Affine Transformations

This entry is about a series of drawings from tilings. It has a few simple math terms, but it's about art, not math. These drawings are dot pattern diagrams I call Self-similar Boundaries.

These dot patterns, structured with tilings dependent on affine transformations, reveal characteristic grids and alignments. An affine transformation is in this case a rotation or scaling followed by a translation. I'm applying the term to the placement of individual tiles, as opposed to patches of tiles. Each dot pattern hints at the design of the tile set because the patterns reveal all possible affine transformations from each tile, with all possible scaling, rotations, and translations predetermined by the tile set. The tile sets are simple polygons. The tilings or tile patches may or may not be symmetrical. They usually have gaps but no overlaps, are always edge-to-edge, and never fill the plane. Each possible tiling from a tile set is a collection of scaling, rotation, and translation of tiles in the set. There is an ordered selection of tile from the tile set, and an affine transformation of each tile to a position edge-to-edge with another tile.

The tile sets always include one or two scaling tiles with a side that matches the length of a polygon in the tile set and a shorter side that determines the affine transformation scaling. The scaling tile may be the only tile in the tile set, or it may be one of several tiles.

These tilings, or tile patches, are one solution to a problem I encountered while generating tilings as structures for a series of vertex pattern diagrams. I needed a way to create greater density changes than I was getting with other tilings. Self-similar tile sets solve the problem. Scaling tiles provide the mechanism.

The three boundary diagrams below are from the same tile set. They represent regular and irregular versions of tilings and dot patterns. The dot patterns reveal the fact that each tile set determines a limited number of possible affine transformations for each tile added. There are usually many possible ways to add a new tile, and the dots represent the vertices of all possible additions. But the design of the tile set limits the pattern of vertices to visible grids and alignments. These are just three examples of the dot patterns I generate from the tilings. You can view the tilings as well here.

Irregular Boundary Diagrams

This boundary diagram is based on a pentagon and three triangles. Two of the triangles are scalene, and the tile structure underneath is irregular, with gaps. The dot pattern transitions from regular to highly irregular. Nevertheless, from the regular bottom of the image to the irregular top the pattern retains some common horizontal and angular alignments.

This is one of several boundary diagrams that I've done combining tiles that would easily be regular tessellations with a transition triangle or two. Previously, I have designed self-similar scaling tile sets that suggest a regular if not gap-less tiling. In this case, the scalene triangles make the transition of pentagons and isosceles triangles from one scale to the next.

Here's a similar example, but with three rhombi and an isosceles triangle making the transition.

Boundary Diagrams

I've created a series of Self-Similar Boundary diagrams. I'm using original self-similar tile sets as the structure for these dot pattern diagrams. I begin the process of creating each diagram by designing a self-similar tile set. Then I create a tiling that emphasizes the transition from the largest tile to a potentially infinite boundary. Finally I generate vertices and extrinsic vertices for the tiling. The vertices or dot pattern recedes from open to closed at the boundary. I've included images of the dot pattern and the tiling structure for the vertices.

Examples of Self-Similar Tile Sets (Part 3)

These are examples of tilings from a variety of polygons. My purpose for making these is art, not math. These tilings, or tile patches, are one solution to a problem I encountered while generating tilings as structures for a series of vertex pattern diagrams. I needed a way to create greater density changes than I was getting with other tilings. Self-similar tile sets solve the problem. See Part 1 and Part 2 for detailed descriptions of two of these examples.

The tile set I describe in Part 1 is closely related to, and can be derived from tilings described by Robert W. Fathauer. Fathauer found two families of self-similar tilings based on segments of regular polygons [See the references below]. He went on to discover a great variety of fractal tilings that, as tilings, are more interesting than most of those I describe here. Unlike Fathauer’s discoveries the tilings below are either trivial, or they require more than one prototile. My method for developing a tiling is not necessarily straight forward or fixed. My purpose is not to generate interesting fractal diagrams, but to develop often asymmetrical structures for vertex pattern diagrams.

I've imposed a few restrictions on these tilings. That is, they should be edge-to-edge, with no overlaps. Tilings may not fill the plane, but should be infinitely scalable at the boundaries, as in a fractal. In some examples there might be gaps, but these gaps might be bounded by infinitely scaleable tiles forming singularities. These are actually tile patches, not tessellations. In most of these examples the tile set could be extended by adding other polygons, especially triangles.

1: A square and a triangle. The triangle is the s=4 case described by Fathauer, and I have retained the square for which the triangle is a segment. In most of these examples I'm showing one possible tiling that might extend indefinitely in a similar way.



2: A pentagon and a triangle. The triangle is the s=10 case described by Fathauer, and I have retained the pentagon that remains after removing five s=10 tiles from a regular decagon. This is the tile set I describe in Part 1.



3: A regular hexagon and two triangles. The isosceles triangle subdivides the equilateral triangle, which subdivides the hexagon. Therefore, a similar tiling could be accomplished with just an isosceles triangle.



4: Two trapezoids and two triangles. Other tilings could be made with just the two trapezoids or one trapezoid and a triangle.



5: A dart and a triangle. In this example the triangle is necessary to continue the scaling when the darts fold in on themselves. Darts alone without the triangles would overlap.



6: Another dart. Unlike the previous example, no triangle is necessary. When the darts fold in on themselves the gap can be bounded by infinitely scalable darts.



7: Another dart and triangle. A right isosceles triangle or square is necessary to fill a gap when the darts fold in on themselves.



8. A hexagon. This example is trivial, but the hexagon prototile can be combined with triangles that subdivide the hexagon to create interesting asymmetries. The hexagon fits inside a regular pentagon. This is the tile set I describe in Part 2.



9: A pentagon and two triangles. The pentagon fits inside a regular hexagon.



10: Golden Rectangles. The lengths of the two longest sides of each L-shaped hexagon are in the golden ratio.



References

Fathauer, Robert W. (2000). "Self-similar Tilings Based on Prototiles Constructed from Segments of Regular Polygons," presented at the Bridges Conference (July 28-30, 2000, Southwestern College, Winfield, Kansas).
http://www.mathartfun.com/shopsite_sc/store/html/Compendium/Bridges2000.pdf
Also see:
http://www.mathartfun.com/shopsite_sc/store/html/Compendium/encyclopedia.html

Self-similarity in Architecture

Most tessellations, especially in architecture, limit the available diversity within the chosen system. Part-to-whole relationships are singular when they could be scalable. Self-similarity solves the problem by allowing a potentially infinite range of parts within one whole. It's possible to design chaotic part-to-whole relationships that are flexible but impractical. Self-similarity restores the practicality while maintaining acceptable flexibility.



Other self-similar tile sets can be based on regular polygons including squares or hexagons. Here’s an animation using three scaled sets of three prototiles: http://joebartholomew.com/aniVertices_HSS9.html

Self-Similar Vertices, Part 2

This is the second of two methods that I use to create dot patterns with self-similar tile sets. My interest in tilings is relative to art, not math. These tilings, or tile patches, are one solution to a problem I encountered while generating tilings as structures for a series of vertex pattern diagrams. I needed a way to create greater density changes than I was getting with other tilings. Self-similar tile sets solve the problem.

The tile set described in Part 1 is closely related to, and can be derived from tilings described by Robert W. Fathauer. Fathauer found two families of self-similar tilings based on segments of regular polygons (see Reference below). One family includes an 18-18-144 triangle that is a segment of a regular decagon. The triangle is the s=10 prototile described by Fathauer. The tile set that I use includes a pentagon as well as the triangle. The pentagon is the shape that remains after removing five s=10 tiles from a regular decagon. The tile set I describe here in Part 2 is also self-similar, but doesn’t include Fathauer’s prototiles.

The method I describe here is based on an infinitely self-similar tile set consisting of a partially concave hexagon and two isosceles triangle prototiles. The hexagon is congruent to a regular pentagon except one side of the pentagon has been made concave with two smaller sides forming interior angles of 72, 252, and 72 degrees in addition to three 108 degree interior angles. The triangles are 72-72-36 and 36-36-108 isosceles triangles.

The initial hexagon and triangle prototiles are sized so the two long sides of the 72-72-36 triangle are equal to the long sides of the hexagon. The single long side of the 36-36-108 triangle is also equal to the long sides of the hexagon. The 36-36-108 triangle fits exactly in the concave area of the hexagon, and together the 36-36-108 triangle and the hexagon form a regular pentagon. The tile set in Part 1 was infinitely scalable pentagon-triangle pairs by setting the next pentagon side equal to the short side of the previous triangle. The tile set describe here is infinitely scalable by setting the next hexagon long sides equal to a short side of the previous pentagon.

Tilings should be edge-to-edge, with no overlaps. Gaps are always fillable. Singularities are not necessary. Tilings may not fill the plane, but could be infinitely scalable at the boundaries, as in a fractal. These are actually tile patches, not tessellations. A radially symmetrical tiling of just hexagons looks like the following figure.

The tilings are a structure for the finished diagram of vertices and what I call extrinsic vertices. This demonstrates the advantage of using a tiling with properties of self-similarity. The changing density of tiles translates to a greater variation in vertex density.

It's possible to create numerous symmetrical tilings with these tiles, but I often choose to create asymmetrical diagrams. The processes, lattices, and patterns I use are not math. I'm influenced by structures in math, science, architecture, and design, but unconstrained by the rigorousness of math. These diagrams have no practical use or purpose other than art.

Reference

Fathauer, Robert W. (2000). "Self-similar Tilings Based on Prototiles Constructed from Segments of Regular Polygons," presented at the Bridges Conference (July 28-30, 2000, Southwestern College, Winfield, Kansas).
http://www.mathartfun.com/shopsite_sc/store/html/Compendium/Bridges2000.pdf
Also see:
http://www.mathartfun.com/shopsite_sc/store/html/Compendium/encyclopedia.html

Self-Similar Vertices

I've been using the process described here to create tilings from a pentagon and triangle. These tilings, or tile patches, are one solution to a problem I encountered while generating tilings as structures for a series of vertex pattern diagrams. I needed a way to create greater density changes than I was getting with other tilings. Self-similar tile sets solve the problem.
My interest in tilings is more art than math. The tile set I’m using is closely related to, and can be derived from tilings described by Robert W. Fathauer (see Reference below). Fathauer found two families of self-similar tilings based on segments of regular polygons. One family includes an 18-18-144 triangle that is a segment of a regular decagon. The triangle is the s=10 prototile described by Fathauer. The tile set that I use includes a pentagon as well as the triangle. The pentagon is the shape that remains after removing five s=10 tiles from a regular decagon. Other hexagonal or square prototiles can be combined with triangles to make self-similar tile sets.

So, this method is based on an infinitely self-similar tile set consisting of pentagons and 18-18-144 triangle prototiles. Tilings should be edge-to-edge, with no overlaps. Gaps are inevitable, but they should allow lining with infinitely scaled tiles. Tilings will not fill the plane but should be infinitely scalable at the boundaries, as in a fractal. These are actually tile patches, not tessellations.

The initial pentagon and triangle prototiles are sized so the long side of the 18-18-144 triangle is equal to the pentagon side. Each subsequent pentagon-triangle pair is scaled so that the next pentagon side is equal to the short side of the previous triangle. Using this scheme, as the boundaries of the tiling grow outwards they form singularities, or gaps surrounded by tiles. The inside edges of these gaps can be continuously and infinitely lined with scaling pentagons and 18-18-144 triangles, or just 18-18-144 triangles.

An interesting feature of these prototiles is that the ratio of the areas of each pentagon to the next smaller pentagon (or triangle to triangle) is always 3.618... or 2 plus Phi.

It's possible to create numerous symmetrical tilings with these tiles, but I often choose to create asymmetrical diagrams. The processes, lattices, and patterns I use are not math. I'm influenced by structures in math, science, architecture, and design, but unconstrained by the rigorousness of math. These diagrams have no practical use or purpose other than art.

Reference:

Fathauer, Robert W. (2000). "Self-similar Tilings Based on Prototiles Constructed from Segments of Regular Polygons," presented at the Bridges Conference (July 28-30, 2000, Southwestern College, Winfield, Kansas).
http://www.mathartfun.com/shopsite_sc/store/html/Compendium/Bridges2000.pdf
Also see:
http://www.mathartfun.com/shopsite_sc/store/html/Compendium/encyclopedia.html