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[[User:IntegralPython/sandbox/Fractal measure| Fractal Measure]]
==Open set condition==
In [[fractal geometry]], the '''open set condition''' ('''OSC''') is a commonly imposed condition on self-similar fractals. In some sense, the condition imposes restrictions on the overlap in a fractal construction.<ref>{{cite journal |last1=Bandt |first1=Christoph |last2= Viet Hung |first2= Nguyen |last3 = Rao |first3 = Hui | title=On the Open Set Condition for Self-Similar Fractals | journal=Proceedings of the American Mathematical Society | volume=134 | year=2006 | pages=1369–74 | issue=5 |doi=10.1090/S0002-9939-05-08300-0 |jstor=4097989 | url=http://www.jstor.org/stable/4097989| url-access=limited}}</ref> Specifically, given an [[iterated function system]] of [[contraction mapping| contractive mappings]] ψ<sub>''i''</sub>, the open set condition requires that there exists a nonempty, open set S satisfying two conditions:
#<math> \bigcup_{i=1}^m\psi_i (V) \subseteq V, </math>
# Each <math>\psi_i (V)</math> is pairwise disjoint.
Introduced in 1946 by P.A.P Moran,<ref>{{cite journal | last=Moran | first=P.A.P. | title=Additive Functions of Intervals and Hausdorff Measure | journal=Proceedings-Cambridge Philosophical Society | volume=42 | year=1946 | pages=15–23 | doi=10.1017/S0305004100022684}}</ref> the open set condition is used to compute the dimensions of certain self-similar fractals, notably the Sierpinski Gasket. It is also used to simplify computation of the packing measure.<ref>{{cite journal| last1=Llorente|first1=Marta|last2=Mera|first2=M. Eugenia| last3=Moran| first3=Manuel| title= On the Packing Measure of the Sierpinski Gasket | journal= University of Madrid | url=https://eprints.ucm.es/id/eprint/58898/1/version%20final(previa%20prueba%20imprenta).pdf}}</ref>
An equivalent statement of the open set condition is to require that the s-dimensional [[Hausdorff measure]] of the set is greater than zero.<ref>
{{cite web |url=https://www.math.cuhk.edu.hk/conference/afrt2012/slides/Wen_Zhiying.pdf |title=Open set condition for self-similar structure |last= Wen |first=Zhi-ying |publisher=Tsinghua University |access-date= 1 February 2022 }} </ref>
===Computing Hausdorff measure===
When the open set condition holds and each ψ<sub>''i''</sub> is a similitude (that is, a composition of an [[isometry]] and a [[dilation (metric space)|dilation]] around some point), then the unique fixed point of ψ is a set whose Hausdorff dimension is the unique solution for ''s'' of the following:<ref>{{cite journal | last=Hutchinson | first=John E. | title=Fractals and self similarity | journal=Indiana Univ. Math. J. | volume=30 | year=1981 | pages=713–747 | doi=10.1512/iumj.1981.30.30055 | issue=5 | doi-access=free }}</ref>
:<math> \sum_{i=1}^m r_i^s = 1. </math>
where r<sub>i</sub> is the magnitude of the dilation of the similitude.
With this theorem, the Hausdorff dimension of the Sierpinski gasket can be calculated. Consider three [[non-collinear points]] ''a''<sub>1</sub>, ''a''<sub>2</sub>, ''a''<sub>3</sub> in the plane '''R'''<sup>2</sup> and let ψ<sub>''i''</sub> be the dilation of ratio 1/2 around ''a<sub>i</sub>''. The unique non-empty fixed point of the corresponding mapping ψ is a Sierpinski gasket, and the dimension ''s'' is the unique solution of
:<math> \left(\frac{1}{2}\right)^s+\left(\frac{1}{2}\right)^s+\left(\frac{1}{2}\right)^s = 3 \left(\frac{1}{2}\right)^s =1. </math>
Taking [[natural logarithm]]s of both sides of the above equation, we can solve for ''s'', that is: ''s'' = ln(3)/ln(2). The Sierpinski gasket is self-similar and satisfies the OSC.
==Hand-eye calibration problem==
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====Separable solutions====
Given the equation {{math|AX{{=}}ZB}}, it is possible to decompose the equation into a purely rotational and translational part; methods utilizing this are referred to as separable methods. Where {{math|'''R'''<sub>A</sub>}} represents a 3×3 rotation matrix and {{math|'''t'''<sub>A</sub>}} a 3×1 translation vector, the equation can be broken into two parts:<ref>
:{{math|'''R'''<sub>A</sub>'''R'''<sub>X</sub>{{=}}'''R'''<sub>Z</sub>'''R'''<sub>B</sub>}}
:{{math|'''R'''<sub>A</sub>'''t'''<sub>X</sub>+'''t'''<sub>A</sub>{{=}}'''R'''<sub>Z</sub>'''t'''<sub>B</sub>+'''t'''<sub>Z</sub>}}
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{{collapse top | Ugandan knuckles}}
Ugandan Knuckles is an [[internet meme]] from January 2018 depicting a deformed version of [[Knuckles the Echidna]]. Players would go in hords to the virtual reality video game ''[[VRChat]]'' to troll other players. The people would say quotes such as "Do you know the way?", which originate from the 2010 Ugandan action film ''[[Who Killed Captain Alex?]]'', as well as "spitting" on other users whom they felt did not know "de way".<ref name="dailydot">{{Cite web|url=https://www.dailydot.com/unclick/ugandan-knuckles-vrchat-meme/|title=How Ugandan Knuckles turned VRChat into a total trollfest|last=Hathaway|first=Jay|date=11 January 2018|website=The Daily Dot|archive-url=|archive-date
===History===
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===Controversy===
Because of its use of a fake Ugandan accent as well as the quotations from ''Who Killed Captain Alex?'', the meme was widely criticized for being racially insensitive;<ref name="dailydot"/><ref name=Polygon2/> ''[[Polygon (website)|Polygon]]'' described it as problematic.<ref name=Polygon2>{{cite web|url=https://www.polygon.com/2018/1/8/16863932/ugandan-knuckles-meme-vrchat|title=
The original creator of the 3D avatar, [[DeviantArt]] user "tidiestflyer", showed regret over the character, saying that he hoped it would not be used to annoy players of ''VRChat'' and that he enjoys the game and does not want to see anyone's rights get taken away because of the avatar.<ref>{{Cite web|url=http://www.gamerevolution.com/news/362289-creator-vrchats-ugandan-knuckles-meme-regrets-decision|title=Creator of
{{collapse bottom}}
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