Editing T-square (fractal)

{{Short description|Two-dimensional fractal}}
{{about|a two dimensional fractal in mathematics||T-square (disambiguation)}}
{{no footnotes|date=May 2014}}
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]].<ref name="Object">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."</ref>

[[Image:T-Square fractal (evolution).png|T-square, evolution in six steps.]]{{clear}}

==Algorithmic description==
[[Image:T_square_fractal_order_8.svg|thumb|256px|T-square of order 8]]

It can be generated from using this [[algorithm]]:

# Image 1:
## Start with a square. (The black square in the image)
# Image 2:
## 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.
## Take the union of the previous image with the collection of smaller squares placed in this way.
# Images 3–6:
## Repeat step 2.


[[File:Golden Square fractal with T-branching 8.svg|thumb|[[Golden ratio|Golden squares]] with T-branching]]
{{multiple image
 | align = 
 | direction = horizontal
 | width = 150
 | header = 
 | image1 = Golden Square fractal 6.svg
 | caption1 = Square branches, related by the [[golden ratio|1/φ]]
 | image2 = Half square fractal 5.svg
 | caption2 = Squares branches, related by 1/2
 | footer = 
}}

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]]."<ref name="Object"/>

==Properties==
The T-square fractal has a [[fractal dimension]] of ln(4)/ln(2) = 2.{{citation needed|date=June 2013}} 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.

The fractal dimension of the boundary equals <math>\textstyle{\frac{\log{3}}{\log{2}}=1.5849...}</math>.

Using mathematical induction one can prove that for each n ≥ 2 the number of new squares that are added at stage n equals <math>4*3^{(n-1)}</math>.

==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:

[[File:V4 ban1 inc2.gif|thumb|none|200px|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:

{|
|- valign=top
| [[File:V4 ban1.gif|thumb|left|200px|Randomly chosen ''v''[i] ≠ ''v''[i-1] + 0]]
| [[File:V4 ban1 inc1.gif|thumb|right|200px|Randomly chosen ''v''[i] ≠ ''v''[i-1] + 1]]
|}

==T-square fractal and Sierpiński triangle==

The T-square fractal can be derived from the [[Sierpiński triangle]], and vice versa, by adjusting the angle at which sub-elements of the original fractal are added from the center outwards.
{|
|- valign=top
| [[File:Sierpiński triangle transforming into a T-square fractal.gif|thumb|left|600px|Sierpiński triangle transforming into a T-square fractal]]
|}

==See also==
*[[List of fractals by Hausdorff dimension]]
*The [[Toothpick sequence]] generates a similar pattern
*[[H tree]]

==References==
{{reflist}}

==Further reading==
* {{cite news| first1=Alioscia |last1=Hamma
|first2=Daniel A. | last2=Lidar
|first3=Simone |last3=Severini
|title=Entanglement and area law with a fractal boundary in topologically ordered phase
|doi=10.1103/PhysRevA.81.010102
|year=2010
|volume=82
|journal=Phys. Rev. A
}}
* {{cite journal|first1=Emad S. |last1=Ahmed
|title=Dual-mode dual-band microstrip bandpass filter based on fourth iteration T-square fractal and shorting pin
|year=2012
|journal=Radioengineering
|volume=21
|number=2
|page=617
}}

{{Fractals}}

{{DEFAULTSORT:T-Square (Fractal)}}
[[Category:Iterated function system fractals]]