FIGURATIVE TESSELLATION METHOD

Free online from my book "Parcelles d'infini"
published in French by Éditions Pour la Science, ISBN 2-84245-075-2, in 2005

With all my thanks to David Bailey www.tess-elation.co.uk for its invaluable assistance concerning the translation.
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Summary

1     Imitating nature

2     Transforming the polygons

3     Assembling the tiles

4     Tracing the structures

5     Using basic polygons

6     Dividing the tiles

7     Tessellating with words

8     Distorting the structures

9     Playing with isometries

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Foreword

"The inventor has, all of a sudden, the distinct feeling
that the designs which he creates (...)
already existed before to have never been thought in the human brain.
Louis de BROGLIE

My song is really ended
when she looks to be made itself alone.
Georges BRASSENS

I have such sensations all the time when I am working on designs
for regular surface fillings.
It seems it is not I who am doing the creating,
but rather that the innocent flat patches over which I am slaving
have their own will, and it is they which guide
the movement of my hand as I draw.
Maurits Cornelis ESCHER

"A long time ago, I chanced upon this domain in one of my wanderings I saw a high wall and a premonition of an enigma, something that might be hidden behind the wall, I climbed over it with some difficulty. However, on the other side I landed in a wilderness and had to cut my way through with a great effort until - by a circuitous route - I came to the open gate, the open gate of mathematics."

"I walk around all alone in this beautiful garden, which certainly does not belong only to me, but whose gate is open to everyone."

Maurits Cornelis ESCHER
Periodic filling of a plan

This method, taken from my book "PARCELLES D’INFINI", offers a walk in the garden of the regular division of the plane that Escher explored in many directions. This area may seem limited judging by the few artworks coming from others. I hope that what follows will make you realize that there are an infinite number of possible figurative motifs. And it is not just copying Escher to make a periodic division of the plane.

… No more, for instance, than copying Jean-Pélerin* that makes a drawing in perspective.

And if, as I hope, you are inspired to do figurative ‘tiles’, throughout the following pages you will find how to give birth to them.
And when, after some difficulties, you successfully make one, you will experience a great feeling of joy and humility before this "PARCEL OF INFINITY".

black line

*. Called the Viator. He wrote the first printed treatise on perspective, published in 1505.

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1   IMITATING NATURE

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There is so much art in nature
that very art consists to well heard and imitate it.

Jacques Bénigne BOSSUET

It is a sad thing to think that nature speaks
and that the human race is not listening.

Victor HUGO

dropped capital the laws of men are ephemeral, those of nature are eternal. Even if our incursions on its paths will be always modest,  this remains a great pleasure and a great honour that to be able to imagine plane variations on its own structures.
The honeycombs of the bees, the fish scales or the sunflowers give us some fine examples of tessellations but it is especially the crystals that nature offers us to show us the way. We find in their structure the 17 symmetry groups allowing to repeat a pattern at infinity. But to draw figurative tessellations it would be preferable to create to us a both simpler and more appropriate classification. This is what we will do.
logo infinitely 1

There are three regular polygons that can divide the plane periodically, these are:

triangle tessellation

The triangle

square tessellation

The square

hexagon tessellation

The hexagon

But fortunately, quantities of irregular polygons can also divide the plane periodically.
       A few examples:

any triangle tessellation

The triangle

right isoceles triangle tessellation

The right isosceles triangle

rhomb 2 angles 120° tessellation

The rhomb 2 angles
120°

rhomb tessellation

The rhomb

parallelogram tessellation

The parallelogram

quadrilateral

The quadrilateral

pentagon 2 sides parallel tessellation

The pentagon 2 sides parallel and equal

hexagon 2 opposites sides

The hexagon 2 opposites sides
parallel and equal, between
2 x 2 sides adjacent and equal

In addition, it is possible to replace the sides of all polygons dividing periodically the plane, by compensated deformations that does not alter their area.
       Examples :

deformed translation

Translations of compensated deformations on parallelogram

Translations of compensated deformations on hexagon with opposites sides parallel

deformed translations

The multiplicity of achievable compensated deformations brings the amazing opportunity to give rise to an infinite number of figurative motifs. .
Examples :

square bird tesselletion

Progressive translations of compensated deformations on squares

square birds tesselletions

Transformations by translations of a same rectangle into six different birds

The base polygon having undergone compensated deformations is named tile.
       Examples :

• Translations on a parallelogram:

fox tiles

Base polygon

Compensated deformation

Tile

• Translation on a square:

witch doctor tiles

Base polygon

Compensated deformation

Tile

• Translation on a concave hexagon:

karate tiles

Base polygon

Compensated deformation

Tile

• Translation on a regular hexagon:

human hexagon tiles

Base polygon

Compensated deformation

Tile

glow tessellation

Glow

foxes at the moon tessellation

Foxes at the Moon

brotherhood tessellation

Brotherhood

leaping witch doctor

Leaping witch doctor

karate tessellation

Karate

A squared paper facilitates the tracing of the tile.
       Example :

dog quadri tiles

Base polygon (this is for us
  a hexagon of which 2 x 2
sides are in prolongation)

Compensated deformations

Tile

A set of tiles is named tessellation :

lucky dog tessellation

Lucky Blanchepatte, my dear gone dog

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2   TRANSFORMING THE POLYGONS

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The geometry is to fine arts
what grammar is to the art of the writer.

Guillaume APOLLINAIRE

More the art is controlled, limited, worked
and more it is free.

Igor STRAVINSKY

dropped capital trying to give birth to figurative motifs without  geometric bases gives very limited results. The polygons are the eggs from which will hatch a multitude of characters.
hexa chick tessellation hexa chick tile infinity logo

There are three possible deformations that can replace the sides of a polygon:
        1. Any (as its name says):

side of polygon before deformation


side of polygon after deformation


symmetry axis


center of rotation

any deformation

2. Axial (any deformation more its reflection in relation to an axis):

axial deformation

3. Rotatory (any deformation more its 180° rotation):

rotatory deformation

The deformation, of any or axial of a side must always be compensated by an identical deformation on another side. The rotatory deformation compensates itself. We call these compensations isometries.

There are four types of isometries :

1. The translation is the simple rectilinear slide of one deformation:

2. The rotation rotation is either rotatory deformation or the pivoting of a deformation around a centre of rotation:

rotatory birds tiles translation birds tile

3. The glide reflection is the reflection of a deformation in relation to an axis, followed by a translation:

4. The symmetrie is the reflection of a deformation in relation to an axis:

symmetrical bird tile glide reflection bird tile

Examples :

• Two translations on a parallelogram:

toucans tessellation

• A 180° rotation and another to 60° on an equilateral triangle:

turning birds tessellation

• A glide reflection as well as a translation on a parallelogram:

swallow tessellation

• Two symmetrical slide reflections and an axial translation on hexagon:

birds legs tessellation

The glide reflection requires some clarification:

- The sides of the polygon may be adjacent (have a common end) or non-adjacent :

A polygon can have two glide reflections. In this case, their vectors axes are either parallel or perpendicular :

glide reflection
adjacentnon-adjacent
Parallelperpendicular




The tracing of the
glide reflections
is particularly
simplified on
a sheet of
squared paper

Let L be the length of the sides of the polygon:

search vector glide reflection

The v and v' points allow to trace the vector axis of the glide reflections. This vector axis indicates the axis of reflection as well as the direction and length of glide.

Example :

perpendicular reflections

The base polygon of this bird is a concave pentagon having one rotatory side and 2 x 2 sides of perpendicular vectors axes.

perpendicular reflections tessellation

ISOMETRIES

DESIGNATIONS
AND ABBREVIATIONS
CONDITIONS
IN THE POLYGON
SIMPLIFIED
REPRESENTATIONS
The translationT2 sides parallel and equal

The rotation

  - 180°

R2

1 any side

  - 120°

R3

2 sides equal and adjacent at 120°

  - 90°

R4

2 sides equal and adjacent at 90°

  - 60°

R6

2 sides equal and adjacent at 60°


The glide reflection
(If there are 2 in the polygon,
we add
' to the 2nd)


G


2 equal sides

The symmetry

S

2 symmetrical sides

There are four types of polygons that can divide the plane periodically:

four polygons tessellation
the trianglethe quadrilateralthe pentagonthe hexagon

The systematic search for all possible combinations between the four isometries and the four types of polygons allows us to lead to a large amount of specific polygons of which we can eliminate those unable to fill the plane.

       Examples :

There then remains 81 specific polygons, called isohedral tiles, of which we also eliminate those where the possibilities of obtaining figurative motif are too small:

       - because having one or more rectilinear sides:

rectilinear sides tiles

detained because









removed because

- because having central symmetry:

central symmetry tiles

Ultimately, there remains 35 modal tiles; 35 parcels of infinity that the magician in you which lies dormant maybe, and will transform into a rabbit, butterfly or dove.

top hat doves transformation tile
doves tessellation

Doves

The need for a framework to this method should not let us forget that there are always opportunities to get out of the framework.
Examples :

      • Escher's splendid "Chinese" has two rectilinear sides:

• Half card-trick has a central axis of rotation:

double queen card escher's chinese
half card trick

Half Card-trick

Here is its basic polygon:

chinese polygon black line

• Same for the octopus:

octopus tessellation
octopus tile divided line

3   ASSEMBLING THE TILES

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It happens sometimes that a person (…)
feels born and grow in her, one day,
a desire very conscious to approach by the imagination,
infinity as close as possible
in the purest way.

Maurits Cornelis ESCHER

Hope nothing from human
if he works for his own life and not for eternity.

Antoine de SAINT-EXUPÉRY

The feeling of infinity
is the real human attribute.

Madame de STAËL

dropped capital assembling the tiles, it is to arrange parcels of infinity  on eternal structures. Assembling the tiles, it is the privilege to play jigsaw puzzles with the universe.
birds rainbow tessellation infinity logo

Tiles similar to a modal tile can fit after rotation and/or glide reflection. They then have a different orientation.

      Examples :

different orientation birds
Glide reflectionRotationGlide reflection + rotation

This set of tiles can then similarly fit in turn by translation, and can then be continued to infinity :

birds sets tessellations
Type 1GType 2Type 2G

All these similar tiles are the transformations of the base modal tile. The whole, we have seen, is called tessellation.

round of lover birds tessellation

This round of lovers is just waiting to grow!

Visually, what characterises first a tessellation is the number of directions taken by the motifs. There are six possible directions, and therefore will be six original types.

Type 1 will have all its tiles in the same direction. Type 2 will have its tiles head to tail. Type 3 will have its tiles in three directions, type 4 in four directions and type 6 in six directions.

Examples :

        • Type 1, the athlete:

• Type 3, the lizard:

type 3 lizards tessellation

• Type 4, the kitten:

type 1 athletes tessellation type 4 kittens tessellation

• Type 2, the siren:

• Type 6, the otary:

type 2 sirens tessellation type 6 otaries tessellation

Then, the tessellations with one or two directions may have their motifs reflected. There will be also a type 1G (G for Glide reflexion) and a type 2G.

• Type 1G, the rooster:

• Type 2G, the dog:

type 1g roosters tessellation type 2g dogs tessellation

Finally, tessellation types 1, 2, 3 and 4 may have their motifs symmetrical. We will have in addition type 1S, type 2S, type 3S and type 4S.

• Type 1S, the gorilla:

• Type 2S, the butterfly:

type 1 gorillas tessellation type 2s butterflies tessellation

• Type 3S, the duck:

• Type 4S, the frog:

type 3s ducks tessellation type 4s frogs tessellation

That is to say 11 types in totality, which will find themselves on 11 structures. In addition, these 11 types have subtypes classified according to their basic polygon. In total, we meet again the 35 modal tiles of the previous chapter.

• The base polygon of the kitten is a right isosceles triangle. We will designate it by the isometries of the sides, that is to say R2 R4 R4:

• The one of the frog is a square. Designation R4S R4S R4S R4S:

frog tile kitten tile

Now here are the designations of basic polygons for:

     • the athlete: T T T T T T
     • the siren: R2 R2 R2
     • the lizard : R3 R3 R3 R3
     • the otary : R2 R6 R6
     • the rooster : T G T G
     • the dog: R2 G G
     • the gorilla: T GS GS T GS GS
     • the butterfly: T R2S R2S T R2S R2S

• The one of the duck is a rhomb with two angles of 120°. We will refer it by R3S R3S R3S R3S because the sides are both of 3 rotations and symmetries:

duck tile

The base polygon of the 'madman' is a regular hexagon. Designation R3 R3 R3 R3 R3 R3:

The use of isometric paper is ideal for tracing tiles with isometries of rotations 3 or 6.

kittens tessellation

Kittens

ducks hexagon tessellation

Ducks

chessboard with frogs tessellation

Chessboard with frogs

madman puzzle tessellation

The madman's jigsaw puzzle

• The base polygon of the "Li'l strong man" is a square. We will designate it by R4 R4 R4 R4:

• The one of Escher (Oh yes! even the Escher's profile is a tile!) is a hexagon. Designation T R2 R2 T R2 R2:

li'l strong man tiles escher hexagon tiles li'l strong man tessellation escher profile tessellation divided line

4   TRACING THE STRUCTURES

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If there is no trick, it is strong...
If there is a trick, it is even stronger!

MYR et MYROSKA

All things, near or far,
secretly are connected to each other
and you cannot touch a flower without disturbing a star.

Francis Joseph THOMPSON
dropped capital the structures are similar to the scenes of magic  theatres, in that they contain all the secrets needed for representation. To know these secrets disenchants only the fool.
35 tiles birds infinity logo



        We have seen that there are 11 structures, but the crystallographers count 17 symmetry groups. In fact, the six missing groups do not allow us to create unique motifs. We will indicate all the same the 6 groups of structures that do not allow to make unique motifs, this after the first 11 groups


The structures are represented using the following elements:

- center of rotation :  

- symmetry axis :  ———— — ——— — ———

- axis of glide reflection : ——————————

TYPE 2G (pgg)  
Rotation 180° + glide reflexion + translation
Structure + base polygon
and its transformations before translations
Example of base
tile
Example of figurative tileExample of arrowhead and
colored tessellation
(minimum of colors)

b (IH53)  Quadrilateral R2 R2 G G
36 / 412 / 77 / 98 / 116

The symmetry group used is that of the International Tables for x-ray crystallography.
The isohedral type defined is the one as given by B. Grünbaum and G. C. Shephard in their book Tilings and Patterns.
When we have several kinds of transformations before translation as in the present case, we try the first: this gives us two tiles. Then we make:
- either the transformation of the two tile set (top example);
- either the transformation of each of the two tiles separately (example below with concave tiles).

The small numbers indicate the angle of rotation (2 = 180°, 3 = 120°, 4 = 90°, 6 = 60°).
The structure of type 1 will be vacant because it contains only translations.
Nineteen basic polygons may be concave. These are: 1b, 1Gb, 1Gc, 1Gd, 1Sb, 2b, 2d, 2e, 2Gb, 2Gc, 2Ge, 2Gf, 2Gg, 2Gh, 2Sb, 2Sc, 3b, 4c et 6d. Exemple :

The figures indicate the number of adjacent vertices.

The 35 types of tessellations will be represented by 35 birds.

All isohedral tessellations can be coloured with a minimum of two or three colors. If all adjacent vertices are of even numbers, two colors are sufficient. If there are vertices of odd numbers, three colors are needed.

TYPE 1 (p1)  
Translations
Structure + base polygon
before translations
Example of base
tile
Example of figurative tile Example of arrowhead and
colored tessellation
(minimum of colors)

a (IH41)  Parallelogram T T T T

38 / 472 / 482 / 492 / 502 / 522 / 73 / 74 / 80 / 105 / 106 / 127 / 128

b (IH1)  Hexagon T T T T T T

182 / 222 / 272 / 282 / 292 / 302 / 722 / 822 / 842 / 872 / 922 / 1112 / 1122 / 1132 / 1142 / 1202 / 1212 / 1292

TYPE 1G (pg)  
Glide reflection + translations
Structure + base polygon
and its transformation before translations
Example of base
tile
Example of figurative tile Example of arrowhead and
colored tessellation
(minimum of colors)

a (IH43)  Parallelogram T G T G

31 / 32 / 7112 / 97 / 108 / 109

b (IH44)  Kite * G G G' G'

19 / 62 / 66 / 67 / 762 / 96 / 102 / A13

*. A kite is a quadrilateral whose one diagonal is perpendicular to each other in the middle).

c (IH2)  Hexagon T G G T G' G'

242 / 262 / 78 / 1102 / 1262 / 1302

d (IH3)  Hexagon T G G' T G' G

17 / 342 / 612 / 632

TYPE 1S (cm)  
Translations
Structure + base polygon
before translations
Example of base
tile
Example of figurative tile Example of arrowhead and
colored tessellation
(minimum of colors)

a (IH68)  Rhomb GS GS G'S G'S *

91 / A14

*. The sides with glide reflection being symmetrical, they are also and necessarily translations.

b (IH12)  Hexagone T GS GS T G'S G'S *

A1

*. The sides with glide reflection being symmetrical, they are also and necessarily translations.

TYPE 2 (p2)  
Rotation 180° + translations
Structure + base polygon
and its transformation before translations
Example of base
tile
Example of figurative tile Example of arrowhead and
colored tessellation
(minimum of colors)

a (IH84)  Triangle R2 R2 R2

51 / 95

b (IH46)  Quadrilateral R2 R2 R2 R2

9 / 88 / 90 / 93

c (IH47)  Parallelogram T R2 T R2

75 / 1152

d (IH23)  Pentagon T R2 T R2 R2


Escher did not make tessellations of this type

e (IH4)  Hexagon T R2 R2 T R2 R2

1 / 5 / 6 / 7 / 8 / 11 / A12

TYPE 2G (pgg)  
Rotation 180° + réflexion glissée + translations
Structure + base polygon
and its transformations before translations
Example of base
tile
Example of figurative tileExample of arrowhead and
colored tessellation
(minimum of colors)

a (IH86)  Isoceles triangle R2 G G


Escher did not make tessellations of this type

b (IH53)  Quadrilateral R2 R2 G G

36 / 412 / 77 / 98 / 116

c (IH51)  Quadrilateral R2 G R2 G

33 / 107 / 124

d (IH52)  Rectangle G G' G G'

39

e (IH25)  Pentagon T R2 T G G

682

f (IH27)  Pentagon R2 G G' G G'

16 / 462 / 592 / 602

g (IH5)  Hexagon T R2 R2 T G G

10 / 582

h (IH6)  Hexagon R2 G R2 G' G G'

2

TYPE 2S (pmg)  
Rotation 180° + translations
Structure + base polygon
and its transformation before translations
Example of base
tile
Example of figurative tile Example of arrowhead and
colored tessellation
(minimum of colors)

a (IH66)  Rectangle T R2S T R2S *

40 / 117

*. The sides with rotation 2 being symmetrical, they are also and necessarily glide reflections.

b (IH69)  Kite R2S R2S R2S R2S *

37 / 89

*. The sides with rotation 2 being symmetrical, they are also and necessarily glide reflections.

c (IH13)  Hexagon T R2S R2S T R2S R2S *


Escher did not make tessellations of this type

*. The sides with rotation 2 being symmetrical, they are also and necessarily glide reflections.

TYPE 3 (p3)  
2 rotations 120° + translations
Structure + base polygon
and its transformations before translations
Example of base
tile
Example of figurative tileExample of arrowhead and
colored tessellation
(minimum of colors)

a (IH33)  Rhomb R3 R3 R3 R3


Escher did not make tessellations of this type

b (IH7)  Hexagon R3 R3 R3 R3 R3 R3

21 / 25 / 432

TYPE 3S (p31m)  
2 rotations 120° + translations
Structure + base polygon
and its transformations before translations
Example of base
tile
Example of figurative tileExample of arrowhead and
colored tessellation
(minimum of colors)

a (IH36)  Rhomb R3S R3S R3S R3S *
532 / 542 / 103 / 123

*. The sides with rotation 3 being symmetrical, they are also and necessarily glide reflections.

TYPE 4 (p4)  
3 rotations 90° + translations
Structure + base polygon
and its transformations before translations
Example of base
tile
Example of figurative tileExample of arrowhead and
colored tessellation
(minimum of colors)

a (IH79)  Right isoceles triangle R2 R4 R4

35 / 118 / 119

b (IH55)  Square R4 R4 R4 R4

15 / 23 / 104

c (IH28)  Pentagon R2 R4 R4 R4 R4

14 / 20 / 423,5

TYPE 4S (p4g)  
3 rotations 90° + translations
Structure + base polygon
and its transformations before translations
Example of base
tile
Example of figurative tileExample of arrowhead and
colored tessellation
(minimum of colors)

a (IH71)  Square R4S R4S R4S R4S*
13 / 452 / 86 / 122 / 125

*. The sides with rotation 4 being symmetrical, they are also and necessarily glide reflections.

TYPE 6 (p6)  
Rotations 180°° + 2 rotations 120° + translat.
Structure + base polygon
and its transformations before translations
Example of base
tile
Example of figurative tileExample of arrowhead and
colored tessellation
(minimum of colors)

a (IH88)  Equilateral triangle R2 R6 R6

44 / 94 / 99 / 100

b (IH39)  Isoceles triangle R2 R3 R3


Escher did not make tessellations of this type

c (IH31)  Kite R3 R3 R6 R6

55 / 56

c (IH21)  Pentagon R2 R3 R3 R6 R6

572 / 70 / 79

The six groups of symmetry do not allow us to make unique motifs:

DesignationStructureExample of figurative tileDesignationStructureExample of figurative tile

pm
(Minimum
2 motifs)

p3m1
(Minimum
3 motifs)

cmm
(Minimum
2 motifs)

p4m
(Minimum
3 motifs)

pmm
(Minimum
3 motifs)

p6m
(Minimum
3 motifs)

The 35
base
polygons

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5   USING BASIC POLYGONS

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Imagination is more important than knowledge.
Albert EINSTEIN

The reason, it is the intelligence in exercise;
the imagination, it is the intelligence in erection.

Victor HUGO

dropped capital vou have the quill and the parchment. You have taken  knowledge of the structures secrets. You now need to appeal to the most important: your imagination. If on an old decrepit wall you see figures appearing more or less of a fantasy nature, you have a good chance of success.

When designing you can randomly choose a basic polygon and trace compensated deformations until you get a rough outline of motif. Or you may already have an idea of the motif to draw, trace it roughly, and adapt on it the type of best suited basic polygon and go back over the compensated deformations.
Of course, you will need to use a lot of the eraser before succeeding with a figurative tile.
Often your motif will be only half successful and so you will need to start over again..



But if really you get nothing satisfying, all is not lost. With the help of your computer, trace again your failed tile. Inside your tile add some motifs not connected, then select all and give a background gradient to all these traces, in the rainbow or tropical sunset style. It only remains to you to baptise the whole of a silly name and you just created a worthy representative of abstract art!

infinity logo

Now that you know how to work the structures, the summary of the previous 35 basic polygons will help you to give birth to figurative tiles. Your hardware: squared paper, isometric paper for types 3 and 6, a pencil and an eraser. That's all! When you will want to multiply the tiles to make tessellations, a computer with a drawing program will be convenient. But that is not required, do not forget that Escher drew all his tilings by hand. Tessellation software like that of Kevin Lee (designer of TesselManiac) will also provide you with valuable assistance for the creation of tiles.

hexus abstract art

Hexus
(Abstract art!) type 1b

elbow to elbow frogs

Elbow to elbow

icarian antipodists

Impossible icarian antipodists
type 2c

wrong way drawing

Wrong way

samson modern art

Samson's sleep
(Modern art!) type 2c

drawing beware fir

Beware… fir!

jealous parrots

Jealous

After this little interlude of more or less successful tiles, we come back to our base polygons.

• Let us take a square of type 1Ga. Trace a few broken lines on two adjacent sides (a). Then let us transfer them on the two other sides using a translation and a glide reflection (b). Let us study the resulting figure: It vaguely has a look of a little man. Let us reduce the left arm, this augments the right arm (c):

Let us modify the left leg, and that improves the definition of the right arm and the right leg (d). Let us refine the exterior details taking account both of what would be the ideal line on one side and what would be the one of its isometry (e). Now let us draw the interior details of the little man. One realizes that we can still improve certain outside line elements (f).

glide reflection boy tile start glide reflection boy tile end

Now just realize of what we made: We have transformed a rigid square into a young boy who runs happily on our sheet of paper!

boys comings and goings tessellation

Comings and Goings

• Let us now take a rectangle, of type 1Ga, and draw a few lines. We soon get the silhouette of a marching service man:

But his silhouette is leaning forward. Transforming the base rectangle into a parallelogram will enable us to straighten it out:

marching service man tile start marching service man tile end

And behold, from a rectangle at the beginning, we get a valiant service man who parades proudly below.

• In fact, a good number of drawings of figurative tiles begin by being... some drawings! Let us take for example the goat below. Let us examine it. It appears clear that there is a possibility of a glide reflection between the front legs and horns:

Then maybe another between a hind leg and the tail. It then only remains to adapt the glide reflections into a kite type 1 Gb:
And here is a brave little goat that comes down from the mountain:

goat glide reflection tile start goat glide reflection tile end

After these three examples, perhaps it seems easy to you to draw figurative motifs. Don't believe it; you need to try and try again, hundred times on the drawing board start your work again.

service man tessellation

Respecful Salutes

mountain tessellation

Mountain

• Let us take again a square of type 1Ga and try a few lines: After the transformation of the square into a rectangle and then a parallelogram there appears an Indian, the hand raised as a sign of peace:

indian tiles

• After the Indians, here are the cowboys with as base polygon a hexagon of type 1Gc:

cowboy tiles

• Let us find again the historic figure of the rodeo cowboy in a concave hexagon type 1Gd:

rodeo cowboy tiles

You will notice that the hat is detached from the rider and his horse. We have there, in fact, a tile composed of two subdivided tiles (see chapter Dividing the tiles).

• Let us stay in America with the creator of rockabilly, Elvis himself, in this regular hexagon of type 1Gc:

elvis presley tiles
indians tessellation

Indians

elvis emulation tessellation

Emulation

cowboy tessellation

Cowboy

rodeo tessellation

Rodeo

• The rhomb type 1Sa is suitable to appear as the silhouette of a bear:

bear tessellation type 1sa bear tiles type 1sa

• But if we lean the bear, the rhomb turns into a parallelogram of type 1a:

bear tessellation type 1a bear tiles type 1a

• And if the bear starts to dance, the base polygon becomes a kite type 1 Gb:

bear tessellation type 1gb bear tiles type 1gb


• The rectangle of type 2Gd easily turns itself into an undulating fish and could make a nice bathroom tiling:

bathroom tessellation rectangle fishe tiles black line

• From a sharp triangle of type 2 is born the pretty curves of a pretty girl:

• The chess piece horse is developed from a quadrilateral of type 2b:

naiade triangle tiles chess piece horse tiles
pretty girl tessellation

Dive in Trouble Water

chess tessellation

Chess

• This big good guy splendidly fills a hexagon of type 2Sc:

• This parallelogram of type 2b turns itself into a quadrilateral to reveal a judo player at the heart of the action:  

big good guy tiles judo tiles

But he has maybe a hidden face...

nice bad tessellation

Nice / Bad

judo tessellation

Judo

• Also in action is this acrobat monkey born from a concave pentagon type 2Ge:

• A type 2Gg hexagon is at the base polygon of the fish below.
But only by its contours, as details of half of the fishes - which should be upside down - was drawn right way up.

acrobat monkeys tessellation fishes upside down right way up tessellation black line

• This inflatable lizard is very convenient to be stored on the beach.
Its underlying polygon is a hexagon of type 2Gh:

• The green tree frog can come and go in a rectangle of type 2Sa:

green tree frogs tessellation inflatable lizards tessellation

• Apart from its ventral fins, the angelfish fits perfectly to a kite type 2Sb:

• This square of type 4b may give birth to a crested bird:

angelfish kite tiles crested bird tiles
angelfish mimetisme tessellation

Mimetisme

chessboard with birds tessellation

Checkerboard with Birds

black line

• You can cut the jumping jack below and assemble it:

jumping jack tile

Perhaps you notice that with the arms and legs in the air this one can also be born from a rhomb type 3Sa. So we can draw its transformations and thus constitute us with an army of jumping jacks !

jumping jack cutting jumping jack tessellation divided line

6   DIVIDING THE TILES

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People never divert themself so well that at the pursuit of eternity.
Jean GIONO

If you want to progress towards infinity,
explore the finite in all directions.

Johann Wolfgang von GOETHE

dropped capital as cells divide themselves to multiply, why not make  tiles to go two by two, see three by three or even more, to tessellate infinity?

The tessellation with this hexagon will give the figure below. Three colors are needed:

hexagons tessellation infinity logo

The tiles of all the types that we have seen can be divided into as many subdivided tiles as we want.

Take for example a hexagon with opposite parallel sides of type 1 b:

divided hexagon tile

Let us divide the hexagon in several ways:

divided hexagons 1 tessellation divided hexagons 2 tessellation divided hexagons 3 tessellation

In fact we can divide the base tile in as many different ways as we wish.
      Division causes changes in the vertices, and so in the minimum number of colors. We have seen that if all adjacent vertices of an isohedral paving are of even numbers, two colors are sufficient, and that if there are vertices in odd numbers, three colors are required.
      It is not the same for the subdivided tiles, as that may require up to four colors.

Example type 1bExample type 3b

divided hexagons 4 tessellation divided hexagons 5 tessellation divided hexagons 6 tessellation

• Here is a tile type 2Gc which resembles nothing much:  

fishes and lizards tessellation fish and lizard tile 1

But when it is divided into two it reveals a fish and a lizard:

fish and lizard tile 2


      • During a walk, pick up a leaf. Draw its outline:

Copy a third and a fourth still making them coinciding with another vertex.

leaf tile 1 leaves tessellation

With tracing paper draw a second leaf by coinciding two of the vertices:

leaves tile 2

With this method you can have the chance of seeing appearing a rabbit among the autumn leaves or a dove among the stars.

rabbits and leaves tessellation
doves and inaccessible stars tessellation

The inaccessible star

• Here are those of the Inaccessible Star: :

• Here are the base polygon and the base tile of the 'rabbit among the autumn leaves':

dove and star tiles rabbit and leaf tiles

You may have noticed that the dove and the star above do not have quite the same outline that the dove at the bottom and the star of the top of the drawing does. It is this appearance/disappearance style of drawing dear to Escher that allows these approximations. These matters are sometimes even more pronounced in the drawings that follow.

various tiles
dream of frog tessellation

Dream of frog

conjuring tessellation

Conjuring

merry christmas happy new year tessellation

Merry Christmas / Happy New Year

contrast tessellation

Contrast

the good and the bad tessellation

The Good and the Bad

dream of paper hen tessellation

Dream of Paper Hen

sea tessellation

Sea

wings and sails tessellation

Wings and Sails

Here is the base tile of 'Two Elements':

two elements tessellation

Two Elements

two elements tiles

• This bird and this fish would be perfect in an appearance/disappearance style drawing, but the result would have been too close to Escher’s Sky and Water.

• Here is the base tile of Child's Eye View: :

parrot and squirrel tiles
child's eye view tessellation

Child's Eye View

This child look that might be the one of Albert Flocon – the brilliant engraver artist - which sets out the rules of the curvilinear perspective, this one giving a more logical and exciting image than the classic perspective for which we have structured space with straight lines that make it sad.

"Learned since childhood, familiar for four centuries,
the traditional image is credited finally for the real image.
It is perhaps an imposture…"
Albert FLOCON / André BARRE
La Perspective curviligne

And now, if we divide a few tiles into several lizards:

• First of all here is the base tile of Lizards Squares, divided into three lizards including two that are identical:

lizards squares tessellation

Lizards Squares

lizards x 3 tiles lizards 6c ttessellation

• And here is the one of the opposite tessellation:

lizards 6c tiles

• The Rose with Lizards is constructed by using tiles of type 4a and 6b between which we have made a few adaptations.

rose with lézards 4a and 6b tessellation

Rose with Lizards

lizards 4a and 6b tiles lizards 3b tessellation

• The drawing below has a type 3b base tile divided into four lizards:

lizards 3b tessellation divided line

7   TESSELLATING WITH WORDS

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The words are chump change of thought.
There are big talkers who pay us in dimes parts.
Others, on the contrary, give only golden louis.

Jules RENARD

The art is beautiful when the hand, head and heart work together.
John RUSKIN

dropped capital let us ask ourselves the question: could the motifs be other things than familiar figures, such as animals, objects or  humans while still having recognizable contours? Well yes, of course! There are the characteres; these signs that men have imagined to communicate and which are, moreover, already in two dimensions.
And even better: there are character sets that are the words and who have as well a specific meaning. We can choose a word for what or who it represents, for its value to us. We can draw it, work it, perfect it... love it as much as a poet.
Why Escher did not make any tessellations with words remains a great mystery. The way has been indicated in a book by Scott Kim. Its title? Inversions. A unique book, at the border between calligraphy and mathematics. We find in, among other wonders, the word "figure" in white on a black background and which the space between letters gradually turns into black letters on a white background, again forming the word "figure" in negative. There is also the word "Lester", with the space between the letters forming the word "Pearl" (these are the Kim's parents). These are not tessellations but it is already good way along.
It was tempting to go this way and make words in Kim's style. It was exciting to go at the end of the road and realize an old childhood dream: to make tessellations with words. And if possible with words dear to my heart..
infinity logo

We have seen that it is easy to divide a tile in two, three or more parts. Well, a tile-word is only the equivalent of a tile divided into as many subdivided tiles it contains letters, accents or dot on the i. It is the assembling of the base tile with its transformations and then the coloring that allows us to read the word in different colors.

• There are no tessellations in the opposite example. But we could cut type 4b tiles like the one below:

l'infini tessellation

L'Infini (The Infinity in French)

l'infini square tile

• Among several early attempts of tessellations with words, here is this one at right:
      An inconclusive attempt. But the result is interesting all the same.

houdini ambigram


      • Honor to the good master. Here is the type 4c base tile used for Escher Pavage (this means Escher Tessellation in French):

escher word tile
escher word tessellation

Escher Tessellation

The transformations of this base tile allow us to read the word "Escher" in four directions.

escher word transformed tile

• We find again this tile-word in decoration around Escher's portrait.

escher word and profile tessellation

Escher


      • Let us take again the tile-word "Escher". And let us color it as below. We see then to appear… Le Jardin d'Escher (The garden of Escher in French).

Escher color tile

Tiling example :

the garden of escher tessellation

The Garden of Escher

escher tiling black line

• Here is the type 2e base tile of Kim :

• Translation axes, rotation axes, symmetry axes, glide-reflection axes; the axes are present in all tessellations. Below you will find the type 6a base tile (equilateral triangle) of Axes:

kim tile axes tile

The transformations of this base tile allow us to read the word "Kim" in two directions.

• En route to a ride to the Étoiles (Stars in French). Here is the used type 6a base tile:


     • Then here is the type 2c base tile of Inversion a particularly dear word to Scott Kim:

étoiles tile inversion tile
scott kim tessellation

Kim

inversion tessellation

Inversion

axes tessellation

Axes

stars tessellation

Stars

• Honor to the natural order, here is the type 4c base tile used for L'Ordre et le Chaos (The Order & the Chaos):

• Yet on the contrary, here is the type 1b base tile of Paradoxe Jour / Nuit (Paradox Day / Night):

ordre tile jour nuit tile

• Of course, it must have a name of musician. To quote geniuses such as Bach or Mozart would certainly make a good impression... but it would not reflect my taste for good old Country music, music that I love and which brings me joy and satisfaction. And do not tell me that there are no geniuses in Country music. Merle Travis not only was one, but even more was a multi-talented genius. He excelled both as author, composer, cartoonist, watchmaker, historian, actor, writer, naturalist, impersonator, designer of guitars, singer and of course as a virtuoso guitarist. We are indebted to him for the guitar style that bears his name, the Travis-pickin’. A style in which the accompaniment, bass and melody are played at the same time.

Here is the base tile of Travis:

merle travis pic merle travis tile black line
order and chaos tessellation

The Order and the Chaos

merle travis tessellation

Travis

travis pickin' tessellation

Travis Pickin'

paradox day night tessellations

Paradox Day / Night

• Brassens... Ah, Brassens! The man with a hundred masterpieces. The perfectionist of the "beautiful language" and chosen expression. The wonderful artisan engraver of words. That the word "Brassens" extends to infinity.

georges brassens pic georges brassens tile

• Depending on that it is white or black, the magic indicates either the art of producing some wonderful effects that are due to natural causes, or the sordid mystification by which some people claim to produce supernatural effects by the intervention of the spirits.
      Here is the type 1a base tile used in Magie blanche / Magie noire (White Magic / Black Magic):

• Among the artists of white magic, Gérard Majax is a worthy representative of them, which does not hesitate to work against the phonies of illusion, the supposedly owners of extraordinary powers and others twisters of teaspoons.

     Here is his type 1b base tile:

magie tile majax tile
georges brassens tessellation

Brassens

the tall oak tessellation

The Tall Oak

white magic black magic tessellation

White Magic / Black Magic

gérard majax tessellation

Majax

• Of course, here is the base tile of the unavoidable Infini (Infinity):

• Then here is the type 2e base tile of Pavé (Tile!). First in two tones then into only one gradient tone.

infini tile pave tile

• This type 6a almost triangular tile is the one of Victorian tessellation:

victorian tile

• If there is a name deserving to tessellate the infinite, it is that of "Einstein". Here is the used type 2e base tile:

albert einstein pic albert einstein tile

"The joy to contemplate and understand,
this is the language that nature inclines to me."
Albert EINSTEIN
How I see the world

infini tessellation

Infinity

Tile pave tessellation

Tile

victorian tessellation

Victorian Tessellation

albert einstein tessellation

Einstein Tessellation

• Erno Rubik is the inventor of the Rubik's Cube, the most diabolical of all puzzles. Here is the type 6d base tile of Rubik's Cube:

• Where is the border between good and evil (Bien et Mal)? Here is the type 1b base tile of Frontière (Boundary):

bien mal tile ernö rubik tile

• Here is the one of Nicolas Pavage 2, type 3a:

• It is, of course, tempting to tessellate with its own name. Here is the type 1b base tile of Nicolas :

nicolas 3a tile nicolas 1b tile

• Finally with words, here is again "Infini" (Infinity) written in a manner that with its transformation by rotation fills exactly a rectangle. We can then assemble the words like bricks.

• Then the same word in a parallelogram:

infini parallelogram tile infini rectangle tile
rubik's cube tessellation

Rubik's Cube

boundary good and evil tessellation

Boundary

nicolas star tessellation

Nicolas tessellation

nicolas painting tessellation

Nicolas

nicolas 2 colors tessellation

Nicolas 2 colors

infinity square tessellation

Infinity Square

infinity cube tessellation

Infinity Cube

divided line

8   DISTORTING THE STRUCTURES

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What is worth to be done is worth to be well done.
Nicolas POUSSIN

The harmony of proportions satisfies the senses.
Saint Thomas d'AQUIN

dropped capital what is better is the enemy of what is good. But the  good is often the friend of what is mediocrity! Let us try to find the better by distorting the structures for dress up concentric circles, spirals, perspective or volume effects. This will allow us to discover new horizons full of wonderful harmonies.
perspective tiles infinity logo

Opposite, here is for example what produce a type 2S structure on a perspective:

black line

• Here is the tessellation (type 2Gb) of Cascade:

• Here is the one of Miel (Honey), type 2Sb:

cascade 2gb tessellation bees 2sb tessellation

• Here is the one of Plouf ! (splash!) type 1Sb:

• The one of Spirale d'infini (Spiral of infinity), type 2d:

plouf 1sb tessellation spirale 2d tessellation
cascade distorted tessellation

Cascade

honey distorted tessellation

Honey

splash distorted tessellation

Splash!

spiral of infini distorted tessellation

Spiral of Infinity

inversion distorted tessellation

Inversion 2

möbius ribbon distorted tessellation

Möbius Ribbon

translation in the infinity distorted tessellation

Translation in the Infinity

albert einstein distorted tessellation

Einstein

nicolas head in the stars distorted tessellation

Head in the Stars

majax distorted tessellation

Majax

pave stars distorted tessellation

Star Tessellation

fire dance distorted tessellation

The Fire Dance

Hello…
It's me
again!

• Opposite here is the original tessellation of type 6b which served for Cercles (Circles). Nine colors are required so as not to have intersecting circles of lizards.

lizards type 6b tessellation

• This same tessellation also served for the Icosaèdre aux lézards (Icosahedron with Lizards) below. An icosahedron is a polyhedron with twenty equilateral triangles for faces. Six colors are necessary to obtain a harmonious distribution. There are five lizards’ noses by the vertex.

• How to put seven lizards' noses by vertex? It is impossible on the normal plane, but not on a hyperbolic disk. The lizards are then reduced to infinity on a circular limit. To ensure the same color to the lizards which follow each other and that two columns of the same color do not intersect, seven colors are needed. In fact, in the disk, if one considers the colors, there are not two identical lizards.

lizards circles tessellation

Circles

icosahedron with lizards tessellation

Icosahedron with Lizards

circular limit tessellation

Circular Limit

divided line

9   PLAYING WITH ISOMETRIES

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It is trust in life,
to compete with the impossible.

Panait ISTRATI

To reach the inaccessible star, this is my quest.
Jacques BREL

dropped capital we cannot go beyond the limits of the possible, but we can push them forward. To do this, what can we do? For  example, have we the ability to alter or add isometries on basic polygons? The answer is yes. We then get super-tiles with amazing features.
infinity logo

• Let us take the 3a base polygon:

Let us add the isometries of the base polygon 1Gb, that is to say two glide reflections. We then get the next polygon:  

rhomb 3a tile rhomb 1gb tile

On a rhomb we have two different isometries of rotation order 3.


Let us draw a motif according to this polygon. For example this "Li'l Wolf" opposite. He looks astounded because he comes to realize that he could tessellate the plane not only according to types 3a and 1Gb, but also in many other ways.

li'l wolf tile

Type 3a tessellation:

li'l wolf 2c tile

Tile type 2c divided into two identical motifs

li'l wolf 3a tessellation li'l wolf 2ge tile

Tile type 2Ge divided into two identical motifs

li'l wolf 6c tile

Tile type 6c divided into two identical motifs

Type 1Gb tessellation:

li'l wolf 1gb tessellation li'l wolf 3b tile

Tile type 3b divided into three identical motifs

li'l wolf 1a tile

Tile type 1a divided into four identical motifs

tessellated rose with wolves

Rose with Li'l Wolves

• Now let us take the 6a base polygon. There we have an equilateral triangle with an isometry of rotation order 6 and one of rotation order 2:

For example this fish:

fish limit tile fish limit 6a tile

From this fish, we can, of course, make a tessellation of type 6a:

fish limit 6a tessellation

Then draw a pattern so that each side of the rotation order 2 be the reduction by half of each compensated deformation of the rotation order 6:

fish limit reduction tile black line

But we can also join to him three smaller identical fishes:

In this way, this gives us a type 6d tile divided into four:    

fish limit x 4 tile fish limit 6d tile
fish family tessellation

But that is not all.
      Let us take again our fish and let him be rotated 6 times. We can add all around a ring of smaller fishes. Then again, add a ring of even smaller fishes, and so on to infinity.
      We then get a perfectly hexagonal figure.

hexagonal limit fishes tessellation

Hexagonal Limit

• In a way reminiscent of the example above, but with a different result, let us take a kite of type 6c:

Type 6c tessellation:

butterflies 6c tessellation kite 6c tile

Let us draw a motif so that both sides of the order 3 rotation are equal to the sides of the order 6 rotation by the tangent of 30° (0.5774). For example this butterfly:

butterfly 6c tile

Type 6d tessellation:

butterflies family tessellation

We can make a tessellation of type 6c but also of type 6d If we add four small butterflies as below:

butterfly 6d tiles
butterflies tessellation part

To construct the Butterfly flake below, we have to make an order 6 rotation. And then, on the periphery, add a ring of small butterflies four times more numerous. Then to this new periphery, make the same three times.
We can, of course, continue to add rings to infinity. Each new ring then, will generate more and more points of infinity as below. All in all… we get an infinity of infinities.

butterflies flake tessellation

Butterfly flake

• As for the butterfly, but reversing the sides of the order 3 rotation, we can make this bird:
On the contrary, it cannot be reduced to infinity. You are obliged to stop at the fifth ring

bird 6c tile

• Let us take the 4S base polygon, that is to say a square with two symmetrical rotations of order 4. Let us add it four rotations of order 2:

Let us draw, for example, this bird:

bird 4s tile squares 4s tiles

We can tessellate the plane according to different types or make a rose or a non-periodic tessellation such the lowest one. We see a teeming mass of birds amongst which some go right their way and a few rarer, illuminate the four horizons.

birds flake tessellation

Birds flake

cardinal points tessellation

Cardinal Points

• Let us see if we can draw a motif which has several base tiles:

This "Good Doggie" will do the trick:

good doggie tile good doggie 3 tiles

Type 3b tessellation:

Type 2Gg tessellation:

Type 1Gc tessellation:

Rose :

good doggies 3b tessellation good doggies 2gg tessellation good doggies 1gc tessellation
good doggies rose tessellation

• And if we find again our lizard friends. In the Smaller and Smaller print (N°35), Escher has used an isosceles right triangle of type 4a to reduce the lizards to infinity according to the diagram below:

This allows us to draw two lizards by triangle instead of one:

lizards limit tiles lizards limit 4a tiles

We can now make a order 4 rotation and reduce the lizards to infinity. The most direct way results an octogon.
In fact, there is not only a couple of different lizards, but four. This allows adaptations.
Escher, in Smaller and Smaller, drew nine different lizards.

lizards limit invert tiles

Let us invert the direction
of triangles as follows:

octogonal limit with lizards tessellation

Octogonal limit

lizards flake tessellation

Lizards flake

hexagonal double limit tessellation

Double limit

I hope you had lots of fun watching this method. This will be the final word: FIN (the END in French).

 Return  to summary the end tessellation

Nota bene :
If you liked this method,
a little word in the "Guestbook"
would make me happy.