멩거 스펀지

1/3분할을 반복하는 0단계,1단계,2단계,3단계의 멩거큐브

수학에서, 멩거 스펀지(Menger sponge), 멩거 큐브, 멩거 유니버셜 커브, 시어핀스키 큐브, 또는 시어핀스키 스펀지^[1]^[2]^[3]는 프랙탈 곡선이다. 1차원 칸토어 집합에 이어서 2차원 시어핀스키 카펫을 3차원으로 일반화했다. 이것은 1926년에 카를 멩거(Karl Menger)에 의해 위상 차원의 개념에 대한 그의 연구에서 처음 묘사되었다.^[4]^[5]

멩거 스펀지는 기존의 차원개념을 정수로 당연시 하던 고정관념에 실수차원이 존재한다는 사실을 증명하는 사례이다.

멩거큐브 단계

큐브의 단위 길이를 '1'로 둘경우 멩거 큐브내에서 ${{1} \over {3}}$ 분할로 3등분하는 단계적 반복으로 증가하는 빈공간의 정육면체의 개수와 빈공간으로 인해 증가하는 큐브의 면적 그리고 상대적으로 줄어드는 부피와의 관계는 아래와 같다.

길이:

L=\left({{1} \over {3}}\right)^{n}

프랙털 차원은

{{ln20} \over {ln3}}=2.726833027...

차원이다.

단계	정육면체-빈공간 정육면체 개수= 남은 정육면체	겉넓이	부피
0	$1-0=1$	$1\times 1\times 6=6$	$1\times 1\times 1=1^{3}=1$
1	$27-7=20$	$\left({{1} \over {3}}\times {{1} \over {3}}\right)\times \left((2\times 20)+(4\times 8)\right)$ $={{(2\times 20)} \over {9}}+{{(4\times 8)} \over {9}}={{72} \over {9}}=8$	${{1} \over {3^{1}}}\times {{1} \over {3^{1}}}\times {{1} \over {3^{1}}}\times 20={{20} \over {27}}=0.740740...$
2	$20\times (27-7)=400$	$\left({{1} \over {3^{2}}}\times {{1} \over {3^{2}}}\right)\times \left((2\times 20^{2})+(4\times 8^{2})\right)$ $=2\left({{20} \over {9}}\right)^{2}+4\left({{8} \over {9}}\right)^{2}={{1056} \over {81}}=13.037037...$	${{1} \over {3^{2}}}\times {{1} \over {3^{2}}}\times {{1} \over {3^{2}}}\times 400={{400} \over {729}}=0.5486968449...$
3	$400\times (27-7)=8000$	$2\left({{20} \over {9}}\right)^{3}+4\left({{8} \over {9}}\right)^{3}={{18048} \over {729}}=24.75720164...$	${{1} \over {3^{3}}}\times {{1} \over {3^{3}}}\times {{1} \over {3^{3}}}\times 8000={{8000} \over {19683}}=0.406442107...$
$n$	$(20)^{n}=N$	$2\left({{20} \over {9}}\right)^{n}+4\left({{8} \over {9}}\right)^{n}$	$\left({{20} \over {27}}\right)^{n}$

총 부피 =부피 x 개수

V=L^{3}\times N=\left({{1} \over {3}}\right)^{3n}(20)^{n}=\left({{20} \over {27}}\right)^{n}

멩거큐브2단계만으로 멩거큐브는 단위정육면체(멩거큐브0단계)면적의 2배를 넘어서며, 부피에서

{{1} \over {2}}

배에 접근한다.

같이 보기

각주

↑ Beck, Christian; Schögl, Friedrich (1995). 《Thermodynamics of Chaotic Systems: An Introduction》 (영어). Cambridge University Press. 97쪽. ISBN 9780521484510.
↑ Bunde, Armin; Havlin, Shlomo (2013). 《Fractals in Science》 (영어). Springer. 7쪽. ISBN 9783642779534.
↑ Menger, Karl (2013). 《Reminiscences of the Vienna Circle and the Mathematical Colloquium》 (영어). Springer Science & Business Media. 11쪽. ISBN 9789401111027.
↑ Menger, Karl (1928), 《Dimensionstheorie》, B.G Teubner Publishers
↑ Menger, Karl (1926), “Allgemeine Räume und Cartesische Räume. I.”, 《Communications to the Amsterdam Academy of Sciences》 . English translation reprinted in Edgar, Gerald A., 편집. (2004), 《Classics on fractals》, Studies in Nonlinearity, Westview Press. Advanced Book Program, Boulder, CO, ISBN 978-0-8133-4153-8, MR 2049443

Read other articles:

Stasiun Rokken (Mie)

Stasiun Rokken六軒駅Stasiun RokkenLokasiOzu-cho, Matsusaka, Mie（三重県松阪市小津町）JepangOperatorJR CentralJalurJalur Utama KiseiSejarahDibuka1894PenumpangFY201084 per hari Sunting kotak info • L • BBantuan penggunaan templat ini Stasiun Rokken (六軒駅code: ja is deprecated , Rokken-eki) adalah sebuah stasiun kereta api di Matsusaka, Prefektur Mie, yang dioperasikan oleh Central Japan Railway Company (JR Central). Stasiun tersebut berjarak 29.1 kilometer dari ...

تاريخ الدين في الولايات المتحدة

هذه المقالة بحاجة لصندوق معلومات. فضلًا ساعد في تحسين هذه المقالة بإضافة صندوق معلومات مخصص إليها. جزء من سلسلة مقالات حولتاريخ الولايات المتحدة التسلسل الزمني ما قبل التاريخ عصر قبل كولومبي حقبة الاستعمار (1776–1789) (1789–1849) (1849 – 1865) (1865–1918) (1918–1945) (1945-1964) (1964-1980) (1964-1980) تاريخ �...

K. Sivan

K. Sivan Ketua Organisasi Penelitian Luar Angkasa IndiaPetahanaMulai menjabat 15 Januari 2018 PendahuluA. S. Kiran KumarPenggantiPetahana Informasi pribadiLahirKailasavadivoo Sivan14 April 1957 (umur 66)Mela Sarakkalvilai,Distrik Kanyakumari, (sekarang Tamil Nadu), IndiaTempat tinggalBengaluru, KarnatakaPendidikan Universitas Madurai Kamaraj (B.Sc.) Institut Teknologi Madras (B.Tech) IISc, Bangalore (M.E.) IIT, Bombay (Ph.D) Sunting kotak info • L • B Kailasavadivoo Siv...

Zhu Quanzhong

Zhu Wen Zhu Quanzhong (Hanzi: 朱全忠, 852-912), terlahir dengan nama Zhu Wen (朱温), adalah seorang gubernur militer pada tahun-tahun terakhir Dinasti Tang. Dia adalah mantan bawahan pemberontak Huang Chao yang kemudian membelot pada Tang setelah kekalahan Huang sudah di depan mata. Sejak itu dia diberikan nama kehormatan Quanzhong yang artinya kesetiaan penuh oleh Kaisar Tang Xizong. Tahun 907, dia menggulingkan Dinasti Tang yang telah lemah dan mendeklarasikan diri sebagai kaisar dari ...

Casato di Wittelsbach

Disambiguazione – Wittelsbach rimanda qui. Se stai cercando altri significati, vedi Wittelsbach (disambigua). Questa voce o sezione sull'argomento storia di famiglia non cita le fonti necessarie o quelle presenti sono insufficienti. Puoi migliorare questa voce aggiungendo citazioni da fonti attendibili secondo le linee guida sull'uso delle fonti. Segui i suggerimenti del progetto di riferimento. Casato di WittelsbachStato Sacro Romano Impero Impero svedese Unione di Kalmar Reg...

Человек прямоходящий

† Человек прямоходящий Научная классификация Домен:ЭукариотыЦарство:ЖивотныеПодцарство:ЭуметазоиБез ранга:Двусторонне-симметричныеБез ранга:ВторичноротыеТип:ХордовыеПодтип:ПозвоночныеИнфратип:ЧелюстноротыеНадкласс:ЧетвероногиеКлада:АмниотыКлада:Синапсиды�...

Rugelach

Pastry RugelachChocolate rugelachTypePastryPlace of originPoland[1][2][3]Region or stateCentral EuropeCreated byAshkenazi JewsMain ingredientsDough: sour cream or cream cheeseFilling: any of raisins, walnuts, cinnamon, chocolate, marzipan, poppy seed, or fruit preserves Media: Rugelach Crescent-shaped rugelach Cut rugelach Rugelach (/ˈruːɡələx/ ROO-gəl-əkh; Yiddish: ראגעלעך, or Yiddish: רוגעלעך, romanized: rugelekh and Hebrew: רוג...

Newspaper endorsements in the 2016 United States presidential primaries

2016 U.S. presidential election Timeline General election debates Parties Polling national statewide by demographics international Newspaper endorsements primary general Russian interference Russia investigation origins counter-narrative Media coverage Social media International reactions Electors Recounts Faithless electors Vote count Republican Party Primaries Candidates Debates and forums Polling national statewide straw polls Endorsements Results Nominee VP candidate selection Convention ...

Pierce Johnson

American baseball player (born 1991) Baseball player Pierce JohnsonJohnson with the Hanshin Tigers in 2019Atlanta Braves – No. 38PitcherBorn: (1991-05-10) May 10, 1991 (age 32)Denver, Colorado, U.S.Bats: RightThrows: RightProfessional debutMLB: May 19, 2017, for the Chicago CubsNPB: March 29, 2019, for the Hanshin TigersMLB statistics (through April 8, 2024)Win–loss record14–16Earned run average4.11Strikeouts259Saves14NPB statistics (through 2019 seaso...

Episodi di Law & Order - Unità vittime speciali (dodicesima stagione)

Voce principale: Law & Order - Unità vittime speciali. La dodicesima stagione della serie televisiva Law & Order - Unità vittime speciali, composta da 24 episodi, è stata trasmessa per la prima volta negli Stati Uniti dalla NBC dal 22 settembre 2010 al 18 maggio 2011. L'episodio La prigioniera costituisce un crossover con la serie televisiva Law & Order: LA, debuttata negli Stati Uniti subito dopo la messa in onda di tale episodio, che ne introduce i protagonisti. In Italia è...

Socialist perspectives on abortion

The examples and perspective in this article may not represent a worldwide view of the subject. You may improve this article, discuss the issue on the talk page, or create a new article, as appropriate. (August 2020) (Learn how and when to remove this message) Socialist perspectives on abortion vary. In the first world, some socialist individuals believe in the legalisation and right to elective abortion care[1] as a public service, whereas others do not believe abortion should be leg...

Pakistani cricket team in Australia in 2016–17

International cricket tour Pakistani cricket team in Australia in 2016–17 Australia PakistanDates 8 December 2016 – 26 January 2017Captains Steve Smith Misbah-ul-Haq (Tests)Azhar Ali (1st, 4th & 5th ODIs)Mohammad Hafeez (2nd & 3rd ODIs)Test seriesResult Australia won the 3-match series 3–0Most runs Steve Smith (441) Azhar Ali (406)Most wickets Josh Hazlewood (15) Wahab Riaz (11)Player of the series Steve Smith (Aus)One Day International seriesResults Australia ...

Oatley, New South Wales

Suburb of Sydney, New South Wales, AustraliaOatleySydney, New South WalesOatley Clock Tower, Frederick StreetPopulation10,486 (2016 census)[1]Established1883Postcode(s)2223Elevation48 m (157 ft)Location18 km (11 mi) south of Sydney CBDLGA(s)Georges River CouncilState electorate(s)OatleyFederal division(s)Banks Suburbs around Oatley: Peakhurst Heights Mortdale Hurstville Grove Lugarno Oatley Connells Point Illawong Como Oyster Bay Frederick Street and Oat...

Alfred Shaker Historic District

Historic district in Maine, United States United States historic placeAlfred Shaker Historic DistrictU.S. National Register of Historic PlacesU.S. Historic district Main buildings, Shaker Village; from a 1915 postcardLocationShaker Hill Road. Alfred, MaineCoordinates43°29′55″N 70°42′41″W / 43.49861°N 70.71139°W / 43.49861; -70.71139Architectural styleFederalNRHP reference No.01000371[1]Added to NRHPApril 11, 2001 Shaker communityCommu...

Сьоме покоління ігрових систем

Ця стаття потребує додаткових посилань на джерела для поліпшення її перевірності. Будь ласка, допоможіть удосконалити цю статтю, додавши посилання на надійні (авторитетні) джерела. Зверніться на сторінку обговорення за поясненнями та допоможіть виправити недоліки. Мат...

Обер-гофмаршал

Н. П. Шереметев, обер-гофмаршал при дворе Павла I с 1797 года по 1800 год. Обер-гофмаршал (от нем. Oberhofmarschall) — придворный чин II класса в Табели о рангах, введенный в 1726 году[1]. До 1726 года чин назывался обер-маршал. Обер-гофмаршал принадлежал к первым чинам Двора. Ему �...

Parc Montsouris

Urban park in Paris Parc MontsourisAlley in Parc MontsourisTypeUrban parkLocation14th arrondissement, ParisCoordinates48°49′20″N 2°20′18″E / 48.82222°N 2.33833°E / 48.82222; 2.33833 (Parc Montsouris)Area38 acres (15 ha)Created2 October 1875Operated byDirection des Espaces Verts et de l'Environnement (DEVE)StatusOpen all yearPublic transit accessLocated near the RER station Cité Universitaire Parc Montsouris is a public park situated in so...

Bill of materials

List used in manufacturing An example of a BOM for a mechanical assembly (in German) A bill of materials or product structure (sometimes bill of material, BOM or associated list) is a list of the raw materials, sub-assemblies, intermediate assemblies, sub-components, parts, and the quantities of each needed to manufacture an end product. A BOM may be used for communication between manufacturing partners or confined to a single manufacturing plant. A bill of materials is often tied to a produc...

Jean-Baptiste Bessières

Vous lisez un « bon article » labellisé en 2015. Pour les articles homonymes, voir Bessières. Pour les autres membres de la famille, voir Famille Bessières. Jean-Baptiste BessièresDuc d'Istrie Le maréchal Jean-Baptiste Bessières. Lithographie de Delpech d'après un dessin de Jean-Baptiste Mauzaisse. Naissance 6 août 1768Prayssac, Quercy, royaume de France Décès 1er mai 1813 (à 44 ans)Weißenfels, royaume de SaxeMort au combat Origine Royaume de France Allégeance &...

American Austin Car Company

American Austin Car Company Rechtsform Company Gründung 1929 Auflösung 1934 Sitz Delaware, USA Branche Automobile American Austin als Coupé … als Roadster … als Kastenwagen … und als Pick-up American Austin Car Company war ein US-amerikanischer Hersteller von Automobilen.[1][2] Inhaltsverzeichnis 1 Unternehmensgeschichte 2 Fahrzeuge 3 Produktions- und Verkaufszahlen 4 Literatur 5 Weblinks 6 Einzelnachweise Unternehmensgeschichte Das Unternehmen wurde 1929 im US-Bunde...

Siluman Srigala Putih

Kota Baliwag

Dacia Sandero

Bira, Bonto Bahari, Bulukumba

Elang tiram

1960 AFC Asian Cup

2025 U Sports Men's Basketball Championship

List of states and union territories of India by fertility rate