Tiling
aka Tessellation



Real world tiling
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Brief Remarks Appearance Links Brief Remarks

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Explanation

Copenhagen cobblestones in Bruges, Belgium Link Bruges

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Savador Dali, playing with a classical deterministic tessellation involving squares and equilateral triangles. 

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Explanation

The definitive collection of regular (deterministic) tessellations: http://www.tilingsearch.org/ It features over 2000 different tiling patterns, many crowd-gathered from Islamic art, others from mathematical theory.
dunbar pics of thirds of hexagons in AustraliaLink/Explanation
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 History of the Cairo Tiling (from David Bailey)
Bev Dunbar and Chrissy Monteleone's photo of  "pentagon used to decorate steps on a veranda in Earlwood, Sydney, Australia, dated about the 1960s."
From David Bailey's extensive site containing many deep analyses of interesting tessellations and their histories.




Dailey's work with tiling (hosted at ello.co/ddailey and on cs.sru.edu/~ddailey)
Generally, in the majority of these, at least, a single tile (shape) is used, but its various orientations are all differently colored. So for example, the  concave equilateral pentagon shown in the first row below, has 10 different orientations (each 36 i for 0 ≤ i < 360 degrees).  I find this makes it easier to identify subtle differences between tessellations.
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Brief Remarks            Appearance Links
Brief Remarks

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Concave equilateral pentagon: an animated introduction
Explore these, here,  at your own pace (using scrollbars and zoom)

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Explanation

Concave equilateral pentagon: animated gallery of options

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Experiments with clusters of the above concave equilateral pentagon. (It turns out there are 422 options for tetraominoes.)

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Fanciful arrangement of clusters of these.

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Link Concave Equilateral Heptagon.

Link One of three quadrilateral segments of an equilateral triangle.

Link Ways of partitioning hexagons into shapes which tile (ideally in ways other than the classical honeycomb/hexagonal tessellation).

Link Quadrilateral half-hexagons.

Link Snowflake-like clusters of half-hexagons. (Pretty!)

Link Showing how to tessellate in an outward spiral (avoiding all but a few hexagonal dominoes) and then replicating the resultant teardrop-pentagon from there.

The clustered teardrop pentagon make some nice patterns, but I'm not sure if it can continue nondeterministically.

Link More experiments with a (larger) clustered spiral teardrop.

Trapezoidal half-hexagon clustered into metatile

Link This particular cluster of half-hexagons forms an 18-sided polygon, which itself tessellates deterministically.

Link/explanation Demonstrating how the L-triomino forms a "rep-tile" -- i.e., makes successively larger  but similar (same shaped) versions of itself. Big triangle adjoined to smaller rhombus Link
Explanation
Triangle adjoining smaller rhombus.
1/24th of an equilateral triangle.  1/12th of a hexagon.
"reptiles" fractally.
See also below (Another way of joining a large triangle with a smaller rhombus.)
double rhombuses and sixths of a hexagon Link Double rhombuses aka sixths of a hexagon V shaped tiles and investigating fault lines Link V's and faultlines.
To what extent can a periodic tiling be completely characterized by its partial faultlines?
I don't know the answer.

Two similar equilateral triangles sharing a vertex and part of an edge

 
 explanation  link Gallery: Big and little triangles sharing an edge and a vertex to make a concave pentagon. Five triangles hooked together into a concave pentagon explanation  link
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Explorations:
Big and little triangles sharing an edge and a vertex to make a concave pentagon.  
Pentagons in Paralleograms Link
Explanation
Parallelograms: clusters of 3 x 4 and 4 x 3. Same tile as above but in curiously heterogeneous, though extensible arrangement.  Note that it consists of differently oriented parallelograms, each composed of the same four basic orientations. Pentagons arranged in parallelograms Explanation
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Same as at left but with animated gradients (for fun)
Equilateral pentagon with two right angles, sliced in half Link
Explanation
Sliced pentagons. Take an equilateral triangle and a square. Hook them together to make a pentagon. Then cut that in half. Sliced Pentagons with heterogeneous fault-lines making a thunderbird Link
Explanation
One of many tessellations involving the sliced pentagons seen at left.
A pentagonal half of a rectangle Link
Explanation
Castile and Leon come to Teotihuacan Radial tiling of S-shaped tile e-Link
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S-shaped tile used radially.
a hexagon split into three heptagons Link
Explanation
Heptagonal Hexatriominoes
Close-up (1 - 5)
Close-up (6 - 11)
Close-up (12 - 18)
Close-up (19 - 22)
radial symmetry with double half hexagon Link
Explanation
Funky Mandala
asterisks made of six double half hexagons Link Asterisks made of double half-hexagons. deconstructed asterisk using dominoes of half-hexagons SVG Link
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Deconstructed asterisk using dominoes made of half-hexagons.
half hexagons made into dominoes Link
Explanation
Gallery of  tilings made with double half-hexagons. seven triangles made into a nonagon Link
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Nonagonal Heptatriominoes: Seven triangles hooked together into a nonagon.
The twelve thirds of a hexagon

severalTilingsoThirdsofHexagon

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Tiling with thirds of a hexagon (Part I) Heptagonal thirds of a hexagon Link
other colors
Heptagonal thirds of a hexagon. A slightly different shape than seen at left. It is also three of the double half hexagons seen above left (e.g., here).
simple tilings with sixths of a hexagon Explanation
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Tiling with sixths of a hexagon. Part 1. various approaches to tiling with ektohexagons Link
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Tiling with sixths of a hexagon, Part 2. This shows that most of the ways that one might try to tessellate with this shape, simply don't work.
ektohexagons part three explanation self contained Explanation
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Tiling with sixths of a hexagon. Part 3. sixths of a hexagon under all twelve thirty degree rotations Explanation
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Tiling with sixths of a hexagon. Part 4. I think I need to write a Part 5.
Bigger hexagons out of fractions of a smaller hexagon Link Reptiling with sixths of a hexagon, by making successively larger hexagons. trichromatic ektohexagonal tilings Explanation
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Nice tiling with ektohexagons. It's four-colored (meaning using four orientations) and faultless (meaning having no continued fault-lines -- unlike 'most' tesselations using this tile).
L tetramino tiling horizontally along a line Link
Explanation
An idea based on the L-tetramino. I tried to replicated it along a base-line and to extend upward from there. Unsure if it can continue. Link
dented octagons Link Dented octagons octagonal crescents Link Octagonal crescents
concave equilateral pentagon Link A concave equilateral pentagon with interior angles of 60, 90, 60, 300 and 30.  Triangle and Rhombus together as hexagon Link Another way of joining a large triangle with a smaller rhombus.
Tiling with half of 34334 prototileExplanation
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A fourth of a square with a bit of family history.Triangles and squares - the 34334 tilingsLink
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Squares with equilateral triangles.
animated gradients in tiling made of V-shapesLink
Explanation
Guangdong-ese Cell Phone Towers. Tiling with V-shapes, made of six equilateral triangles. I don't remember if these tile any way other than as dominoes made into parallelograms.






Tessellation-like things (packings, knot tilings, paradoxes, etc)
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Brief RemarksAppearanceLinksBrief Remarks
paradoxical knot tiling (carbon mesh) Link
Explanation
Super-strong carbon mesh with quinary memory. (Simple nondeterministic paradoxical tessellation). See here for more on paradoxical tiling.     morphing the edges of a square into bezier curves Link Deforming a square tessellation (using SVG animate)

shimmering moire tessellation

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Making moire-like patterns out of overlayed and animated squiggles.

Link Dynamic tessellations formed from shape-morphing animations.
tangled tessellation Link
Explanation
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Positron emission tomography 31 knot tiles enumerated systematically Animated Explanation
Scrollable Explanation
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Systematic enumeration of Knot-tiles
Tapestry of Paradox Explanation
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Paradoxical tiling








Animations of tessellations (already studied by others) including morphs.

HistoryOfEurope.gif

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Hooking squares and equilateral triangles together, replicating and animating (just for fun)        Twirling 34334 pattern space Link
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Twirling the SVG pattern space of the (3,4,3,3,4) tessellation of squares and triangles.
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More work with the (3,4,3,3,4) tiling, showing that it tiles nondeterministically. Explanation The (3,4,3,3,4) tiling is the first non-deterministic tiling I worked with, some forty years ago. I need to develop something deeper to show some of my investigations.

Link A fun concave decagon that hooks with itself.  Also this demonstrates how to make a rectangular segment that can be used as an SVG <pattern> block to avoid having to replicate or re-<use>, hence allowing large-scale animations that are faster than the equivalent built of numerous vectors.

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Same as at left, but making the SVG <pattern> space visually obvious.

Link Morphing between two tessellations by expanding the borders (the grout) and celebrating them as full partners in tiling.

Link Bi-stable rotation of SVG pattern space

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Fun use of animated gradients in simple triangular tiling.

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Explanation
triangle with twelvefold symmetryLink
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Clock tiles   



Additional links (Ello):




Additional links (granite):

AskKnot.svg basicknots.svg concavePentas2.svg CubicFoamMeta.svg CubicFoam.svg CuFoam.svg EulerFoam3.svg eulerfoam4expanded.svg EulerFoam4.svg EulerFoam.svg  flag.svg Knots31.svg Knots31W.svg KnotsFollow.svg KnotsMake.svg KnotsSimpleTry6.svg pentagonCuteAnim.svg pentagonCuteCSS.svg pentagons9CSS.svg PentaPensa.svg Polygons.svg QFoamEx.svg QuartercircleBezs.svg