imported>Chunbum Park |
imported>John Stephenson |
| (142 intermediate revisions by 5 users not shown) |
| Line 1: |
Line 1: |
| == '''[[Hausdorff dimension]]''' ==
| | {{:{{FeaturedArticleTitle}}}} |
| ''by [[User:Melchior Grutzmann|Melchior Grutzmann]] (and [[User:Brandon Piercy|Brandon Piercy]] and [[User:Hendra I. Nurdin|Hendra I. Nurdin]])
| | <small> |
| | | ==Footnotes== |
| ----
| |
| | |
| In [[mathematics]], the '''Hausdorff dimension''' is a way of defining a possibly fractional exponent for all figures in a [[metric space]] such that the dimension describes partially the amount to that the set fills the space around it. For example, a [[plane (geometry)|plane]] would have a Hausdorff dimension of 2, because it fills a 2-parameter subset. However, it would not make sense to give the [[Sierpiński triangle]] [[fractal]] a dimension of 2, since it does not fully occupy the 2-dimensional realm. The Hausdorff dimension describes this mathematically by measuring the size of the set. For self-similar sets there is a relationship to the number of self-similar subsets and their scale.
| |
| | |
| === Informal definition ===
| |
| Intuitively, the dimension of a set is the number of independent parameters one has to pick in order to fix a point. This is made rigorously with the notion of ''d''-dimensional (topological) [[manifold]] which are particularly regular sets. The problem with the classical notion is that you can easily break up the digits of a real number to map it bijectively to two (or ''d'') real numbers. The example of space filling curves shows that it is even possible to do this in a continuous (but non-bijective) way.
| |
| | |
| The notion of Hausdorff dimension refines this notion of dimension such that the dimension can be any non-negative number.
| |
| | |
| Benoît Mandelbrot discovered<ref>B.B. Mandelbrot: ''The fractal geometry of nature'', Freemann '''(1983)''', ISBN 978-0-716-711-865</ref> that many objects in nature are not strictly classical smooth bodies, but best approximated as fractal sets, i.e. subsets of '''R'''<sup>''N''</sup> whose Hausdorff dimension is strictly greater than its topological dimension.
| |
| | |
| | |
| === Hausdorff measure and dimension ===
| |
| Let ''d'' be a non-negative real number and ''S'' ⊂ ''X'' a subset of a metric space (''X'',''ρ''). The ''d''-dimesional Hausdorff measure of scale ''δ''>0 is
| |
| :<math> H^{d*}_\delta(S) := \inf \{\sum_{i=1}^\infty r_i^d : S\subset\bigcup_{i=1}^\infty B_{r_i}(x_i), r_i\le\delta \}</math>
| |
| where B<sub>''r''<sub>''i''</sub>(''x''<sub>''i''</sub>)</sub> is the open ball around ''x''<sub>''i''</sub> ∈ ''X'' of radius ''r''<sub>''i''</sub>. The ''d''-dimensional Hausdorff measure is now the limit
| |
| :<math> H^{d*}(S) := \lim_{\delta\to0+} H^{d*}_\delta(S)</math>.
| |
| As in the Carathéodory construction a set ''S'' ⊂ ''X'' is called ''d''-measurable iff
| |
| :<math> H^{d*}(T) = H^{d*}(S\cap T)+ H^{d*}(T\cap X\setminus S)</math> for all ''T'' ⊂ ''X''.
| |
| A set ''S'' ⊂ ''X'' is called Hausdorff measurable if it is H<sup>''d''</sup>-measurable for all ''d''≥0.
| |
| | |
| ''[[Hausdorff dimension|.... (read more)]]''
| |
| | |
| {| class="wikitable collapsible collapsed" style="width: 90%; float: center; margin: 0.5em 1em 0.8em 0px;"
| |
| |-
| |
| ! style="text-align: center;" | [[Hausdorff dimension#Literature|notes]]
| |
| |-
| |
| |
| |
| {{reflist|2}} | | {{reflist|2}} |
| |}
| | </small> |
Latest revision as of 10:19, 11 September 2020
The Irvin pin. The eyes have always been red, but there are
urban legends about the meanings of other colors.
A pin from another company, possibly Switlik or Standard Parachute. This style is common in catalogs and auctions of military memorabilia.
The Caterpillar Club is an informal association of people who have successfully used a parachute to bail out of a disabled aircraft. After authentication by the parachute maker, applicants receive a membership certificate and a distinctive lapel pin.
History
Before April 28, 1919 there was no way for a pilot to jump out of a plane and then to deploy a parachute. Parachutes were stored in a canister attached to the aircraft, and if the plane was spinning, the parachute could not deploy. Film industry stuntman Leslie Irvin developed a parachute that the pilot could deploy at will from a back pack using a ripcord. He joined the Army Air Corps parachute research team, and in April 1919 he successfully tested his design, though he broke his ankle during the test. Irvin was the first person to make a premeditated free fall jump from an airplane. He went on to form the Irving Airchute Company, which became a large supplier of parachutes. (A clerical error resulted in the addition of the "g" to Irvin and this was left in place until 1970, when the company was unified under the title Irvin Industries Incorporated.) The Irvin brand is now a part of Airborne Systems, a company with operations in Canada, the U.S. and the U.K.[1].
An early brochure [2] of the Irvin Parachute Company credits William O'Connor 24 August 1920 at McCook Field near Dayton, Ohio as the first person to be saved by an Irvin parachute, but this feat was unrecognised. On 20 October 1922 Lieutenant Harold R. Harris, chief of the McCook Field Flying Station, jumped from a disabled Loening W-2A monoplane fighter. Shortly after, two reporters from the Dayton Herald, realising that there would be more jumps in future, suggested that a club should be formed. 'Caterpillar Club' was suggested because the parachute canopy was made of silk, and because caterpillars have to climb out of their cocoons and fly away. Harris became the first member, and from that time forward any person who jumped from a disabled aircraft with a parachute became a member of the Caterpillar Club. Other famous members include General James Doolittle, Charles Lindbergh and (retired) astronaut John Glenn.
In 1922 Leslie Irvin agreed to give a gold pin to every person whose life was saved by one of his parachutes. By 1945 the number of members with the Irvin pins had grown to over 34,000. In addition to the Irvin Air Chute Company and its successors, other parachute manufacturers have also issued caterpillar pins for successful jumps. Irvin/Irving's successor, Airborne Systems Canada, still provides pins to people who made their jump long ago and are just now applying for membership. Another of these is Switlik Parachute Company, which though it no longer makes parachutes, still issues pins.
