T-square (fractal)

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In mathematics, the T-square is a two-dimensional fractal. It has a boundary of infinite length bounding a finite area. Its name comes from the drawing instrument known as a T-square. [1]

Mathematics Field of study concerning quantity, patterns and change

Mathematics includes the study of such topics as quantity, structure (algebra), space (geometry), and change. It has no generally accepted definition.

Fractal A self-similar pattern or set in mathematics

In mathematics, a fractal is a subset of a Euclidean space for which the Hausdorff dimension strictly exceeds the topological dimension. Fractals tend to appear nearly the same at different levels, as is illustrated here in the successively small magnifications of the Mandelbrot set; because of this, fractals are encountered ubiquitously in nature. Fractals exhibit similar patterns at increasingly small scales called self similarity, also known as expanding symmetry or unfolding symmetry; if this replication is exactly the same at every scale, as in the Menger sponge, it is called affine self-similar. Fractal geometry lies within the mathematical branch of topology.

T-square technical drawing instrument

A T-square is a technical drawing instrument used by draftsmen primarily as a guide for drawing horizontal lines on a drafting table. It may also guide a set square to draw vertical or diagonal lines. Its name comes from its resemblance to the letter T. T-squares come in varying sizes, common lengths being 18 inches (460 mm), 24 inches (610 mm), 30 inches (760 mm), 36 inches (910 mm) and 42 inches (1,100 mm).

Contents

T-Square fractal (evolution).png

Algorithmic description

T-square. T-Square fractal (8 iterations).png
T-square.

It can be generated from using this algorithm:

  1. Image 1:
    1. Start with a square. (The black square in the image)
  2. Image 2:
    1. At each convex corner of the previous image, place another square, centered at that corner, with half the side length of the square from the previous image.
    2. Take the union of the previous image with the collection of smaller squares placed in this way.
  3. Images 3–6:
    1. Repeat step 2.


Golden squares with T-branching Golden Square fractal with T-branching 8.svg
Golden squares with T-branching
Golden Square fractal 6.svg
Square branches, related by the 1/φ
Half square fractal 5.svg
Squares branches, related by 1/2

The method of creation is rather similar to the ones used to create a Koch snowflake or a Sierpinski triangle, "both based on recursively drawing equilateral triangles and the Sierpinski carpet." [1]

Koch snowflake fractal and mathematical curve

The Koch snowflake is a mathematical curve and one of the earliest fractals to have been described. It is based on the Koch curve, which appeared in a 1904 paper titled "On a Continuous Curve Without Tangents, Constructible from Elementary Geometry" by the Swedish mathematician Helge von Koch.

Sierpinski carpet Sierpiński mathematical shape

The Sierpinski carpet is a plane fractal first described by Wacław Sierpiński in 1916. The carpet is one generalization of the Cantor set to two dimensions; another is the Cantor dust.

Properties

The T-square fractal has a fractal dimension of ln(4)/ln(2) = 2.[ citation needed ] The black surface extent is almost everywhere in the bigger square, for once a point has been darkened, it remains black for every other iteration; however some points remain white.

In mathematics, more specifically in fractal geometry, a fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a pattern changes with the scale at which it is measured. It has also been characterized as a measure of the space-filling capacity of a pattern that tells how a fractal scales differently from the space it is embedded in; a fractal dimension does not have to be an integer.

The fractal dimension of the boundary equals .

Using mathematical induction one can prove that for each n ≥ 2 the number of new squares that are added at stage n equals .

The T-Square and the chaos game

The T-square fractal can also be generated by an adaptation of the chaos game, in which a point jumps repeatedly half-way towards the randomly chosen vertices of a square. The T-square appears when the jumping point is unable to target the vertex directly opposite the vertex previously chosen. That is, if the current vertex is v[i] and the previous vertex was v[i-1], then v[i] ≠ v[i-1] + vinc, where vinc = 2 and modular arithmetic means that 3 + 2 = 1, 4 + 2 = 2:

Chaos game method of creating a fractal, using a polygon and an initial point selected at random inside it

In mathematics, the term chaos game originally referred to a method of creating a fractal, using a polygon and an initial point selected at random inside it. The fractal is created by iteratively creating a sequence of points, starting with the initial random point, in which each point in the sequence is a given fraction of the distance between the previous point and one of the vertices of the polygon; the vertex is chosen at random in each iteration. Repeating this iterative process a large number of times, selecting the vertex at random on each iteration, and throwing out the first few points in the sequence, will often produce a fractal shape. Using a regular triangle and the factor 1/2 will result in the Sierpinski triangle, while creating the proper arrangement with four points and a factor 1/2 will create a display of a "Sierpinski Tetrahedron", the three-dimensional analogue of the Sierpinski triangle. As the number of points is increased to a number N, the arrangement forms a corresponding (N-1)-dimensional Sierpinski Simplex.

Randomly chosen v[i] [?] v[i-1] + 2 V4 ban1 inc2.gif
Randomly chosen v[i] ≠ v[i-1] + 2

If vinc is given different values, allomorphs of the T-square appear that are computationally equivalent to the T-square but very different in appearance:

Randomly chosen v[i] [?] v[i-1] + 0 V4 ban1.gif
Randomly chosen v[i] ≠ v[i-1] + 0
Randomly chosen v[i] [?] v[i-1] + 1 V4 ban1 inc1.gif
Randomly chosen v[i] ≠ v[i-1] + 1

See also

Toothpick sequence geometric fractal-like pattern

In geometry, the toothpick sequence is a sequence of 2-dimensional patterns which can be formed by repeatedly adding line segments ("toothpicks") to the previous pattern in the sequence.

H tree

In fractal geometry, the H tree, or T-branching, is a fractal tree structure constructed from perpendicular line segments, each smaller by a factor of the square root of 2 from the next larger adjacent segment. It is so called because its repeating pattern resembles the letter "H". It has Hausdorff dimension 2, and comes arbitrarily close to every point in a rectangle. Its applications include VLSI design and microwave engineering.

Related Research Articles

Hausdorff dimension invariant

In mathematics, Hausdorff dimension is a measure of roughness and/or chaos that was first introduced in 1918 by mathematician Felix Hausdorff. Applying the mathematical formula, the Hausdorff dimension of a single point is zero, of a line segment is 1, of a square is 2, and of a cube is 3. That is, for sets of points that define a smooth shape or a shape that has a small number of corners—the shapes of traditional geometry and science—the Hausdorff dimension is an integer agreeing with the usual sense of dimension, also known as the topological dimension. However, formulas have also been developed that allow calculation of the dimension of other less simple objects, where, based solely on their properties of scaling and self-similarity, one is led to the conclusion that particular objects—including fractals—have non-integer Hausdorff dimensions. Because of the significant technical advances made by Abram Samoilovitch Besicovitch allowing computation of dimensions for highly irregular or "rough" sets, this dimension is also commonly referred to as the Hausdorff–Besicovitch dimension.

Self-similarity

In mathematics, a self-similar object is exactly or approximately similar to a part of itself. Many objects in the real world, such as coastlines, are statistically self-similar: parts of them show the same statistical properties at many scales. Self-similarity is a typical property of artificial fractals. Scale invariance is an exact form of self-similarity where at any magnification there is a smaller piece of the object that is similar to the whole. For instance, a side of the Koch snowflake is both symmetrical and scale-invariant; it can be continually magnified 3x without changing shape. The non-trivial similarity evident in fractals is distinguished by their fine structure, or detail on arbitrarily small scales. As a counterexample, whereas any portion of a straight line may resemble the whole, further detail is not revealed.

Sierpiński triangle fractal composed of rep-tiled triangles

The Sierpinski triangle, also called the Sierpinski gasket or Sierpinski sieve, is a fractal and attractive fixed set with the overall shape of an equilateral triangle, subdivided recursively into smaller equilateral triangles. Originally constructed as a curve, this is one of the basic examples of self-similar sets–that is, it is a mathematically generated pattern that is reproducible at any magnification or reduction. It is named after the Polish mathematician Wacław Sierpiński, but appeared as a decorative pattern many centuries before the work of Sierpiński.

Fractal art

Fractal art is a form of algorithmic art created by calculating fractal objects and representing the calculation results as still images, animations, and media. Fractal art developed from the mid-1980s onwards. It is a genre of computer art and digital art which are part of new media art. The mathematical beauty of fractals lies at the intersection of generative art and computer art. They combine to produce a type of abstract art.

Menger sponge 3d fractal

In mathematics, the Menger sponge is a fractal curve. It is a three-dimensional generalization of the one-dimensional Cantor set and two-dimensional Sierpinski carpet. It was first described by Karl Menger in 1926, in his studies of the concept of topological dimension.

Dragon curve fractal constructible with L-systems

A dragon curve is any member of a family of self-similar fractal curves, which can be approximated by recursive methods such as Lindenmayer systems.

Iterated function system

In mathematics, iterated function systems (IFSs) are a method of constructing fractals; the resulting fractals are often self-similar. IFS fractals are more related to set theory than fractal geometry. They were introduced in 1981.

Sierpiński curve

Sierpiński curves are a recursively defined sequence of continuous closed plane fractal curves discovered by Wacław Sierpiński, which in the limit completely fill the unit square: thus their limit curve, also called the Sierpiński curve, is an example of a space-filling curve.

Hilbert curve space-filling curve

A Hilbert curve is a continuous fractal space-filling curve first described by the German mathematician David Hilbert in 1891, as a variant of the space-filling Peano curves discovered by Giuseppe Peano in 1890.

In geometry, pinwheel tilings are non-periodic tilings defined by Charles Radin and based on a construction due to John Conway. They are the first known non-periodic tilings to each have the property that their tiles appear in infinitely many orientations.

Hexaflake

A hexaflake is a fractal constructed by iteratively exchanging hexagons by a flake of seven hexagons; it is a special case of the n-flake.

Vicsek fractal

In mathematics the Vicsek fractal, also known as Vicsek snowflake or box fractal, is a fractal arising from a construction similar to that of the Sierpinski carpet, proposed by Tamás Vicsek. It has applications including as compact antennas, particularly in cellular phones.

An n-flake, polyflake, or Sierpinski n-gon, is a fractal constructed starting from an n-gon. This n-gon is replaced by a flake of smaller n-gons, such that the scaled polygons are placed at the vertices, and sometimes in the center. This process is repeated recursively to result in the fractal. Typically, there is also the restriction that the n-gons must touch yet not overlap.

Mosely snowflake

The Mosely snowflake is a Sierpiński–Menger type of fractal obtained in two variants either by the operation opposite to creating the Sierpiński-Menger snowflake or Cantor dust i.e. not by leaving but by removing eight of the smaller 1/3-scaled corner cubes and the central one from each cube left from the previous recursion (lighter) or by removing only corner cubes (heavier). In one dimension this operation is trivial and converges only to single point. It resembles the original water snowflake of snow. By the construction the Hausdorff dimension of the lighter snowflake is

The Fibonacci word fractal is a fractal curve defined on the plane from the Fibonacci word.

Ulam–Warburton automaton

The Ulam–Warburton cellular automaton (UWCA) is a 2-dimensional fractal pattern that grows on a regular grid of cells consisting of squares. Starting with one square initially ON and all others OFF, successive iterations are generated by turning ON all squares that share precisely one edge with an ON square. This is the von Neumann neighborhood. The automaton is named after the Polish-American mathematician and scientist Stanislaw Ulam and the Scottish engineer, inventor and amateur mathematician Mike Warburton.

References

  1. 1 2 Dale, Nell; Joyce, Daniel T.; and Weems, Chip (2016). Object-Oriented Data Structures Using Java, p.187. Jones & Bartlett Learning. ISBN   9781284125818. "Our resulting image is a fractal called a T-square because with it we can see shapes that remind us of the technical drawing instrument of the same name."

Further reading

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