Although
on the face of it the title would suggest a tessellation book 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.
As a generalised statement, the advice given 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
understanding of the underlying issues concerning the creation of
life-like motifs. Very few books concern themselves with this matter, even in
passing, and which lies at the crux of the problem of designing high quality
motifs.
Broadly, upon having absorbed each chapter, a series of
activities are then given, suitable for the above age range.
Chapter
1, Introduction to Tessellations, contains illustrations of
tessellations in the real world as a historical context.
Chapter
2, Tessellations in our World, shows tessellations from the natural
world.
Chapter
3, Geometric Tessellations, details the underlying mathematical aspects behind
tessellations.
Chapter
4, Symmetry and Transformations in Tessellations, discusses mirror symmetry
and translation, glide reflection, and rotation.
Chapter
5, Tips on Designing and Drawing Escheresque Tessellations, gives a
series of tips on designing Escher-like tessellations
Chapter
6, Special Techniques for Solving Design Problems, gives a series of
techniques for improving on a initial tessellation
Chapter
7, Tessellations Based on Square Tiles, a step by step guide to creating
tessellations
Chapter
8, Other Tiles Based on a Square Grid, utilises right triangles and kites
with a series of templates
Chapter
9, Tessellations Based on Equilateral Triangles, utilises equilateral
triangles with a series of templates
Chapter
10, Tessellations Based on 60°-120° Rhombus Tiles, utilises rhombuses
with a series of templates
Chapter
11, Tessellations Based on Hexagonal Tiles, utilises hexagons with a
series of templates
As
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, ‘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.
A 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.
As 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:
Quote
1
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.
I
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.
Another 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.
An 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 me.
Also, 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 interior details
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 utilise 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 utilised 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
Excellent!
This 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
The
different motifs should be commensurably scaled, unless a special effect is
desired
Excellent!
This 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
This
really 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 particular motifs.
Tip
6, Page 54
Stylize
the 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 55
Choose
a style that fits your taste and abilities
This
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
Choose
colours that suit your taste and that bring out the tiles
Excellent!
However, 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, 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
Distorting
the entire tessellation
Excellent! This 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
Breaking
symmetries
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
otherwise.
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
utilising an essentially unsuitable one.
Technique
4, Page 67
Splitting
and moving vertices
Excellent!
This is a device of some importance, albeit of a secondary aspect to the
initial design, in that it can be utilised 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
motif.
Critiques
of the Individual Tessellations:
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.
Dinosaur
(T-Rex), Page 37
A
lower standard 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.
In the
Garden, Page 37
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.
Horned
Lizard and Gila Monster, Page 38
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 poor, not following the map-colouring rule.
Ankylosaurus,
Page 39
A
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.)
Pteranodons,
Page 39
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 pteranodons 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.
Squids,
Rays, and Sea Turtles, Page 40
A
themed tessellation, with a sea theme. A better tessellation, in that the
outlines are more recognisable.
Firemen,
Page 42
A
lower quality type of motif, of heads. Of its type, reasonable.
Winged
Dragon, Page 49. Also see page 56 (Wireframe)
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.
Rays,
Page 50, 115 (Wireframe)
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).
Beetles,
Moth, and Bumblebees, Page 51
A
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. Average.
Cats
and Mice. Page 52
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 of is
one of quality – are the motifs good enough here? A further drawback is the
incommensurate scale 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.
Monkeys
(and Bananas), Page 52
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,
although I have less of an issue with this.
Frogs,
Page 54
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.
Cats,
Page 54 (Wireframe)
A
somewhat unusual portrayal, with the cat shown on its back. Of its type, quite
good. Note that the elements here are distinct from the body, save for one leg.
Also note the shading, where contiguous regions share the
same shade.
Rabbits,
Page 64 (Wireframe)
Quite
pleasing, although the motif is seen from the front, rather than the ideal
side. A drawback is the ears, which merge.
Chameleon,
Page 64 (Wireframe)
Quite
pleasing, in that a recognisable motif is discernable, of the ideal
representation, from the side, and furthermore with all its elements readily
identifiable.
Pteranodons,
Page 65
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.
Frogs
(and Toads, Snakes), Page 66 (Wireframe)
I don’t
like this. 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 unrecognisable.
Seahorse,
Page 79 (also see page 63 and coloured version, back cover) (Wireframe)
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.
Also see a six-fold version, page 106.
Seahorses
and Eels, Page 80 (Wireframe)
A
themed tessellation, with two related motifs. 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.
Flowers
1, Page 88 (also see coloured version, back cover) (Wireframe)
Of
its type, quite pleasing, however, the category itself, of an ‘amputation’ type
is of a decidedly lower tariff.
Seahorses
(Six-fold Seahorse), Page 106 (Wireframe)
Very
Pleasing. As can be seen, this utilises 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.
Manta
Rays, Page 115 (Wireframe)
As
such, an exotic creature, of which most people will be unfamiliar with its
exact shape. However, upon picture research, this looks broadly correct.
Geese,
Page 116 (Wireframe). Aso see frontispiece
Quite
pleasing. Of note here is premise of ‘same orientation’ that 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 utilizes 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.
Men
in Coats, Page 118 (Wireframe)
I
like this. From the outline alone it is recognisable as a human figure, despite
the arms merging with the body. Every element is in proportion. This is arguably
the best tessellation here.
Largemouth
Bass, Page 120 (Wireframe)
Quite
pleasing, as by the outline alone, if not readily apparent, it can be said to
be undoubtedly fish-like.
Dragonflies
(Bug Reflections), Page 124 (Wireframe)
Of
its type, reasonable. 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, relatively
easy.
Summary
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’ 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 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’.
Last Updated: 30 September 2009