from that period would give us more exact information about the date. If I locate it or any related correspondence, I can share it with you.

As can be seen, the date was thus likely in 1971. With the family provenance, one can indeed confirm the account on the blog. Admittedly, there is nothing here of a groundbreaking nature, albeit even so, it is pleasing to find the background to his rendition in the print. Further, I have finally been able to establish at least one book in Escher’s library, a long-held ambition! Many pictures show his bookshelves, but do not detail his books. As such, no other books are (surprisingly) seemingly known! Does any reader know of others? Was an inventory made? And what happened to Escher’s library?


Further, I then decided to investigate the print in more detail, with published references to it. As to references, I include all the major books on Escher, and other ‘likely candidates’, whether the prints above are included. This is so that I can say conclusively that the material has been examined, and is either discussed or not, and so save re-examination of a later date when I will have forgotten what I have surveyed. For other more ‘minor’ works, although indeed on Escher, there are simply so many that I don’t have the time to examine all. This being so, I will look for Flatworms upon normal, occasional re-reading.  As a broad brush statement, none of these are particularly in-depth. As such, Flatworms can be described as a lesser discussed print (fairly or unfairly) in that it is not discussed extensively as with others that have greater exposure, or of more obvious popular appeal, such as Day and Night. I might just add that Planaria are fascinating creatures, of which the work of James McConnell, in particular, is well worth reading.





Bell, Marc. Marc Bell Presents the Magical World of M. C. Escher. Boca Raton Museum of Art January 20–April 11, 2010

P. 154 shows the print also repeats the text from Escher in Graphic Work. There is no insight by Bell (or others).


Buchsbaum, Ralph. Animals Without Backbones. University of Chicago Press. Eleventh Impression 1947. First published 1938. (May 2019). Available on the Internet Archive:


Of peripheral Escher interest. Said (and confirmed by Sherry Buchsbaum, the daughter of the author in a reply to a blog posting, below, to be the book that Escher used for his Flatworm drawing references. Although obviously non Escher per se, it is included here in relation to him.

See Chapter 10 p. 109  and Chapter 12 p. 124. The book itself has acquired a degree of fame in the Planaria world. From Amazon: Animals Without Backbones has been considered a classic among biology textbooks since it was first published to great acclaim in 1938. It was the first biology textbook ever reviewed by Time and was also featured with illustrations in Life. Harvard, Stanford, the University of Chicago, and more than eighty other colleges and universities adopted it for use in courses. Since then, its clear explanations and ample illustrations have continued to introduce hundreds of thousands of students and general readers around the world to jellyfishes, corals, flatworms, squids, starfishes, spiders, grasshoppers, and the other invertebrates that make up ninety-seven percent of the animal kingdom.


Ernst, Bruno. The Magic Mirror of M. C. Escher. Tarquin Publications 1985 (first published 1972).

Shows the print, p. 96 and relatively detailed discussion, p. 97. However, this is almost wholly of the structural aspect, with flatworms mentioned only in passing, albeit to make a specific point. Escher also added his own commentary to Ernst’s view, albeit, again, this was of structural matters, and not on the flatworms.


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

In accordance with the book, of works and commentary, Escher added p. 20. … however, when this building [of tetrahedra and octahedra] is filled with water, flatworms can swim in it


Hart, George W. ‘Bringing M.C. Escher’s Planaria to Life’. Bridges, 2012, 57-64.

In short, an article inspired by Escher’s print ‘Flatworms’, with the print having common connections to Hart’s interest in sculpture, and in particular here that of octahedra and tetrahedra. Begins with a brief discussion on the print, with references to the above polyhedra, and also of the flatworm, and then more extensively Hart’s own work in the field, concentrating on the polyhedral aspect per se. Hart, in general, is more concerned with his special interest, rather than flatworms. However, he does indeed make one interesting unconnected point in the introduction, commenting on none of Escher's’ trademarks’ being here, but is rather a portrayal of a plausible, albeit unfamiliar, scene. Simply stated nothing per se new on the print.


Locher, J. L. The World of M. C. Escher. Abradale Press Harry N. Abrams Publishers Inc. New York 1988. First Published 1971.

P. 239 shows the print and basic information, without discussion. No other detail.


Locher, J. L. (general editor). Escher The Complete Graphic Work. Thames and Hudson 1992. First published 1982. Translated from the Dutch Leven en Werk von M.C. Escher. Meulenhoff 1981.

Entry 431. Basic information concerning the print. No other detail.


Locher, J. L. The Infinite World of M. C. Escher. Abradale Press/Harry N. Abrams Inc. New York. First published 1984.

P. 138 shows the print and basic information, without discussion. No other detail.


McConnell, James V. ‘Confessions of a scientific humorist’. impact of science on society, Vol. XIX, No. 3 July-September 1969, 241-252.

Of James McConnell interest, re Escher-flatworms, albeit there is nothing here on Escher, but rather of his (admirable) humour. McConnell’s piece was part of a seemingly special edition on humour and science, from Unesco.


McConnell, James V. Article Title Unknown. Worm Runner’s Digest Vol. XVI No. 2, December 1974, pages unknown. WANTED

Of Escher reference, at least of the cover, of which after this  there are many uncertainties here. I do not have the journal in my possession, and quite where I got this reference from is unclear; I may have found it independently, although I doubt it. Be all as it may, an article in The Unesco Courier of April 1976, shows the cover of the WRD, illustrated with Escher’s Flatworm print (a topic of recent (May 2019) interest). Quite what, if indeed there is an Escher related article here  is unclear.


McConnell, James V. ‘Worm-Breeding With Tongue in Cheek or the confessions of a scientist hoisted by his own petard’. The Unesco Courier, April 1976, pp. 12-15, 32

As such, the Escher aspect here is only of illustrations; there is not any reference in the text. More exactly, this shows shows the cover of the WRD of 1974, illustrated with Escher’s Flatworm print (a topic of recent (May 2019) interest). The Courier piece is an interesting read in many ways. There is no Escher discussion as such in it, although the Flatworms print is shown on p. 13, with the premise on flatworms (a most interesting creature, I might add. I had no idea of the fascinating science on it). As an aside, I very much enjoy McConnell’s humour.


Price, Jeffrey. M. C. Escher Amazing Images. Catalog of Rare Original Prints and Drawing. (Privately published book/catalogue).

Unpaginated, but ‘self paginated’ p. 55. With his own commentary and insight. Of note is an accompanying sketch (which I had forgotten about!), of a tessellating flatworm, derived from an arrow tessellation. This is plainly indicative of a flatworm, but oddly, given the relatively good quality, Escher did not proceed with this.



Not Mentioned:


Coxeter, H. S. M; M. Emmer, R. Penrose, and M. L. Teuber, Eds. M.C. Escher: Art and Science. Amsterdam: North-Holland 1986.

Without going through all 402 pages (as it lacks an index), the print does not appear to be discussed.


Fellows, Miranda. The Life and Works of Escher. Parragon Book Service Limited, 1995.


Forty, S. M C Escher. Taj Books 2003.


Anon. M. C. Escher 29 Master Prints. Harry N. Abrams, Inc. Publishers New York 1983. Edited by Darlene Geis.


Ford, Karin (translator) and Janet Wilson, editor. English Language version. Escher on Escher. Exploring the Infinite. Harry N. Abrams, Inc. 1989. With a contribution by J. W. Vermeulen. Compiled by W. J. van Hoorn and F. Wierda. Originally published under the title Het oneindige

Without going through all 158 pages (as it lacks an index), the print does not appear to be discussed. Pp. 155-158 gives a list of Escher illustrations, of which there is no mention.


Schattschneider, Doris. Visions of Symmetry. Notebooks, Periodic Drawings, and Related Work of M. C. Escher. New York. W. H. Freeman and Company 1990. Revised edition 2004.


Schattschneider, D. and M. Emmer (editors). M. C. Escher’s Legacy. A Centennial Celebration. Springer. First edition 2003, paperback 2005.

Without going through all 457 pages (as it lacks an index), the print does not appear to be discussed. Pp. 456-458 give a list of Escher illustrations, of which there is no mention.


Thé, Erik (Designer). The Magic of M. C. Escher. Joost Elffers Books Harry N. Abrams 2000. Foreword by W. F. Veldhuysen. Introduction by J. L. Locher.

Not mentioned.




‘Sysopje’. Seemingly by ‘sysopje’ from his email address. Described as:

Waterworld. A website about nature in the Netherlands. There are 1600 web pages about animals, vegetables, herbs, trees, wild plants.


Shows Escher’s print, with brief commentary:

M.C. Escher and the flatworm. Escher was fascinated by the behaviour of the flatworm.

A flatworm is completely flat, and has no idea of up and down. It knows only light and dark and seems to be propelled by magic. This all puzzled Escher and he thought about the strange universe were flatworms would rule there world.




From Wikipedia:

Everything2 (styled Everything2), or E2 for short, is a collaborative Web-based community consisting of a database of interlinked user-submitted written material. E2 is moderated for quality, but has no formal policy on subject matter. Writing on E2 covers a wide range of topics and genres, including encyclopedic articles, diary entries (known as "daylogs"), poetry, humor, and fiction.

Dutch artist M.C. Escher produced a lithograph in 1959 named Flat worms (Platwormen) (viewable online at http://www.tabletoptelephone.com/~hopspage/Flatworm.jpg), depicting a structure formed from alternating tetrahedronal and octahedronal bricks with cute l'il flatworms slithering all over them. He noted that such a structure would be impractical for humans as the resulting surfaces produce neither vertical walls nor horizontal floors, but if it filled with water it would work dandy as a home for flatworms. Kibo (of Usenet infamy) points out the unutilised possibility here - Escher using worms in his works prior to the scientific establishment of both their regenerative and maze-running abilities - and hypothesizes that if only he'd known, Escher would have depicted planarian worms regenerating in the forms of moebius strips while running geometrically impossible labyrinths. Anyone with a passing familiarity with Escher's work can agree that this wouldn't be atypical for the Dutch mindbender.

As ever, I am interested in hearing of thoughts and comments on this page.

5. Whirlpools

An eclectic selection of print references to Escher’s print Whirlpools, for purposes of research of an article by Peichang Ouyang, with advanced mathematics based on this print. Of course, this also serves as an investigation into Whirlpools in its own right. No claim is made to exhaustiveness. However, it is certainly comprehensive! For ease of finding the quoted text, this is shown in blue. Comments to self are in red. I divide the text into three parts, according to depth and relevance. First, in-depth discussions, second, discussed, but not extensively; and third, none or minor references, essentially in passing. Upon an initial overview of the references, there is perhaps surprisingly little in-depth discussion/analysis for such a major print. Four references of note stand out:
To Infinity and Beyond, by Eli Maor, pp. 175176.
Magic Mirror of M. C. Escher by Bruno Ernst, pp. 106 (general discussion)  107 (Print) 108 (A mention of the print in passing).
Escher The Complete Graphic Work, by J. L. Locher, pp. 89 (Letter to Arthur), 95 (Print), 153 (A mention of the print in passing), 342 (Notes). (Of note is that Escher discusses the print in relative depth in a letter to his son Arthur, including his well-known quote on Whirlpools, I doubt whether “the public” will ever understandin full, not curtailed, as in others.)

The Graphic Work of M. C. Escher, pp. 1415. (Escher discussion.)


Text in blue: Text repeated as in the book/article

Text in red: My comments

1. In-depth discussions

To Infinity and Beyond, by Eli Maor

P. 147 (print, in colour, with caption)

P. 175

The same idea [Path of Life II] found its supreme expression in Whirlpools (1957; Plate VI, see p. 147), which in my opinion is the most beautiful of all of Escher’s works. Two systems of logarithmic spirals, running parallel to each other, emanate from the top center, and after an infinite number of revolutions attain their maximum size at the center of the print; then they diminish again until they reach the lower center. Along these spirals, red and grey fish swim peacefully—the red fish originating from the lower center and moving towards the upper, the grey ones in the opposite direction. The entire picture can be turned through 180° about its center—viewing it upside down will merely turn the red fish into grey ones and conversely. (Even Escher’s signature appears twice—in the lower right corner and again upside down in the upper left.) Most of Escher’s ideas are embodied here in the most

P. 176

masterful way—his fascination with the infinite, for the two centers are infinitely remote and thus forever inaccessible to our fish; his lifelong obsession with tessellations, congruence, and similarity (not only does each red fish have an exact counterpart among the grey ones, but the fish completely fill the space around and between the spirals); and finally, his extraordinary talent for depicting movement, change, cycle, and rhythm. It is perhaps symbolic that Escher was commissioned by the city of Utrecht to use the same design in a mural for one of the city’s cemeteries. He himself painted this large mural, 3.7 meters in diameter.

[Side caption]

On his print, Whirlpools (1957): I designed a division of the plane consisting solely of fish that ‘move' in black spirals towards the centre (symbolizing death or dying), while a series of white fish ‘move’ outwards from the same centre (life, birth). The attractive, and at the same time difficult, thing is the diminution of the fish figures into infinity. The outer fish will be about five feet long and I want to try and reduce their size consistently until they are mere specks of about half an inch in length. (Comments made by Escher regarding the commission he received for a large mural based on this print.)

P. 176 Side Caption, but no picture

Again on Whirlpools: I am using a new printing technique based on a very amusing twofold-rotation-point system. It is difficult to explain in words, but what it amounts to is that from each of the blocks (probably three) that I have to cut, I make only half of the surface that they have to fill together; the other half is produced by repeating the blocks after they have been turned a hundred and eighty degrees. I doubt whether ‘the public’ will ever understand, let alone appreciate, what fascinating mental gymnastics are required to compose this sort of print.

This gives Escher’s quote on the print in full.? Check

The Magic Mirror of M. C. Escher, by Bruno Ernst, pp. 106108

P. 106

The impressive woodcut Whirlpools (1957) (figure 238) came into being prior to the Path of Life prints. The same construction is used here as was used for the spirals, while a number of possibilities inherent in this framework were not utilized. Only two spirals are drawn simultaneously in the upper and lower constructions and they both move in the same direction. These spirals are in line with the backbones of two opposing series of fish, and at the center one construction merges with the other.

The grey fish are born in the upper pool and, growing larger, keep swimming further outward. They start on their journey (already diminishing in size) toward the lower pool, where after an endless

series of reductions, they disappear at the central point. The red fish swim in a contrary direction, from the lower pool to the upper.

The whole picture is printed from two blocks only. The one from which the grey fish of the lower part are printed is used over again to print the red fish of the upper part. This is why we see Escher’s signature and the date twice on the same print.

Toward the end of the year in which Whirlpools appeared,

[Image next page, P. 107, in colour]


Escher received a commission from the city of Utrecht for a mural in the main hall of the third municipal cemetery. This is done as a circular painting with a diameter of 3.70 meters. Not only did Escher produce the design but he carried out the actual painting himself (page 59). This wall painting is almost an exact replica of one half of Whirlpools.

Escher The Complete Graphic Work, by J. L. Locher, General Editor

P. 89

Letter to Arthur, Baarn, 2 November 1957

The Utrecht town council has invited me to design a mural for the town cemetery… While I am waiting to start this job, I am working on a double spiral in woodcut [Whirlpools], in three colours [cat. 424], along the same line of thought as that of the mural. I am using a new printing technique based on a very amusing twofold-rotation-point system. It is difficult to explain in words, but what it amounts to is that each of the blocks (probably three) that I have to cut, I make only half of the surface that they have to fill together, the other half is produced by repeating the blocks after they have been turned a hundred and eighty degrees. I doubt whether “the public” will ever understand, much less appreciate, what fascinating mental gymnastics are required to compose this sort of print.”

P. 95 print in colour

In ‘Vision of a Mathematician’, by Bruno Ernst, pp. 135154, on Section 7, ‘The Infinite’,

P. 153

Initially the results were unsatisfactory… Division (cat. no. 411) ... and Smaller and Smaller (cat. no. 413)...In the woodcut Whirlpools of 1957 (cat. no. 423); see also page 95) he used a better solution.

P. 342 Catalogue entry


Signature in the print: bottom right

Date in the print: top left

Added signature: bottom left

Other additions: eigen druk (own printing), bottom right

In the first state, the signature is missing

The Graphic Work of M. C. Escher, pp. 1415

P. 14 (print in colour)

WHIRLPOOLS, woodcut printed from 2 blocks, 1957, 45 x 23.5 cm

Closely related to the foregoing picture (Sphere Surface with Fishes), there is here displayed a flat surface with two visible cores. These are bound together by two white S-shaped spirals, drawn through the bodily axes of, once again, fishes swimming head to tail. But in this case they move forward in opposite directions. The upper core is the starting point for the dark coloured series, the component members of which attain her greatest size in the middle of the picture. from then on, they come within the sphere of influence of the lower core, towards which they keep on swirling until they disappear within it. The other, light-coloured, line makes the same sort of journey but in the opposite direction. As a matter of special printing technique, I would point out that only one wood-block

P. 15

is used for both colours. These having been printed one after the other on the same sheet of paper, and turned 180 degrees in relation to each other. The two prints fill up each other’s open spaces.

The book gives different translations of the titles from the native Dutch Draalkolken, with Drehstrudel (German), Tourbillons (French), and Virvar (Swedish). Indications are promising for searching under these terms.

Invaluable as it gives Escher own view on the print.

P. 27

Grafiek en tekeningen M. C. Escher, 1960

8. DRAAIKOLKEN, houtgravure van twee blokken gedrukt, 1957, 45 x 23.5 cm.

Twee kernen zijn met elkaar verbonden door twee witte S-vormige lijnen, getrokken door de lichaamsassen van kop-aan-staart zwemmende vissen. Twee stromen van vissen dus ook, elk met zijn eigen kleur, die zich in tegengestelde richting voortbewegen. Van oneindig klein geleidelijk in omvang toenemende, verwijderen de dierfiguren van elke stroom zich spiraalsgewijze van het punt waar zij geboren werden. In het midden bereiken zij hun maximale grootte en komen vervolgens in de invloedsfeer van weer kleiner naarmate zij er dichterbij komen en gaan er tenslotte in teloor.

Als druktechnische bijzonderheid zij hier opgemerkt, dat er slechts  een blok is gesneden waarmee, na elkaar op het zelfde stuk papier, de beide kleuren zijn afgedrukt. Het totaal der figuren van een kleur is namelijk congruent aan, maar 180 graden gedraaid ten opzichte van dat der andere kleur. De beide afdrukken dekken elkanders open tussenruimtes volkomen.

Graphics and drawings M. C. Escher Translation

8. Vortices, wood engraving printed in two blocks, 1957, 45 x 23.5 cm.

Two cores are connected by two white S-shaped lines drawn by the body axes of head-to-tail swimming fish. So too are two streams of fish, each with its own color, moving in opposite directions. From infinitely small to gradually increasing in size, the animal figures of each stream spiral away from the point where they were born. In the center they reach their maximum size and then fall back into the sphere of influence of smaller again as they get closer to it and finally get lost.

As a printing-technical special feature it should be noted here that only one block has been cut with which both colors are printed one after the other on the same piece of paper. The total of the figures of one color is congruent to, but rotated 180 degrees with respect to that of the other color. Both prints completely cover each other's open spaces.

2. Discussed, but not extensively

The World of M.C. Escher, by J. L. Locher (editor)

Preparatory drawing p. 228; print p. 229, black and white

Coxeter’s essay ‘The Mathematical Implications of M. C. Escher’s Prints’, p. 53

In a general discussion on similarities:

In Whirlpools (Cat. 224) any two fish of the same colour are related by a loxodrome homography (if we ignore the insufficient distortion of the upper fish). Two loxodromes (a loxodrome is the inverse of an equiangular spiral) running down the backbones of the fish, have been drawn with remarkable accuracy. Moreover, any two fish of different colours (i.e., one dark, one pale are related by a hyperbolic antihomography (the product of inversions in three circles such that two are nonintersecting while the third is orthogonal to both).

Some advanced discussion by Coxeter!

Wolfram MathWorld

A path, also known as a rhumb line, which cuts a meridian on a given surface at any constant angle but a right angle. If the surface is a sphere, the loxodrome is a spherical spiral. The loxodrome is the path taken when a compass is kept pointing in a constant direction. It is a straight line on a Mercator projection or a logarithmic spiral on a polar projection (Steinhaus 1999, pp. 218-219). The loxodrome is not the shortest distance between two points on a sphere.

Visions of Symmetry, by Doris Schattschneider

Pp. 249250 (print in colour), 321322

While working on the life passage designs (funeral), Escher also completed the graphic work Whirlpools, which is a double spiral. In this print, each fish is born in the eye of a whirlpool, follows a coiled path outward while growing in size, then follows a spiral path in the opposing direction toward the vortex of the second whirlpool, until disappearing into

P. 250

its tunnel. Escher carved only one block to print the colors in this woodcut; the figures printed in one color are identical to those of the contrasting color. By rotating the block 180°, the second spiral of fish neatly interlocks with the first; the carved date and initials MCE appearing at the top and bottom of the print.

P. 322

...While designing the Utrecht murals, Escher also made Whirlpools, in which the same fish from the vignette now follow a single path that begins and ends in a tightly coiled spiral.   

Escher on Escher, edited by Janet Wilson, English-language edition, p. 40 (of cancelled 1964 lecture notes)

P. 40 (print in colour)

The left print [Whirlpools] presents two nuclei with infinitely small figures. They are linked by a red and blue row of fishes swimming head to tail and moving in opposite directions. The whole red trail has exactly the same shape as the blue one. When rotated 180 degrees around an axis in the center, they cover each other’s open interspaces. Since I employed the woodcut technique to make this print, I needed only a single block, with which both colors were printed.

The Magic of M. C. Escher, by Erik Thé (designer)

Pp. 176177 (print p. 176, in colour, cancelled blocks p. 177, in colour)

No commentary as such, but shows ‘progressive proofs from cancelled woodblocks’.

Gives the Escher quote but without source:
‘I doubt whether “the public” will ever understand, much less appreciate, how many gymnastics of the brian, fascinating to me, have preceded the construction of such a picture.

On Whirlpools, 1958

Of note here is the curtailed (and slightly different) quote, which is not made clear, shown in full in Escher The Complete Graphic Work. For ease of comparison, I show the relevant part below:

I doubt whether “the public” will ever understand, much less appreciate, what fascinating mental gymnastics are required to compose this sort of print.”

3. None or minor references, essentially in passing

M. C. Escher’s Legacy, by Doris Schattschneider et al

Likely, no references. As the book lacks an index, and without any obvious indication as by the chapter descriptions, it is a time-consuming task to read each page, of which save for a skim I refrain from.

The Magical World of M. C. Escher, by Marc Bell (Boca Raton exhibit)

P. 151, print in colour with caption only, no discussion.

M. C. Escher Amazing Images, by Jeffrey Price

Not mentioned at all!

M. C. Escher Art and Science, by H. S. M. Coxeter et al

As the book lacks an index, and without any obvious indication as by the chapter descriptions, it is a time-consuming task to read each page, of which save for a skim I refrain from. However, I found one reference, by Haresh Lalvani, p. 191, but this is no more than the title in passing.

M. C. Escher’s Universe of Mind Play (catalogue)

P. 128, print, in black and white, no discussion

M. C. Escher 29 Master Prints, by William Wegman, designer

Not in.

M. C. Escher, by Sandra Forty

Not in.



Erik Kersten, 7 November 2017, Whirlpools

Early November 1957 Escher finished his woodcut and wood engraving Whirlpools. He used a new printing technique for it, cutting one block which he printed on the same piece of paper in two colours.


[Caption] M.C. Escher, Whirlpools, wood engraving and woodcut in red, grey and black, printed from two blocks, November 1957

Two rows of fish swimming head to tail fill the space. The red row has exactly the same shape as the grey one, but has been turned 180 degrees. Starting infinitely small, the fish gradually grow, removing themselves from their origins. They are at their largest in the centre, where they come within reach of the other core. They start to revolve around it, get smaller and finally dissolving into oblivion. Escher wrote to his son Arthur about it*:

‘I doubt whether the “public” will understand, let alone appreciate, what fascinating mental gymnastics are required to compose this sort of print.’

Wood engraving and woodcut printed in colors, 1957, signed in pencil and inscribed 'eigen druk', on Japan paper, framed

image: 438 by 235 mm 17¼ by 9¼ in

sheet: 546 by 317 mm 21½ by 12½ in


[*] Wim Hazeu, M.C. Escher, Een biografie, Meulenhoff, 1998, blz. 391

Of note here is the curtailed (and slightly different) quote, which is not made clear, shown in full in Escher The Complete Graphic Work.

John Golden and others


At first glance I'd say this looks like an iterated loxodromic Möbius transformation with the two vortices as fixed points. But after a quick experiment I have to concede that this first impression was wrong. – MvG Mar

The light and dark blue logarithmic spirals are suitably translated and scaled versions of


in polar coordinates (r,θ).

To confirm the OP's suspicions and MvG's comment, Escher appears to have hand-interpolated near the center. It appears there is no conformal transformation sending the spirals to loxodromes, compare the following image of loxodromes under a Möbius transformation:

Somewhat advanced!


Google Translated. The graphic artist and death, by Chris den Engelsman.

The design for the wall painting. In 1957, in the same period as the design for the wall painting, Escher works on the woodcut and engraving Vortices [Whirlpools], for which he even invented a new way of printing. He prints both colors of this woodcut with the same block, when printing one color the block is rotated 180 degrees relative to the print of the other color. The woodcut representation consists of two S-shaped streams of fish, each stream in its own color, moving in opposite directions. From infinitely small, gradually increasing in size, the fish spiral away from the point of birth. In the middle they reach their maximum size, and they enter the sphere of influence of the other core. They begin to revolve around it, become smaller again and finally lose in infinity. It seems only a small step from the presentation of this woodcut to the design that Escher would make for the Tolsteeg cemetery in Utrecht. The interpretation of the design seems to fit almost seamlessly into the work that Escher was already working on. 

A relatively brief mention in passing in the context of spirals in the context of death. It is not entirely clear if this is Engelman’s own viewpoint, or has compiled the above from other sources, or perhaps it’s even a blend. Be that as it may, from this and other writings, he impresses me greatly.

Craig Kaplan, https://isohedral.ca/escher-like-spiral-tilings/

Again, this isn’t a new idea. Escher experimented with it in his print Whirlpools, and others such as Jos Leys have created similar designs.

A mention in passing, on Möbius transformations.


...Remarks: Representation of loxodromes

Picture and basc detail only, such as the title, year etc. One comment of note is that of a loxodromes supposition, but is unlikely.

Popular Culture

Book and Journal Covers of Whirlpools (3)

Cortazar, Julio. Fine del gioco. Einaudi Tascabili. Classici Moderni 1151, Einaudi, Torino, 2003 

Dobzhansky, Theodosius. The Biology of Ultimate Concern. Meridian; Meridian edition Fontana, 1971

Rochat, Denise. Contrastes. Grammaire du français courant. Prentice Hall, Second edition, 2009

Translated. Contrasts. Grammar of Fluent French

Record covers: None!

Escher’s Spiral print chronology:

Spirals (wood engraving, 1953)

Whirlpools (woodcut, 1957)

Sphere Surface with Fish (woodcut, July 1958)

Sphere Spirals (woodcut, October 1958)

Path of Life I (woodcut, March 1958)

Path of Life II (woodcut, March 1958)

Path of Life III (woodcut, 1966)

Created 24 June 2019.

24 June 2019. Research 1 - Flatworms

21 May 2020. Research 2 - Escher's Spiral Works

11 March 2021 (First incarnation). Research 3 - Whirlpools. Last updated 26 May 2021

23 July 2021. Escher's missing periodic drawing No. 135