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Tessellations
Tessellation Definition
A tessellation is created when a shape isrepeated over and over again covering a
plane without any gaps or overlaps.
Another word for a tessellation is a tiling .
Tiling Definition
When you fit individual tiles together withno gaps or overlaps to fill a flat space like aceiling, wall, or floor, you have a tiling.
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What areTessellations
The word'tessera' in latinmeans a smallstone cube.They were usedto make up'tessellata' - themosaic picturesforming floorsand tilings inoman
buildingsThe term has
become morespecialised andis often used torefer to picturesor tiles, mostlyin the form ofanimals andother lifeforms, whichcover thesurface of a
plane in asymmetricalway withoutoverlapping orleaving gaps.
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Symmetry
!f you look at a completedtessellation, you will see the originalmotif repeats in a pattern. "nemathematical idea that can beemphasi#ed through tessellations issymmetry.
There are $% possible ways that a pattern can be used to tile a flatsurface or 'wallpaper'.
&olya $%
ways to tilea surface
(scher read &)lya's $*+ paperon plane symmetry groups.(scherunderstood the $% planesymmetry groups described in themathematician &)lya's paper,even though he didn't understandthe abstract concept of the groupsdiscussed in the paper.
etween $* / and $* + (scher produced colored drawingswith a wide variety of symmetrytypes while working on possible
periodic tilings. 0e adopted ahighly mathematical approachwith a systematic study using anotation which he inventedhimself.
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Four Types of Symmetry in aPlane
There are ways of moving a motif to
another position in the pattern. Thesewere described by (scher.
Translation eflection otation
1lide eflection
Translation eflection otation 1lide eflection
A translationis a shape thatis simplytranslated, orslid, acrossthe paper anddrawn againin another
place.
Thetranslationshows thegeometricshape in thesamealignment asthe original2 itdoes not turnor flip.
A reflection is ashape that has
been flipped.3ost commonlyflipped directly tothe left or right4over a 5y5 a6is7 orflipped to the topor bottom 4over an565 a6is7,reflections canalso be done at anangle.
!f a reflection has been donecorrectly, you candraw an imaginaryline right throughthe middle, and thetwo parts will besymmetrical5mirror5 images.To reflect a shapeacross an a6is is to
plot a special
otation is spinning the pattern around a point,rotating it. A rotation, orturn, occurs when anob8ect is moved in acircular fashion around acentral point which doesnot move.
A good e6ample of arotation is one 5wing5 ofa pinwheel which turnsaround the center
point. otations alwayshave a center, and anangle of rotation.
!n glide reflection, reflectionand translation are usedconcurrently much like thefollowing piece by (scher,0orseman. There is noreflectional symmetry, nor isthere rotational symmetry.
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corresponding point for every point in theoriginal shape.
&entomino 9hapes
A pentomino is the shape of five connected checkerboards:uares. There are only twelve different pentomino shapes
Rectangles
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Triangles
Hexagons
(scher painted this study of a tile fromthe Alhambra.
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(scher works based on a circle
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Tesselation to thelimit of a circle.
IslamicTil
es
Theseare star
patterntiles.
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0ere
aremore!slamic star
patter n tiles.
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Tessellationswith a3-Deffect
sch
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scher
sch
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