Escher-Type Drawings by E. R. Ranucci and J. L. Teeters
This book is aimed apparently primarily at a
school-age level, of approximately the 12-14+ year age range. As such, it has a
pleasing structure, as it progresses from first principles of basic
tessellations to Escher-like tessellations in a sequential manner, albeit
whether it is quite necessary for the early chapters to be so relatively
lengthy, containing elementary symmetry matters that have been discussed
elsewhere in a multitude of books is doubtful, the material covering nothing
new. However, a case can be made for a self-contained book containing this
material, gathered in one place, of which I have no qualms about.
Annoyingly, who actually created the many Escher-like tessellations designs is
not always explicitly stated – two main authors appear on the cover, Ranucci
and Teeters, whilst on the title page two further people are credited, Dale
Seymour and Bob Fuller. Joseph Teeters is credited on some, but not all of the
drawings. I now address each chapter in turn, relatively briefly for the non
Escher-like aspects, and in more depth where these Escher-like art is shown.
Chapter 1 Plane Tessellations, Pages
Starts with the basics, of regular and semi-regular tessellations, and
generalises quadrilaterals, pentagons and hexagons, and some simple geometry
aspects as to why certain polygons tile.
Chapter 2 Transformations and Symmetry,
This discusses the three basic transformations, of translation, rotations
Chapter 3 Altering by Translation, Pages
This addresses the beginnings of Escher-like creation, by the alteration of
the sides of various tessellating polygons by translations, with in some
instances this illustrated by life-like motifs, unfortunately of a very poor
standard, and a recreation of some of Escher’s own tessellations. Here, the
method given for the like-like tessellations is somewhat simplistic - a
‘squiggly’ line is given, from which, when duly applied to the prepared
tessellation, is then added suggestive life-like markings. However, what is
lacking here is any suggestion of refinement of the original line - that given
remains implicitly inviolate. Such matters effectively serve to restrict the
quality of the subsequent motif. However, as a general premise, this is indeed
a viable way in which to begin. Such examples are additionally accompanied by
showing examples of (non life-like) tessellation and symmetry, and as such are
not per se concerned with the creation process.
Chapter 4 Altering by Rotation, Pages
This continues on the altering theme, this time by rotation with a few
examples of Escher’s own tessellations and the author's own.
Chapter 5 Altering by Reflection and
Rotation, Pages 103-117
This continues on the altering theme, this time by reflection and rotation with
a few tessellations of the authors own devising, and this time (inconsistently)
excluding Escher’s own examples.
Chapter 6 Analysing Escher's
Tessellations, Pages 123-134
This analyses four of Escher’s tessellations in terms of how he likely
created them, in order of sequence Nos. 20 (Birds), 1 (Creature), 96 (Swans),
and 21 (Imp). The selection appears to have been arbitrary, and includes three
examples (Nos. 20, 1, 21) that are relatively complex which are beyond the
understanding of the intended audience. Indeed, some of these even confuse
Ranucci, as he gives spurious constructions.
Regrettably, Ranucci errs here in the analysis of No. 20. Although I will not
go into detail here, he draws the wrong conclusions from his analysis.
Paradoxically, he apparently inadvertently gives the correct fundamental
construction in Fig.6-4 and then promptly asserts that this is not the true
construction! What he thinks is the correct fundamental region of Fig.6-6 is
incorrect… The analysis of No. 20 is somewhat involved, of which I will discuss
separately elsewhere. To choose this particular tessellation for analysis was
unwise, as it is beyond the understanding of the intended audience.
No.1 is poorly explained, and again requires a more lengthy description.
Ranucci states, page 128 … a quadrilateral ABCD. Paradoxically, some of
Escher’s earlier examples are relatively complex in their construction (as
here), involving what I term as ‘overlapping of lines’, and the explanation
given is far too simple, and again is inappropriate for the intended audience.
No. 21 is again lacking. Ranucci states:
basic cells for the drawing are distorted hexagons
Whilst Escher can be seen to have overlaid his motifs with a rhombus grid, rather than a distorted
The construction given for No. 96 is true as to how Escher did this, albeit the
impression given is one of ‘inviolate’ lines, achieved at the ‘first instance’
of drawing, which is not how Escher did this – he refined the initial line.
Chapter 7 Using Imagination and Special
Techniques, Pages 137-149
This chapter is the main aspect for this review, and so is thus discussed
in greater depth than with other chapters, discussing as it does the creation
of original Escher-like tessellations. In contrast to other books, this is at
least a concerted effort, of both depth and variety of method, and is by far
the most informative as regards the design process, albeit the examples arising
from this are not up to Escher’s standard. Two different methods are given:
(i) of deforming a given polygon, (Ranucci calls this ‘using imagination’) and
(ii) of ‘adapting’ a non-tessellation figure into a tessellation. (Ranucci
describes this as a ‘special technique’).
The deforming method, although somewhat brief, meets with my full approval, and
is undoubtedly how Escher went about his. Firstly, a given tessellation, Fig. 7-1
is altered, with opposite sides of a hexagon translated. This is then examined,
with the question posed by Ranucci, page 138 concerning Fig. 7-2:
the imagination is required. What does this outline remind you of?
Six possible designs are given, five of which are of the type I describe as
'cartoon-like heads' (a lower category of difficulty), and one of an American
football player, albeit the latter in a somewhat ‘rough and ready’ state at
this initial stage. For example, the head is unnaturally pointed (with what
Teeters then describes as a 'triangular hat’), the shoulder protrudes at an
unnatural angle, and the legs and feet are poorly defined. Although this does
indeed broadly resemble a (American) football player, a more correct anatomical
appearance would ideally have been accomplished, along with all the
accoutrements of a football player i.e. appropriate clothing, such as a helmet.
The football player is then separated as Fig. 7-4, with the intention of
modifying, i.e. improving the existing design. Teeters states that
player is running. The triangular hat needs to be changed to a helmet. The legs
need to look more like legs. The arm on the left needs to be more like the one
on the right... and so on.
This is good advice, as it shows that Teeters understands the issues, by
assessing the weaknesses and addressing these concerns, and not simply
accepting the first design, a typical beginner’s shortcoming. Such changes are
thus duly made, albeit unfortunately, for the sake of the design process, the
‘experimental phases’ are not shown. Still, it should be obvious as to how he
proceeded. The refined tessellation, Fig. 7-5, can thus be seen to be a
distinct improvement, as when compared to the original of Fig. 7-2, the figure
is now more anatomically correct and appropriate - for example, the
protuberance of the shoulder has been removed, and the 'triangular hat' is now
a helmet, thereby reinforcing the impression of a football player. However,
weaknesses remain, in that the figures’ right leg is lacking in definition.
This definition aspect will typically be found in tessellation matters, in that
no matter how hard one tries, some elements will remain weak, of which the
question the tessellator has to bear in mind is to the degree of acceptability
– does the tessellation overall broadly retain artistic integrity? In this
A different method given by Teeters, with Fig. 7-6 is to begin with a given,
non-tessellating figure (coincidentally of another American footballer), it
being 'adapted' to a tessellating tile, Fig. 7-7.
completely different approach is to start with a design in mind and to use your
knowledge of mathematics of tessellation to create a figure that tessellates.
Let’s consider the football player again. You may have seen a photograph or an
artist’s sketch that could be a beginning. The general outline of what you have
in mind might be in a more triangular shape like the one in Figure 7-6.
This method is less frequently encountered, and indeed is the only example of
its type in the book. Whether this is a ‘better’ method than the other is a
moot point. The decisive test is in the final tessellation – namely in the
quality of motif.
The adaptation method has a weakness in that one has to first simplify it into
a tessellating polygon, and then perform a symmetry operation on it to
accommodate the new lines. By contrast, using the deform method; this aspect is
alleviated, as the tessellation operation (when undertaken appropriately)
performed is certain to tessellate.
The premise of this method is that is different from that previously given.
Stated simply, this (adaptation) can be described as ‘top down’, whilst the
former is ‘bottom up’. The adaptation will result in a motif that of necessity
will be weaker than the initial one (of a real motif), whilst the ‘bottom up’
will always be striving for improvements. As such, I am reluctant to too
dogmatic as to which is the best, as I have shied away from using the ‘top
down’ method. Preferences aside, the tessellation Ranucci shows as a result of
this method is indeed another fine example, albeit again I have minor quibbles,
with the right leg lacking definition, but again, this broadly retains artistic
integrity, and so shows that this method is viable.
Teeters briefly introduces yet another technique during its creation (that can
be applied to both methods), namely that of 'overlapping', illustrated by Fig. 7-9
and Fig. 7-10. (This is a device Escher frequently used in his earliest
tessellations, of 1938, but noticeably less so from 1940 onwards.)
Disappointingly, these aspects are discussed in a most brief manner, it being
summed up in a handful of lines and diagrams, whereas a more in-depth treatment
Teeters concluded the chapter with some further analyses of Escher’s
tessellation No.113 (Fish and Boat), and unfortunately gives a spurious
interpretation of the construction, that is not only long-winded, but wrong.
Various unit cells are given as potential underlying polygons, before he
eventually settles on a parallelogram as definitive. No. In actuality, this was
based upon a hexagon, and can clearly be recreated – just locate the vertices
of both motifs, and join accordingly. The hexagon can be either convex or
Chapter 8 Worksheets for Developing
Techniques, Pages 151-170
Broadly, ‘things to try’, based on chapters 2-5.
Chapter 9 Selected Grids, Pages 171-187
A variety of grids to use for experimentation purposes.
Critiques of the Original
Unacceptable, Poor, Reasonable, Good,
1. Man with ‘Banner Board’. Page 1
Reasonable. Although not shown entirely as a complete
tessellation, the implication is indeed of that premise. This was probably
designed with a specific purpose, signifying the chapter number (and is used
for succeeding ones), of a figure holding a ‘banner board’, and as such, is a
relatively pleasing example. This is let down somewhat by the drawing, as it is
clear that it is intended for the figure to be mirror symmetrical, but some
careless drawing (of shoulders and head) departs from this. However, this is a
minor drawback, relating to accuracy in drawing, and is not to the detriment of
the intrinsic quality of the concept, and could easily be corrected.
2. ‘Woman’? Page 44
Unacceptable. A shape with a few woman-like head
elements added. This is truly dreadful! The only remotely woman-like aspect of
this is the head. The mouth or nose (I can’t tell what this is representing)
looks like a bird’s beak…. And as for the body… Anybody who does examples like
this and perhaps thinks they are good does not understand tessellation.
3. ‘Clown’? Page 44
Unacceptable. A shape with clown-like elements added.
Again, this is dreadful. The only remotely clown-like aspect of this the head.
What is the region below the head…?
4. Birds. Page 47
Poor. A shape with one bird-like element (eye) added. Birds? No. This is just a
shape with an eye. No other interior details are included…
5. Unidentifiable? Page 48
Unacceptable. A shape with a dot for an eye and mouth-like elements added.
What sort of a creature is this? Dreadful.
6. Head (with feather in cap). (Teeters)
Reasonable. This head is reasonable within the context of a lower degree of
tariff. A shape with head-like elements added. A reasonable head, in that the interior
details are at least clear. However, it is of a lower quality category, lacks a
defining outline necessary for such wireframe types, and it fails miserably the
silhouette test, the test of quality.
7. Wrestlers (Teeters) Page 51
Reasonable. This does indeed represent a pair of figures in a typical wrestling
pose. However, the figures have shortcomings, with general awkwardness and
anatomic inconsistencies, with the legs of the left hand side figure being
8. Flying Owl (Teeters). Page 52
Reasonable. Broadly owl-like, albeit with reservation. Pleasingly, rather
than a generic bird, this is designed specifically of an owl, in that the ears
are showing, a typical identifying feature, as in a real-life example. A
drawback is that the wings are out of alignment. Being a wireframe type, would
benefit from outlining.
9. Dog (Teeters). Page 52 (and repeated
Reasonable. Broadly dog-like, with an outline that clearly resembles a dog,
albeit with reservations, as it is still anatomically deficient, especially of
the front legs which are somewhat awkward, and mouth region which is awkward.
Even so, although not without faults, it is one of the more worthy examples,
and has some degree of artistic integrity.
10. Spaceship. Page 53
Unacceptable. A shape with supposedly ‘space-like’
elements added. Really? This looks like a shape to me with obscure markings…
11. Dog Head, of Doberman Pinscher and An
Old-Fashioned Shoe. Page 97
Reasonable. Both motifs are shown in combination, the head being of a lower
tariff of difficulty. Unfortunately, the presentation of the symmetry is poor,
with no apparent ordering - the motifs appear to have been placed randomly… The
Doberman pinscher, albeit of only a ‘cartoon head’, is reasonably good, in that
it is identifiable as a dogs head. The shoe is somewhat less so in quality, as
there is no space between the heel and the sole as in a real-world example.
Furthermore, to a less important degree (as sometime incongruous motifs are
unavoidable), the above motifs in combination are somewhat incongruous (albeit
Escher included some like incongruent examples in his own work) – there is no
connection between these. Better would be to have the respective motifs as
12. Mr. Frog, Mr. Golliwog, Knight, The
Executioner, Ms. Fish, H. Dawg (and an unstated ‘long nose’). This also appears
on the front cover, in colour. Pages 98 and 99
Unclassified. A somewhat confusing sequence of two related pages, with
single tiles with motifs on page 98, and a combination of tessellations in a
triangular composition on page 99. Unfortunately, these do not marry, as
Ranucci and Teeters give 6 different motifs, page 98, of which they use three
for the triangular composition on page 99, whilst introducing an additional
motif not on page 98. What can one say… To ease the analysis, I discuss each
Various cartoon-like motifs of six different motifs, of four ‘heads’ of various
motifs, and two whole animals, a frog and fish. Two of these are very pleasing
examples of their type, namely Mr. Golliwog (which bears no resemblance to a
golliwog…) and H. Dawg, whilst the others are less so – the Mr. Frog especially
is very poor, a shape with frog-like elements added…
A composition of a 4-unit sided triangle of 16 equilateral triangles, based on
three of the preceding motifs, with, oddly, bizarrely even, of one new
addition, of ‘long nose’, which incidentally is quite good quality.
Disappointingly, the composition shown is very poor, as although four different
motifs (‘Long Nose’, Ms. Fish, Mr. Frog, and H. Dawg) are shown repeated four
times, there is no apparent orderly placement of these, resulting in an uneven
arbitrary distribution of motifs. Also, the artist selects inferior examples in
the composition, including the very poor Mr. Frog, whilst excluding the
excellent Mr. Golliwog…
13. Heads - Two Different Heads and
‘Unidentifiable Creature’. Page 100
Poor. A tessellation of two different tiles with three different motifs…. A
shape with head-like elements added. Truly dreadful. Shapes with eyes, nothing
more. The third motif is probably intended as a ‘complete’ animal, albeit
unidentifiable, and is unlike the other two in its nature. This being so, why
include in an obvious incongruous composition? Furthermore, again, the
composition is lacking…
14. Bird. Page 113
Reasonable. A better tessellation. The bird is broadly recognisable in
15. St. Bernard Dog (Teeters). Page
Reasonable. A better tessellation, albeit with reservation. A reasonable
portrayal of a dog is shown, and more specifically of a specific breed as well,
with the pleasing factor of a rum barrel beneath the St. Bernard’s collar
adding to the interpretation. However, the lower part of the dog, with legs,
belly is entirely mere surface decoration. Furthermore, the position adopted is
not a true representation of a sitting dog (if this interpretation is correct),
albeit this is masked to some extent by the relative quality of the interior
16. Heads - 6 Different Heads - Freckle
Face, Football Player, Animal, Young Girl, Weird Sailor, Big Mouth. Page 138
Unclassified. Although not shown as a tessellation per se (aside from the
football player on the next page), the various possibilities are clear in their
tessellation premise. Five heads (of a cartoon-like appearance) and one whole
football figure are shown, of various qualities. ‘Weird Sailor’ (albeit nothing
like a sailor...) is probably the best.
17. Football Player (Also see my
comments of Chapter 7). Page 139
Reasonable-Good. A better tessellation. Here the authors refine the
football player of Figure 7-4, and it must be admitted, successfully. This
actually resembles a football player. Although the resulting figure is not
outstanding, it is considerably better than the initial drawing of Figure 7-4,
and is one of the better ones, and is outlined.
18. Football Player (Also see my
comments of Chapter 7). Page 142
Reasonable. A better tessellation, albeit with reservation. Again, another
success in relative terms. The figure is in proportion, and has no stand out
weaknesses. However, both arms are mere surface decoration, and a leg is
Ideally, this would have been coloured in either three or four colours, as two
colours as here are inadmissible for map colouring, but this is a relatively
minor matter in the context of the book.
Overall, the book has many shortcomings. Quite simply, the authors plainly do
not truly understand the issues of composing Escher-like tessellations, as they
include many examples that are unworthy. The number of quality tessellations
amount to a mere handful, with dogs on pages 52 and 114, and football players
on pages 139 and 142 being the best. Conspicuous throughout are those of the
‘head’ type, the easiest type to compose, requiring little or no ability.
However, some of these, within a whimsy, cartoon-like category, are quite
pleasing, such as Mr. Golliwog or H. Dawg, page 98. The authors do not question
why Escher only showed a single instance of this type (and myself none. There
is a reason…). Noteworthy is the exclusion as to what it is that makes birds
and fish so suitable for motifs, to the exclusion of other motifs. This would
entail understanding the issues, which the authors plainly lack.
A lesser criticism is that occasionally Teeters shows compositions which are
particularly poor, in both concept (with unrelated motifs, albeit this is a
lesser criticism), and symmetry arrangements, with the motifs added at whim, in
an arbitrary manner, with no consideration for symmetry. This aspect, in
contrast, is a major criticism. This is very surprising, given Ranucci’s
Not only is there general poor practise, but there is mathematical
inexactitude, and furthermore in more than one place, such as with the bird and
fish analysis on pages 146-149 and 'Day and Night birds, pages 123-126. Again,
this is inexcusable.
Although the authors purport to show how Escher did his tessellations, this is
only very loosely so. When analysing/recreating Escher’s tessellations, the
impression given is that of a single instance of a line drawn in a ‘fully
refined’, state, whereas Escher reworked
the initial line. Oddly, the authors state this was what they did for their own
tessellation of the football player, but do not do this for their other
However, aside from the above, overall the book is an ideal introduction for
the intended age range if due allowance is made for the above aspects. In all
likelihood, one would not expect the intended audience to be overly concerned
with the production to an Escher-like standard, and although written for a
younger age group, this should not necessarily mean that a less than exact
approach should be adopted than would otherwise occur.
Agree or disagree? Email me
updated: 26 September 2009. 25 June 2012 minor revision. 14 July 2014: Corrected attribution of 'Saint Bernard' to 'St. Bernard'