M. C. Escher - Research


Aside from my interest in Escher as regarding the mathematical and especially the tessellating aspect is that of what I term as ‘other matters’ in his oeuvre. As such, these above have dedicated pages, in various capacities, of which the page here is of a less substantial nature, albeit within an individual entry matches the above in depth and substance. In the fullness of time, the material here may indeed reach the threshold of a more dedicated page being required. Meanwhile, I place here my ‘other matters’ on Escher researches under a broad title.

1. Square Limit

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

3. Flatworms

1. 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




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.


Research 1 - 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.

Created 24 June 2019.

24 June 2019. Research 1 - Flatworms

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