Designing and Drawing Tessellations,
Although on the face of it the title would suggest
a book on tessellations per se, the premise is of overwhelmingly towards
designing Escher-like tessellations. A better title would have been to include
a reference to ‘life-like motifs’ or ‘Escher-like’, although admittedly the
cover picture is of a life-like motif.
a generalised statement, the advice given by Fathauer is very good indeed, with
many useful hints and tips as to the ways and means of creating Escher-like
motifs. Although primarily aimed at a school-age level (12-16), anyone with an
interest in creating Escher-like tessellations will find it advantageous, as it
broadly addresses the all-important understanding of the underlying
issues concerning the creation of life-like motifs. Very few books concern
themselves with this matter, even in passing, and of which lies at the crux of
the problem of designing high-quality motifs, and so this aspect in particular
is warmly welcomed.
Broadly, upon the reader having absorbed each
chapter, a series of activities are then given, suitable for the above age
brief overview of the chapters:
1, Introduction to Tessellations, contains illustrations of
tessellations in the real world as a historical context, leading onto
forerunners of Escher.
2, Tessellations in our World, shows tessellations from the natural
3, Geometric Tessellations, details the underlying mathematical aspects
4, Symmetry and Transformations in Tessellations, discusses mirror
symmetry and translation, glide reflection, and rotation.
5, Tips on Designing and Drawing Escheresque Tessellations, gives a
series of tips on designing Escher-like tessellations
6, Special Techniques for Solving Design Problems, gives a series of
techniques for improving on a initial tessellation
7, Tessellations Based on Square Tiles, a step by step guide to creating
8, Other Tiles Based on a Square Grid, uses right triangles and kites
with a series of templates
9, Tessellations Based on Equilateral Triangles, uses equilateral
triangles with a series of templates
10, Tessellations Based on 60°-120° Rhombus Tiles, uses rhombuses with a
series of templates
11, Tessellations Based on Hexagonal Tiles, uses hexagons with a series
such, one could argue that Chapters 1-4 could very well be excluded on the
grounds of familiarity, as previous books have covered the same ground. Still,
for an easy, and convenient, ‘basics to hand’ covering of such matters, there
is nothing to fault here. However, for those familiar with such matters, the
book only really begins much later, with Chapter 5, this concerning the
Escher-like aspect, with a series of tips on drawing and designing. Chapter 6
then gives a series of techniques, in effect attempts at improving upon the
initial tessellation. Chapters 7-11 are concerned by more specific matters,
with tessellations based on specific tiles that frequently occur.
feature of the tessellations is that many are shown as wireframes, which as I
discuss elsewhere, in Essay ** renders the motifs as not as readily
identifiable as could otherwise be. This could, and should, have been shown in
different shades, especially as the tessellations were drawn on a computer,
from which such matters are quite easy to accomplish. Cost seems to have been a
consideration; there are no colour pictures in the book.
stated above, the essence of the book is Chapters 5 and 6. Chapter 5 consists
of a series of tips, or statements, with illustrations, giving advice on the
creation process. Chapter 6 then essentially refines the tessellation with a
variety of special techniques. As such, as they are the core of the book, I now
examine these tips and techniques in relative detail, firstly with a quote from
the introductory text of Chapter 5, and then the following tips, and then
Chapter 6, of techniques.
Chapter 5 Specifics:
Designing an Escheresque tile is easier if you choose motifs to be
flexible. Some animals can be twisted around and distorted quite a bit, while
others can’t be. Lizards, birds and fish seem to work particularly well, and
M.C. Escher used these motifs for many of his tessellations. As a motif people
are especially challenging. Not only are they not very flexible, but the human
eye is more sensitive to whether people look right than to whether or not other
animals that we don’t see as often look right.
take issue here with the term ‘flexible’. I am not certain as to what the
author intends here – does he mean inanimate (non-flexible) and animate
(flexible) motifs? Inanimate motifs, although indeed generally inflexible, are
not necessarily non-flexible. For instance, a book can be shown opened or
closed, a car can have the doors, bonnet, or boot opened, and so have a degree
of flexibility. Admittedly, most inanimate motifs lack this, for example a
soccer ball, or a house. Just by changing the qualifier to animate or non-animate
clears the issue, and removes any uncertainty.
aspect I take issue with here is the specific motifs used to illustrate this
point. Broadly, with few exceptions, any creature can be twisted around,
and adopt different poses. For example, in contrast to Fathauer’s assertion
that a human figure is, in so many words, ‘not flexible’, it can adopt a whole
myriad of poses, such as standing, lying down, jumping, stretching, crouching
or bending in half. Other creatures have different degrees of flexibility.
Birds can have their wings raised, lowered. Fish can have their body twisted,
and fins of just about any shape. However, all this does not explain why
birds and fish are easy motifs to tessellate with, in contrast to human figures
which are relatively difficult.
exception to this ‘flexible’ rule would be creatures with a shell, such as
turtles, of which by their very nature such poses above are impractical.
However, it is still possible to design plausible motifs with shells.
Therefore, as argument per se, this ‘flexible’ description does not convince
I take issue with lizards being the first motif mentioned; as this gives the
impression that this is one of the easiest motifs to accomplish. This motif is
found less frequently in tessellation, with birds and fish dominating, and so I
would emphasise these motifs, and would have lizards third in this list.
Tip 1, Page 49
To the extent
possible, the outline of the tile should suggest the motif even without
Exactly! I cannot
over emphasise the importance of this point. This is the essence of life-like
tessellation, a true test of understanding of the issues and of one’s ability
to compose motifs that are recognisable by their outline or silhouette alone.
Furthermore, this point is clearly illustrated with two examples, of which I
describe as inferior and superior, thus clearly showing the differences in
quality. In silhouette form, the inferior T. Rex (left) is not particularly
suggestive of a recognisable motif. In contrast, the superior winged dragon
(right) is instantly recognisable as of a motif in profile, albeit of an
imaginary nature. Note that although the T. Rex is not readily identifiable,
with the addition of interior detail it can be made into a ‘respectable’ motif
of relative quality.
Tip 2, Page 50
The tiles should
make orientational sense
As such, I take
issue with this statement, as it is open to interpretation, albeit it does
contain merit, but in a different context. The author is apparently referring
to what I prefer to term as views, in that the motifs should be
consistent, such as all seen from above, or from the side, but never combined.
If indeed shown combined, then such a presentation is obviously inconsistent.
This also implies that the motifs are ideally portrayed in an ‘upright’
orientation. However, in the course of designing tessellations, one can use
various symmetries, some of which naturally ‘force’ the tessellation motif to
appear in many different orientations. For example, Escher has tessellations
that have as any as six orientations. Does this make the ‘six orientation’ type
inferior to a ‘single’ orientation per se? I think not. Certainly, this issue
has implications when a story is being told, as with Escher's prints, in which
he used elements of his tessellations. For example, Day and Night, where
it is of crucial importance for the motifs to be upright, as otherwise the
composition is absurd. However, this is a different matter from a tessellation
per se, where such concerns do not come into play.
Tip 3, page 51
When using more
than one motif, choose motifs that go together
shows true understanding of the issues. Although it is possible to have
creatures in a ‘multi-motif’ tessellation that have no connection, such as a
dog and a book, such tessellation are lacking in an aesthetic sense. Better is
to have those with connections to each other, such as a dog and bone or with
natural contrasts, such as angels and devils. However, although desirable, this
should not be undertaken at the expense of quality of motif. A lesser quality
example is not made ‘better’ by just having connections to each other.
Tip 4, Page 52
motifs should be commensurably scaled, unless a special effect is desired
shows true understanding of the issues. Although it is possible to have
creatures in a tessellation that are not commensurably scaled, such as a dog
larger than a human, such tessellations are lacking in an aesthetic sense.
Better is to have those that are more or less true to scale. Again, this should
not override matters of quality though.
Tip 5, page 53
For real world
motifs, it is important to use source material to get the details right
concerns the drawing aspect, and is sound advice. As one’s tessellation will
mostly concern the animal world, of which there is a great many different
motifs, with literally thousands of different species, who can claim to be an
expert in drawing all possible creatures? For example, although a tessellating
motif may look like a certain creature, say, a Gila monster, how many people
could draw this accurately enough for it to be regarded as true without
reference to a picture? Not many. Certainly, not myself. Even Escher himself
used reference material.
Also, as the motif can adopt different poses, say with a bird in flight with
different wing positions, or a fish twisting, of which again appropriate
reference material is desirable.
If one is serious about creating life-like motifs, one should bias this
learning towards birds and fish, as tessellations mostly consist of these
Tip 6, Page 54
design. Don’t try to make it too realistic
I take issue with
this. Again, it’s not clear as to what the author has in mind. The outline? The
interior design? On the contrary to the given statement, the desire of utmost
realism, as regards the outline, should be the main aim. However, although
a laudable aim, this ideal is seldom accomplishable. The author gives a frog
example to illustrate his point. As can be seen, the frogs leg/foot consists of
a series of very fine detail, with many convoluted curves that are simply not
‘appropriate’ for tessellation purposes. Of necessity, these feet/legs have to
be simplified if it is to tessellate. As can be seen, upon so composing a
tessellation, these are now heavily stylized, but to an acceptable degree, as
the motif is still broadly recognisable as a typical frog. Incidentally,
Escher’s own frogs are shown in the same stylized manner as here. In short,
make the outline as realistic as possible; and only stylized if not.
Tip 7, Page
a style that fits your taste and abilities
refers to the style, as regards the degree of realism and stylization of the
finished artwork, either by hand or computer, and which can range from a
wireframe, cartoon, to a photorealist finish, with nuances in between. To
illustrate the point, six different examples are given with different drawing
styles. However, the issue of ‘recognition’, as in a ‘complete’ tessellation,
with many motifs is not addressed. For example, a tessellation with a
photorealist finish, which may be thought to be the epitome as regards realism,
when shown as a actual tessellation will be found to be most unclear, as the
various colourations tend to mask the individual motifs. Therefore, a generally
stylized finish is to be preferred, of generally one coloured motifs.
Tip 8, Page 56
that suit your taste and that bring out the tiles
the author somewhat dilutes this by stating in the text that it is usually
desirable, but not essential to use different colours for adjacent tiles…For
me (and Escher), it is essential; as such uncoloured (wireframe) types lack
what I term as recognition. For example, to compare colour and non-colour
types, I use the seahorse tessellation. This is shown uncoloured page 63, and
coloured on the back cover. Is the motif readily identifiable with no colour? I
think not. When coloured, immediately so.
As a rule, colour your tessellation. By so doing, this will aid recognition of
the motifs, as the colour defines the motif, in contrast to a wireframe example
which is implied here.
Another area of ambiguity is in colouring adjacent motifs the same colour.
Generally, I frown upon this, as it does not aid the recognition. However, on
occasions, it is possible to override this rule. An example of my own is of the
Penrose tiling, of birds and fish, where I have two motifs. These are coloured
in two colours. However, it can be seen that this violates the map-colouring rule
of non-contiguous colouring, where three colours are necessary in this
instance. However, as three colours are incompatible with two motifs, and so
there is a trade of between minimum numbers of colours and map colouring.
However, as a rule, recognition of the motif should come first.
Chapter 6 Specifics:
Technique 1, Page 64
technique distorts the ‘finished’ tessellation slightly, either by stretching
or shearing, with the aim in mind of improving upon the original. However, the
examples here, of rabbits and a chameleon barely require this feature. Indeed,
at first glance the chameleon is no different.
To give perhaps a better illustration, beginning with a square, let us say that
this may be suitable, but not ideal, for a proto human figure. By its very
nature, the motif will be somewhat dumpy. If this is then stretched to a
rectangle, then a more realistic human figure will appear.
However, although the premise is good, this technique is very infrequently
required. Only on a handful of occasions will it be found necessary.
Technique 2, Page 65
This could perhaps
more better to have been included in Technique 4, Splitting and Moving. As
such, this is of limited use, and will only be used very infrequently.
Technique 3, Page 66
Splitting a tile
into two or more smaller tiles
Excellent! This is
a device that is of some importance as regards design, and is in common usage
(in contrast to other techniques mentioned above). I frequently use this, of
which Escher did so likewise. Therefore, this is of more interest than
Although the principle is correct, this is unfortunately illustrated with a
relatively poor quality tessellation that does not serve the argument well, of
frogs, toads, and snakes, but it will suffice for discussion purposes. The
premise here is that the frogs and toads are of an acceptable standard, whereas
the snake is not. Accepting this premise as so, can this be made better?
Instead of adding snake-like detail to essentially unsuitable raw material, a
different approach is possible, with the tile spitting procedure. This is now
divided into two identical tiles from which a ‘better’ creature emerges, a
tadpole. In reality, this is not really any better than the snake, but the
principle is germane.
Frequently, a situation will arise where in a tessellation of ‘multiple tiles’
where for one tile the motif will be of good quality, but for the other tile
nothing suitable can be found. By so ‘tile splitting’ this thus gives a
possibility of a better motif, and so a higher quality tessellation than
otherwise, rather than using an essentially unsuitable one.
Technique 4, Page 67
Excellent! This is
a device of some importance, albeit of a secondary aspect to the initial
design, in that it can be use frequently to improve upon the initial
tessellation. Escher frequently made use of this somewhat more advanced
technique, especially at the beginning of his studies, perhaps somewhat
surprisingly, given the relative degree of complexity involved. Broadly, the
procedure involves ‘releasing’ the motif from its original vertices, and by so
doing thus, in theory at least having the potential to improve the original
Critiques of the
A pleasing variety
of motifs are shown, with no less than 43 different motifs, this in contrast to
the more commonly found birds and fish, to the general exclusion of others.
However, this as an ideal per se should not be at the expense of quality – a
quality motif, say a bird, is worth more than a hundred (or more) poor quality,
of rarely shown motifs. Quality is everything…
Many of the tessellation shown here are in outline (wireframe) mode, a type of
presentation that I dislike, preferring coloured, or shaded examples. Although
this is fine for studies, I do not consider it so for finished work. However,
this aspect does not affect the quality of the motif per se, as it could be
coloured. For example, some of the best examples namely Men in Coats, Seahorses
are presented as such. Where examples of this type occur, I simply state
wireframe, with the aim in mind of signifying my disapproval of presentation.
1. Dinosaur (T-Rex), Page 37
Reasonable. A lower
tariff viewpoint tessellation, in that its outline is not immediately
recognisable as a dinosaur. Better would have been from the side. That being
so, the artist nonetheless manages to create a respectable tessellation. The
shading is poor, not following the map-colouring rule.
2. In the Garden, Page 37
Poor. I don’t like
this. A themed tessellation, of three garden-based insects, with bugs,
ladybirds and leaves. Although as a concept aesthetically satisfying, as might
be imagined, this is of a lower quality, due to the self imposed themed
restrictions. Indeed, one of the motifs is unidentifiable.
3. Horned Lizard and Gila Monster, Page 38
Poor. I don’t like this. As can be seen, both of the creatures have legs that
are merged with the body, thus rendering their outline indistinct. A decidedly
lower quality example. The shading is also poor, not following the
4. Ankylosaurus, Page 39
passable Ankylosaurus, although the view shown is not the best, as this would
be better portrayed from the side. Pleasing elements include the plate at the
rear of the tail, echoing the real motif. However, this would be lost on most
people, as whom, save for a dinosaur expert, would know this? (I myself only
knew this for a picture search for these comments.)
5. Pteranodons, Page 39
Poor. I don’t like this. Strictly speaking, this is not a Pteranodon, in that
creature has amongst its elements a longish neck and legs, where this has these
elements more restricted. Also, it’s shown from the above rather than ideally
from the side, where its ‘Pteranodon elements’ would be more readily visible.
Another example where the elements merge with the body. Of lower quality. Also
note the shading, where contiguous regions share the same shade. However, in
this instance, this was probably undertaken to emphasise the glide reflection
symmetry, and so has a purpose.
6. Squids, Rays, and Sea Turtles, Page 40
Reasonable. A themed tessellation, with a sea theme, with squids, rays, and sea
turtles. A better tessellation, in that the outlines are more recognisable.
7. Firemen, Page 42
Reasonable. A lower
tariff of difficulty type of motif, of heads. Of its type, reasonable.
8. Winged Dragon, Page 49. Also see page 56 (Wireframe)
Good. A good tessellation, the motif being
readily recognised in silhouette, which as stated above, is the true test of
quality, and is shown in its most typical representation, from the side.
However, it is of a lower tariff of degree difficulty, being an ‘imaginary’
creature. Would benefit by shading/colouring.
9. Rays, Page 50, 115 (Wireframe)
Poor. I am not convinced by the portrayal of the rays here. As such, these seem
highly contrived, of which after a picture search was unable to find any that
adopt this position (although this is not to say that they might). Would
benefit by shading/colouring.
10. Beetles, Moth, and Bumblebees, Page 51
themed tessellation, with three insect motifs. As such, although the motifs are
identifiable with the detail, in silhouette this is less so, as the outlines
lack the necessary indentation.
11. Cats and Mice, Page 52
Poor. A themed tessellation, with two
related motifs, cats and mice. Although at first glance a respectable
tessellation, with both cats and mice readily seen, I have reservations here.
The legs of both motifs merge with the body (as does the mouse’s tail), and so
is not ideal. As such, I am dubious of this type – although of a theme, and so
aesthetically pleasing as a concept, the trade-off is one of quality – are the
motifs good enough here? A further drawback is the incongruous scales used,
although I have less of an issue with this. Also, the colouring is a concern,
which has adjacent areas the same colour, albeit again, of lesser concern than
the quality per se.
12. Monkeys (and Bananas), Page 52
Poor. A themed tessellation, with two related motifs, of monkeys and bananas.
Although at first glance a respectable tessellation, with both monkeys and
bananas readily seen, I have reservations here. Three of the four elements of
the monkey merge with the body. A further drawback is the incommensurate scale
of the respective motifs, although I have less of an issue with this.
13. Frogs, Page 54
Reasonable. A better tessellation, in that the silhouette is instantly
recognisable, with the elements (limbs) made distinct. Although the legs/feet
are shown somewhat stylized, this is a necessity, and should not be thought of
as a weakness in this case. Also note the shading, where contiguous regions
share the same shade.
14. Cats, Page 54
Reasonable-Poor. A somewhat unusual portrayal, with the cat shown on its back.
Of its type, quite good, although again with considerable surface decoration.
Also note the shading, where contiguous regions share the same shade.
15. Rabbits, Page 64 (Wireframe)
Reasonable. Quite pleasing, although the motif is seen from the front, rather
than the ideal side. A drawback is the ears, which merge. Would benefit by
16. Chameleon, Page 64 (Wireframe)
Good. Pleasing, in that a recognisable motif
is discernable, of the ideal representation, from the side, and furthermore
with all its elements readily identifiable. Would benefit by shading/colouring.
17. Pteranodons, Page 65
Reasonable. Strictly speaking, this is not a Pteranodon, but rather a generic
dinosaur, in that former creature has a longish neck and legs, where this has
these elements considerably more restricted. Perhaps I’m being over fussy here
as to title. Still, if a specific creature is given, one should expect a degree
of resemblance. Also, it’s shown from the front, rather than ideally from the
side, where its Pteranodon elements would be more readily visible. Certain
aspects of this merge. Average. Also note the shading, where contiguous regions
share the same shade.
18. Frogs (and Toads, Snakes), Page 66
Unacceptable. I don’t like this, albeit themed. The
frog’s front and back legs merge with the body, whilst the snake is most
contrived, with a pseudo coiled effect. The tadpoles are vague, essentially
benefit by shading/colouring.
19. Seahorse, Page 79 (also see page 63 and coloured
version, back cover) (Wireframe)
Good. A very pleasing tessellation, of
undoubted high quality. Attention has been paid to fine detail, and so thereby
this aids recognition. Indeed, all the elements are readily visible, such as
the elongated nose, fins, curly tail eye region, all of which echoes the
subtleties of the real motif. Would benefit by shading/colouring.
Also see a six-fold version, page 106.
20. Seahorses and Eels, Page 80 (Wireframe)
Reasonable. A themed tessellation, with two related motifs. However, I am not
convinced by the eel drawing. Is this truly eel-like? Of note is the quality of
the seahorse, which is of a lesser quality than with single example
tessellations of this motif on pages 79 and 106, this caused by the demands of
the themed tessellation. Would benefit by shading/colouring.
21. Flowers 1, Page 88 (also see coloured version, back
Reasonable. Of its type, quite pleasing, however, the category itself, of an
‘amputation’ type, is of a decidedly lower tariff. Would benefit by
22. Seahorses (Six-fold Seahorse), Page 106 (Wireframe)
Good. Very pleasing. As can be seen, this uses
many of the same lines as of the four-fold example (page 79), and is also
instructive in showing how an initial suitable line can be used again with
different symmetries. Would benefit by shading/colouring.
23. Manta Rays, Page 115 (Wireframe)
such, an exotic creature, of which most people will be unfamiliar with its
exact shape. However, upon picture research, this looks broadly correct. Would
benefit by shading/colouring.
24. Geese, Page 116 (Wireframe). Also see frontispiece
Good. Quite pleasing. Of note here is
premise of ‘same orientation’ that in this instance I consider to the detriment
of the tessellation. Note that this tessellation consists of a single tile, it
appearing in two orientations. Consequently, if the same interior was employed,
one motif would be upright, the other upside down. As has been previously stated,
the author does not favour such ‘orientation discrepancies’, of which to
overcome this uses a different interior for the ‘upside’ down tile. However,
the query is to whether this justifiable, as I consider that bird to be
noticeably weaker. Would benefit by shading/colouring.
25. Men in Coats, Page 118 (Wireframe)
Good. I like this. From the outline alone it is recognisable broadly as a human
figure, despite the arms merging with the body. Every element is in proportion.
This is arguably the best tessellation here. Would benefit by
26. Largemouth Bass, Page 120 (Wireframe)
Reasonable. Quite pleasing, as by the outline
alone, if not readily apparent, it can be said to be undoubtedly fish-like. Would
benefit by shading/colouring.
27. Dragonflies (Bug Reflections), Page 124 (Wireframe)
However, examples of this type, with ‘imaginary’ bugs, are quite easy to do,
and are of a lower tariff of difficulty. This is despite them being of three
motifs, of which as detailed elsewhere in my essays, is generally difficult to
accomplish, but in instances of this type, is relatively easy. Would benefit by
As such, I am broadly
positive about this book, albeit with some reservations. Pleasingly, the author
has addressed some of the key issues underlying the creation of Escher-like
motifs, such as the outline or silhouette test; this being a key aspect. This
is something which is conspicuous by its absence in other people’s writings (I
even include Escher in this), and so on that account the book is warmly
welcomed. However, even here, other key issues have been omitted, such as why
birds and fish are so suitable, as well as an assessment as to inherent worth
of different motifs. (Some motifs are more ‘worthy’, or of higher tariffs than
others.) Furthermore, there are other aspects I am less keen on. Prominent
throughout is the insistence on ‘themed tessellations’, of which I consider
detracts from issues of quality of motif. Numerous instances are shown where
such themed examples lack inherent quality. Although as an ideal the aim is
laudable, the tessellations of this type are much lower in quality, as might
readily be imagined by such restrictive choices. Here, I believe that this
factor has overridden matters of quality assessment, with many examples of at
best questionable quality, and at worst unworthy of the art.
Although many different motifs are shown, of which again is a laudable aim, in
the interests of variety, many of these fail the outline or silhouette test, the
true test of quality. Again, I believe this desire to show many different
motifs have overridden matters of quality assessment, with many examples
unworthy of the art.
Nonetheless, there are indeed some of true worth here, such as the Men in Coats, Chameleons, and Seahorses.
Updated: 30 September 2009. Revised 10 July 2012