Pinterest

Today
Watch
Shop
Explore
When autocomplete results are available use up and down arrows to review and enter to select. Touch device users, explore by touch or with swipe gestures.

Aperiodic tiling

I am interested in substitution tilings of pieces with sides of unit length, especially those created by pattern blocks, Fractile-7 or Tessel-8 pieces. Some of…
More
·
510 Pins
 42w
Collection by
Jim Millar
Super tile at iteration level 3 of Schaad's quasiperiodic tiling version 1.
Super tile at iteration level 3 of Schaad's quasiperiodic tiling version 2.

6-fold inflation=3 (Pattern Block)

15 Pins

8-fold 2cos(pi/8)=sqrt(2+sqrt2)) (Tessel-8)

7 Pins
Mandala and 4 iterations of tiling using just squares. (Early original)
Mandala and first 4 iterations of a nonperiodic tiling based on square and pi/3 and pi/6 rhombs and 4 pointed stars (instead of pi/4 rhombs and 4-pointed stars). (Early original.)

8-fold flexed to 4-fold sqrt(2+sqrt2))

2 Pins
Mandala and first 2 iterations of nonperiodic Ammann tiling.
Ammann-Beenker substitution tiling with fractal super tiles, substitution rules

8-fold Ammann 1+sqrt (2) (Tessel-8)

5 Pins
Ammann tiling flexed to use only squares. (Original)
My first tiling with pentablocks--an Amman tiling of course! (What did you think it would be?). The purple rhomb is the same size as the green rhomb. The purple rhomb and the pink rhomb are the thin tile in the Ammann tiling. The green rhomb is the square in the Ammann tiling.
My first Ammann tiling using pattern blocks. (original)

8-fold Ammann flexed to 4–fold

12 Pins
Playing with 8-fold rhombs from Tessel-8. Mandala from 3rd octagonal rhomb-square tiling. (Original)
Fractalized version of the eightfold Watanabe-Ito-Soma tiling. Eightfold star of rhombus tiles.
Square from the third iteration of the first aperiodic tiling using squares and 45 degree rhombs. (This is the same as the Watanabe-Soma-Ito 8-fold tiling.)

8-fold Square-Diamond 2+sqrt(2) (Tessel-8)

10 Pins
Mandala from substitution tiling based on https://www.pinterest.com/pin/507851295472073513/ but using pattern blocks. (Original)
Mandala based on substitution tiling from https://www.pinterest.com/pin/507851295472073497/ but using pattern blocks. (Original)
Square from second iteration of https://www.pinterest.com/pin/507851295472073497/ but using pattern blocks. (Originally posted at https://www.pinterest.com/pin/507851295450828041/ )

8-fold Square-Diamond flexed

6 Pins
Rhomb square octagon
Fractalized rhombus tille at iteration level three of the Octagon-Square-Rhombus tiling. This tiling is related to the fractalized Watanabe-Ito-Soma tiling.
Fractalized Octagon-Square-Rhombus tiling. The substitution rules at first and second iteration level.

8-fold Octagon-Square-Diamond 2+sqrt (2) (Tessel-8)

28 Pins
Third iteration of the square from a square diamond octagon substitution tiling. (Original)
First two iterations of a square diamond octagon substitution tiling. (Original)

8-fold Square Diamond Octagon 2*sqrt(2+sqrt(2))+sqrt(2-sqrt(2) (Tessel-8))

2 Pins
Octagon and first iteration of an 8-fold substitution tiling #2. All vertices are fixed points. (Original)
Square and first 2 iterations of 8-fold substitution tiling 2. All vertices are fixed points. (Original)
Diamond and first 2 iterations of 8-fold substitution tiling 2. All vertices are fixed points. (Original)

8-fold diamond square octagon 4+2*sq(2) (tesselgons)

5 Pins
Second 5-pointed star that emerges in a Penrose tiling of pi/5 rhombs.
First 5-pointed star that emerges in a Penrose tiling of pi/5 rhombs.

10-fold Penrose (1+sqrt5))/2 (Pentablocks)

2 Pins
A second Penrose tiling using Fractile-7 rhombs (original). The black rhomb is the same shape as the yellow rhomb. The black and blue rhombs replace the Penrose fat rhombs, and the yellow and red rhombs replace the Penrose thin rhombs.
Another portion of the Penrose tiling using pattern blocks. (Original)
The lattice formed by the edges of the Penrose tiling rhombs is not rigid. Demonstration using arbitrary rhombs, 4 degrees of freedom. (original)

10-fold Penrose flexed

7 Pins
Super tiles of thin and fat diamonds at iteration level 6.

10-fold 2cos(pi/10)=sqrt((5+sqrt(5))/2)

6 Pins

11-fold

2 Pins
Mandala with 4 iterations from a diamond-triangle substitution tiling (original)
Green and blue triangles with 5 iterations from a diamond-triangle substitution tiling. (Original)
Diamond with first 5 iterations from a diamond-triangle substitution tiling. (Original)

12-fold Diamond-Triangle 2cos(pi/12)=sqrt(2+sqrt(3)) (Pattern Block)

3 Pins
Substitution rules with super tiles up to iteration level 4
Thin rhombus super tile at iteration level 6.
Triangle super tile at iteration level 6.

12-fold 2cos(pi/12)=sqrt(2+sqrt(3)) (Pattern Block + Tesselgon Stars)

14 Pins
Mandala and 4 iterations from the square-triangle-diamond-3star nonperiodic tiling twisted to 6-fold tiling using triangles. (early original)
Mandala and 4 iterations of the square-triangle-diamond-3-star nonperiodic tiling using pi/9 pieces (from the 18-fold tiling). (early original)

12-fold Square-Diamond-Triangle-3star flexed to 6-fold

2 Pins
Mandala and first 2 inflations from tiling #42 with inflation factor 1+sq(3). All tiles have unit edges. Reflective symmetry. (Original)
Inflation rules for tiling #42 using 2 squares with inflation factor 1+sq(3), all tiles unit edges. Reflective symmetry. (Original)
Mandala and first 2 inflations for tiling #40 with inflation factor 1+sq(3), all tiles unit edges. Reflective symmetry. (Original)

12-fold Square-Triangle-Diamond 1+sqrt(3) (Pattern Block)

28 Pins
Three iterations of the mandala from a nonperiodic tiling, but using triangles and pi/3 rhombs. (Early Original) Each color has a different expansion rule in the design.
Three iterations of the mandala from a nonperiodic tiling, but using pi/9 pieces from the 18-fold tiling. (Early Original) Each color has a different expansion rule in the design.

12-fold Square-Triangle-Diamond 1+sqrt(3) flexed to 6-fold

2 Pins
dodecagonal tiling
Mandala from the third diamond-triangle nonperiodic tiling using pattern blocks, inflation level 2. (Original)
A second diamond-triangle nonperiodic tiling using pattern blocks. The diamonds reflect each other and the triangles are each reflectively symmetric, (original)

12-fold Diamond-Triangle 2+sqrt (3) (Pattern Block)

11 Pins
Sequence showing the second iteration triangle of diamond triangle tiling, with diamond alternately thickening (tan) or thinning (lavender) until becoming either pi/3 rhomb or 0 rhomb. Green and blue triangles reflect each other. (Original)
The first diamond triangle tiling, with diamond alternately thickened/thinned to become either pi/3 rhomb or zero rhomb. Iterations 0-2. (Original)
Mandala based on diamond triangle tiling with diamond either thickened to pi/3 rhomb or 0 rhomb. Iteration 2. (Original)

12-fold Diamond-Triangle flexed to 6-fold

3 Pins
Mandala and 2 iterations from square-triangle substitution tiling 58, based on purple triangle. Mandala for tiling 59 is the same to two iterations, differences are only seen at the third iteration. (Original)
Square-triangle tilings 58-59, substitution rules. (Original)
Square-triangle tiling 14, third iteration mandala from the green triangle. (Original)

12-fold Square-Triangle 2+sqrt (3) (Pattern Block)

105 Pins
Square-triangle tiling with square deforming to pi/3 rhomb to pi/6 rhomb to 0 rhomb. Iteration level 2 of the tilings. Each can be made with pattern block shapes. (Original)
Third iteration of the square. From a triangle-square substitution tiling with the square deformed into a pi/3 rhomb. (Original)
First two iterations of a square-triangle tiling with the square deformed into a pi/3 rhomb. (Original)

12-fold Square-Triangle flexed to 6-fold

3 Pins
Mandala from the sixth simple nonperiodic tiling using pattern blocks, inflation level 2. (Original)
A mandala from the 18th simple nonperiodic tiling using pattern blocks. (Original)
A larger square from the 18th simple nonperiodic tiling using pattern blocks. (Original)

12-fold Square-Triangle-Diamond 2+sqrt(3) (Pattern Block)

44 Pins
Second mandala with a tiling using pattern blocks and 3 pointed stars, with 4-fold symmetry. Second iteration. (Original)
First mandala from a tiling using pattern blocks and 3 pointed stars, with 12-fold symmetry. Second iteration. (Original)
Two mandalas from a tiling using pattern blocks and a 3 pointed star, first iteration of each. (Original)

12-fold Square-Triangle-Diamond-3star 2+sqrt(3) (Pattern Block + Tesselgon Stars)

3 Pins