tessellations by recognizable figures -...

M.C. Escher Reptiles puzzle.

From EscherMathIn this section we will explore somemethods for creating Escher liketessellations. We will use the geometry wehave developed in the previous sections tocreate tessellations by recognizablefigures.

Contents

1 Explorations2 Introduction3 Escher's Polygon Systems4 Tessellating With Translations5 Tessellating With Reflections

5.1 A Two Tile Pattern5.2 A cm Pattern

6 Tessellating With GlideReflections7 Tessellating With Rotations

7.1 Rotations withtranslation7.2 Rotation about themidpoints of sides7.3 Rotation about a vertex

8 Other Interesting Methods9 Heesch Types10 Relevant examples fromEscher's work11 Related Sites

Explorations

Escher-Like Tessellations ExplorationsEscher Tessellations Using Geometer’s Sketchpad

IntroductionA tessellation, or tiling, is a division of the plane into figures called tiles. The most common tessellationstoday are floor tilings, using square, rectangular, hexagonal, or other shapes of ceramic tile, but manymore tessellations were discussed in the Tessellations by Polygons chapter.Escher's primary interest in tessellations was as an artist. He wanted to created tessellations byrecognizable figures, images of animals, people, and other everyday objects that his viewers wouldrelate to. He used these figures to tell stories, such as the birds evolving from a rigid mesh of triangles tofly free into the sky in Liberation. In Predestination, flying birds and fish are born from the same black andwhite tessellation, fly into three dimensionality, and then meet when the fish completes the "predestined"killing of the bird.In his native Dutch, Escher called these tessellations 'Regelmatige Vlakverdeling', and collected them inhis sketchbook. His sketches drew inspiration from geometric patterns created by Islamic artists. TheIslamic religion forbids the creation of representational images, and so Islamic artists have developed a

broad vernacular of decorative patterns over centuries of work. Escher writes[1]:

What a pity that the religion of the Moors forbade them to make images! It seems to me that

Tessellations by Recognizable Figures - EscherMath http://euler.slu.edu/escher/index.php/Tessellations_by_Recognizable_Figures

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M.C. Escher, Liberation, 1955

they sometimes came very close to the development oftheir elements into more significant figures than theabstract geometric shapes that they created. No Moorishartist has, as far as I know, ever dared (or didn't he hit onthe idea?) to use as building components concrete,recognizable figures borrowed from nature, such as fishes,birds, reptiles, or human beings. This is hardly believable,for recognizability is so important to me that I never coulddo without it.

Though Escher's goal was recognizability, his tessellations beganwith geometry, and as he grew more accomplished at creatingthese tessellations he returned to geometry to classify them. All ofEscher’s tessellations by recognizable figures are derived fromjust a handful of geometric patterns. There are several differenttechniques that Escher used, and sometimes he combinedtechniques as well, but all involve a transformation from a simplegeometric pattern to a complicated, recognizable figure.The simplest example of an Escher tessellation is based on asquare. Start with a simple geometric pattern, a square grid, andthen change that ever so slightly.

In this example each vertical edge of the grid was deformed to look like a lightning bolt. Then, eachhorizontal edge was redrawn as a bent curve. Note that all the vertical pieces were changed in the sameway, and that all horizontal pieces look the same as well.The bends were not introduced with any intention of creating a particular shape, but now that the newpattern is drawn it becomes a sort of inkblot test. What do you see? Some will think it looks likeinterlocked men, others may see birds. Decorating each of the newly formed tiles can emphasize a


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particular interpretation, and creates a tessellation by recognizable figures, in the style of Escher. Thismethod also works well if you start with a geometric tessellation by rectangles or parallelograms, as wewill see in the section #Tessellating With Translations.Escher wrote that creating these tessellations is a "compelling game", and like any game it is fun once youlearn the rules. Practice making your own tessellation based on squares, and make more tessellationsusing the different techniques of this section as you learn them.

Escher's Polygon SystemsEscher created his tessellations by using fairly simple polygonal tessellations, which he then modifiedusing isometries. This chapter gives a brief overview of Escher's own categorization system fortessellations and contains instructions for creating tessellations by recognizable figures using some ofEscher's simpler techniques.Escher organizes his tessellations into two classes, systems based on quadrilaterals, and triangle systemsbuilt on the regular tessellation by equilateral triangles. The bulk of Escher's tessellations are based onquadrilaterals, which the novice will find much easier to work with. The less common triangle systemsare easily identified because three or six motifs will meet at a point, and the entire tessellation will haveorder 3 or order 6 rotation symmetry.For a complete discussion of Escher's systems, read Visions of Symmetry (Chapter 2), which alsoreprints each page of Escher's notebook "Regular Division of the plane into asymmetric congruentpolygons". Here, we give only a brief summary of his quadrilateral systems.Within the quadrilateral systems, Escher's categorization has two factors: the type of polygon and thesymmetries present in the tessellation. Specifically, he assigns a capital letter to each type of polygon anda Roman numeral to each type of symmetry. The polygons are:

A - ParallelogramB - RhombusC - RectangleD - SquareE - Isoceles Right Triangle

Note that each is a special type of quadrilateral except for E, the isoceles right triangle. These45°-45°-90° triangles fit together to form a square grid with order 4 rotation symmetry.Escher's ten different systems of symmetry do not correspond exactly to the wallpaper groups. Thewallpaper group for a figure describes the isometries of the figure. Escher's systems also describe theisometries, but are additionally concerned with the relative positions of the different motifs. There arefive wallpaper groups that have no reflections and no order 3 rotations, and all of Escher's symmetrysystems correspond to one of these five wallpaper groups.

Symmetry types for quadrilateral systemsSystem Translations Rotations Glide-Reflections

ITranslation in both transversaland diagonal directions

none none

IITranslations in one transversaldirection

2-fold rotations on the vertices;2-fold rotations on the centersof the parallel sides

none

IIITranslations in both diagonaldirections

2-fold rotations on the centersof all sides

none

IVTranslations in both diagonaldirections

noneGlide-reflections in bothtransversal directions

VTranslations in one transversaldirection

none

Glide-reflections in onetransversal direction. Glide-reflections in both diagonaldirections

VITranslations in one diagonaldirection

2-fold rotations on the centersof two adjacent sides

Glide-reflections in onediagonal direction. Glide-reflections in both transversaldirections, but only in thedirection of the sides withoutrotation point

VIInone

2-fold rotations on the centersof the parallel sides

Glide-reflections in onetransversal direction. Glide-reflections in both diagonaldirections.


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Sketch #96 (Swans)

VIIInone

2-fold rotations on the fourvertices

Glide-reflections in bothtransversal directions

IXnone

4-fold rotations on diagonalvertices; 2-fold rotations ondiagonal vertices

none

Xnone

4-fold rotations on threevertices; 2-fold rotations on thecenter of the hypothenuse

none

Escher created at least one tessellation with each of thepossible systems in his categorization. His sketches wereorganized into five folio notebooks, the Regular Division ofthe Plane Drawings. Each of these drawings is carefullynumbered and marked with Escher's categorization.For instance, in Sketch #96 (Swans), notice the system IV-Ddenoted below the sketch. System IV-D means that theunderlying geometric tessellation is based on a square and thatthere must be translations in both diagonal directions, norotations, and glide-reflections in both transversal directions.It is hard to see what Escher means by 'transversal directions'.In this sketch #96, you need to turn the sketch at an angle so asto see rows and columns of touching swans, alternating blackand white colors. These strips alternate a swan with it'smirror image, so swans along these strips (the 'transversaldirections') are alternately reflected. It's a different usage ofthe term 'glide reflection' than we're used to seeing. In fact,using the mathematical definition of glide reflection this sketchhas two different types of glide reflection symmetry, both inthe vertical direction.

Tessellating With TranslationsThe simplest and most flexible tessellations are Escher's Type I systems, which can be based on aparalellogram, rhombus, rectangle or square.

Schematic for Escher's Type I Tessellation Systems

To create one of these tessellations, follow these steps:

Draw a parallelogram. This is easy on graph paper, as you can count squares to ensure theopposite sides are parallel and the same length.

1.

Alter the top edge of one parallelogram by replacing it with a curved or crooked line.2.Translate that edge to the bottom of the paralleogram.3.Alter the left edge of the parallelogram4.Translate the left edge to the right side.5.

This gives a figure which tessellates, and with luck its outline will suggest a recognizable motif that you


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Creating a tile for asystem I-A tessellation.

(See Sketch #105(Pegasus)).

Regelmatige vlakverdeling, Plate I.

can develop with further alterations to the edges. Finish by creating morecopies of the motif by translation.The resulting tessellation has symmetry group p1.

Escher's Regelmatigevlakverdeling, Plate I is anillustrated description of thisprocess. A grid of parallelogramsappears in panels 1-4, thendevelops bent or curved edges inpanels 5-7. Finally, with theaddition of detail, the tile becomesa bird or a fish.Escher made many sketches usingsystem I. Some good examples tolook at include Sketch #38(Moths), Sketch #73 (Flying Fish),Sketch #74 (Birds), Sketch #105 (Pegasus), Sketch #106 (Birds),Sketch #127 (Birds), and best of all Sketch #128 (Birds) whereit is very easy to see how the bird motif developed from a squaretile.

Tessellating With ReflectionsFigures with bilateral symmetry are naturally easier to make into recognizable figures, because manynatural forms have bilateral symmetry. To create a tessellation by bilaterally symmetric tiles, we need tostart with a geometric pattern that has mirror symmetries. However, these mirror symmetries should notlie on the straight sides of the polygon tiles. If they do, the straight sides must remain straight and there isno longer flexibility to make a recognizable figure.

A Two Tile PatternThis is a very simple method for generating a tessellation by twodifferent tiles. Each of the two tiles has bilateral symmetry.Begin with a tessellation by rectangles. The vertical mirrorsymmetries down the centers of the rectangles will remain in thefinal tessellation.

Draw a rectangle.1.Alter one side, for example the left side.2.Alter half of the top edge, and half of the bottom edge.3.Reflect the side and both half-edges across the central vertical mirror line.4.

Repeat the resulting figure in a checkerboard pattern, leaving spaces which form the other tile of thetessellation.

Notice that the horizontal strips of tiles form frieze patterns with pm11 symmetry, which explains why thehorizontal translation is by two tiles - the vertical mirror lines must be spaced at half the translationlength.


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Sketch #91 (Beetles)

A cm PatternThis pattern begins with a tessellation by rhombuses.

Draw a rhombus.1.Alter one side, for example the top left side.2.Translate from the top left to the bottom right side.3.Reflect both top left and bottom right across the vertical center line of therhombus to finish all four sides.

4.

The resulting figure tessellates in a pattern similar to wood shingles, and gives atessellation with symmetry group cm.Escher's Sketch #91 (Beetles) uses this technique.

Tessellating With Glide ReflectionsThis simple arrangement of parallelograms is a good starting point for creating tessellations with glidereflection symmetry:


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Creating a tile for a systemV tessellation. (See Sketch

#97 (Bulldogs)).

The pattern has horizontal translation symmetry, and vertical glide reflection. To create an interestingtessellation from it:

Draw a parallelogram (or rectangle).1.Alter the top edge of the parallelogram by replacing it with acurved or crooked line.

2.

Glide-reflect that edge to the bottom of the paralleogram.3.Alter the left edge of the parallelogram4.Translate the left edge to the right side.5.

This gives a figure which tessellates. Repeat identical copies of it to theleft and right, and repeat mirror image copies above and below.The resulting tessellation has symmetry group pg. Escher would describethis as a Type V system, although it doesn't fit exactly into hiscategorization. Along with Sketch #97 (Bulldogs), Escher used thistechnique in Sketch #108 (Birds) and Sketch #109 (Frogs). Another goodexample is Sketch #17 (Parrots), though it is a slight variant.For a shape that lends itself even more towards recognizable figures, divide each parallelogram into twohalves by drawing its short diagonal. Then, erase the horizontal edges to form a tessellation by "kite"shapes:

Alternately, draw the long diagonals and erase horizontal edges to form a tessellation by "dart" shapes:

To create a tessellation using the kite or dart pattern above, follow these steps:

Draw the kite or dart pattern by starting with parallelograms.1.Alter one top edge of the kite or dart by replacing it with a curved or crooked line.2.Glide-reflect that edge to the bottom of the kite or dart.3.Alter the other top edge of the kite or dart.4.Glide-reflect that edge to the bottom of the kite or dart.5.


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Creating a tile for a system Vtessellation based on kites. (See

Sketch #66 (Winged lions)).

Escher classified this sort of tessellation as Type IV. Good examplesof tessellations based on the kite shape are Sketch #62 (Sniffers),Sketch #66 (Winged lions), Sketch #67 (Horsemen), and Sketch #96(Swans). The birds in Sketch #19 (Birds) are only slightly alteredfrom the dart scaffolding, although Escher's visible grid of rhombusessuggests he went about the construction in a completely differentmanner.Escher wrote in his summary chart that Type IV tessellations havetranslations in both diagonal directions and glide-reflections in bothtransversal directions. This means that motifs that share a side arereflected images, and motifs that touch at corners (diagonally) aretranslated images.

Type IV tessellations based on the rhombus (left) and on th

Tessellating With RotationsThere are many ways to use rotation symmetry as the basis for a tessellation, and only some simpler oneswill be described in this section.

Rotations with translationEscher's Type II tessellations begin with a grid of parallelograms (squares, rectangles, and rhombusesalso work). The tile will be distorted into a shape that can tessellate using 2-fold rotations at all fourvertices and 2-fold rotations in the centers of one pair of opposite sides, as shown by the red dots in thefigure below.

Escher Type II tessellations,

To create a tessellation using this technique;

Draw a parallelogram, rhombus, rectangle, or square.1.Alter the top edge of the parallelogram by replacing itwith a curved or crooked line.

2.

Translate that edge to the bottom of the parallelogram.3.Alter half of one remaining side by replacing it with acurved or crooked line.

4.


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Creating a tile for a system IItessellation based on parallelograms.

(See Sketch #75 (Lizards)).

Rotate that half side by 180° to complete the side.5.Repeat steps 4 and 5 with the remaining straight side.6.

This gives a figure which tessellates. Repeat identical copies of itby translating up and down, and repeat rotated copies of it to theleft and right. This results in a tessellation with symmetry group p2. Examples in Escher's work includeSketch #75 (Lizards), Sketch #115 (Fish and birds), and Sketch #117 (Crabs).

Rotation about the midpoints of sidesAll triangles tessellate, and all quadrilaterals tessellate. The pattern for these general tessellations is builtfrom 180° rotations about the midpoints of the polygon sides. Altering half of each side and filling in theother half by rotation will also give a tessellating shape. In fact, this technique quite general and worksfor many geometric tessellations.

Draw a 3- or 4-gon1.Create a midpoint on each of the sides.2.Modify one half of each of the sides.3.Rotate each side 180° about the midpoint.4.

Fill in the tessellation by rotations. Because 180° rotations were used but neither reflections nor glidereflections, the resulting patterns will have symmetry group p2.

Example starting from a triangle.


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Example starting from a kite-shaped quadrilateral.

For Escher, these were Type III tessellations. The best example is Sketch #88 (Seahorse), becauseEscher's geometric scaffolding for the sketch is also in his notebook. In the scaffolding, the underlyingshape appears to be a triangle but should really be viewed as a quadrilateral with two sides in a straightline, giving four vertices and four midpoint rotation centers.Another good example is Sketch #9 (Birds). In Sketch #9 (Birds), each bird is derived from aquadrilateral which you can find by connecting the points with four birds coming together. In Escher'sscaffolding for the sketch, there is a visible grid of paralleograms which he obviously used to lay out thepicture. His scaffolding is an easier way to build this type of tessellation by hand, and relies on a theoremof Euclidean geometry:

The four midpoints of the sides of any quadrilateral form a parallelogram

Escher would have drawn the grid of parallelograms, constructed the midpoints of each side of theparallelogram, and then altered the parallelogram to the bird form allowing the sides to bend and cornersto move.Other examples of Type III tessellations are Sketch #90 (Fish) and Sketch #93 (Fish), where in the latterthe eyes and mouths of the fish destroy the rotation symmetry of the silhouette.

Rotation about a vertexStarting with a pattern of squares can produce a resulting tessellation with an order 4 rotation andsymmetry group p4.

Draw a square.1.Modify one side.2.Using 90° rotations at the vertices, copy the modified side to the other three sides.3.

Fill in the tesselation by 90° rotations:


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Sketch #25 (Reptiles)

Tessllation built with rotations around a vertex.

Escher made no tessellations using this technique, but did do something similar with his Type Xtessellations. In these, he would divide the square into two 45°-45°-90° triangles, and use a 180° rotationaround the midpoint to modify the dividing line. Examples are sketches Sketch #35 (Lizards), Sketch #118(Lizards), and Sketch #119 (Fish).Rotation about a vertex can be applied to a regular hexagon as well, andEscher used this as the basis for one of his most successfultessellations, Sketch #25 (Reptiles). Altering the hexagon shape willbreak some of the many symmetries of the hexagon grid, so it isimportant to carefully identify which symmetries will remain in the finaltessellation. In this example, the only symmetries used are order 3rotation symmetries of three types, marked in the picture below as thered, blue, and green dots. Each edge of the grid touches exactly one ofthese rotation centers, so three edges of each hexagon are free to alterand the other three are forced by the choice of symmetry.The steps for creating the tile in this pattern are:

Draw a hexagon, and mark three of its corners (here theyare red, green, and blue).

1.

Choose a marked corner, and alter a side that touches it.2.Rotate the altered side to the other edge at that corner.3.Repeat with the other two marked corners.4.


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Sketch #76 (Birds and horses)

Other Interesting MethodsCutting a tile into two pieces is a simple way to get addedflexibility. The dividing line rarely needs to obey any symmetries,and so can be drawn freely. A good example of this is Escher'sSketch #76 (Birds and horses), which could be started as a TypeIV or Type V grid of parallelograms each divided into two. Otherexamples based on translation only are Sketch #52 (Birds andfrogs) and Sketches #47-50 which are the basis for Verbum.

Pentagon tessellation. Escher's favorite Alhambra pattern.

Heesch TypesEscher's tessellations are sometimes described according to theirHeesch type. There are symbols for tessellations based ontriangles, quadrilaterals, pentagons, and hexagons. Thesetessellations are all isohedral, i.e for any two tiles there is asymmetry mapping one tile to the other. The most general type oflabelling for the tile is to use the following convention:

C: the edge has 180 degree rotational symmetry with itself.Label that edge C.T: an edge has translational symmetry with another edge. Label both T.G: An edge has glide reflectional symmetry with another edge. Label both G.Cn: An edge has (360/n) degree rotational symmetry with an adjacent side.Label both Cn

In the standard notation the convention is to start the Heesch type with the a minimal label, where T < C <C3 < C4 < C6 < G < G1 < G2. For instance, we would write TCTC and not CTCT because we want to startoff with the "lowest" possible label.For triangles the following 5 labelings will create a tessellation: CCC, CC3C3, CC4C4, CC6C6 ,CGG .For quadrilaterals the following 11 labelings will create a tessellation: TTTT, CCCC, TCTC, C3C3C3C3,C4C4C4C4, C3C3C6C6 ,CCGG, TGTG, CGCG, G1G1G2G2, G1G2G1G2.For pentagonal tiles the following 5 labelings will create a tessellation: TCTCC, TCTGG, CC3C3C6C6

,CC4C4C4C4, CG1G2G1G2.For hexagonal tiles the following 7 labelings will create a tessellation: TTTTTT, TCCTCC,TG1G1TG2G2, TG1G2TG1G2, TCCTGG, C3C3C3C3C3C3, CG1CG2G1G2.

Relevant examples from Escher's work

Regelmatige vlakverdeling, Plate IRegular Division of the Plane Drawings, particularly those which also show the underlyinggeometric tessellation:

Sketch #67 (Man on horse) and related work Visions of Symmetry pg. 111.Sketch #96 (Birds)Sketch #127 (Birds)Sketch #128 (Birds)

Related Sites


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David Bailey's World of Escher-like Tessellations (http://www.tess-elation.co.uk/)Tessellation Database (http://www.tessellation.info) by Snels-Design.Escher in the Classroom (http://britton.disted.camosun.bc.ca/jbescher.htm) by Jill Britton.

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