跳去內容

三不互扣環

出自維基百科，自由嘅百科全書
三不互扣環
L6a4
交叉數6
Hyperbolic volume7.327724753
棍數9
Conway notation.1
A–B 表示法6Template:Sup sub
ThistlethwaiteL6a4
其他
alternating, hyperbolic

喺數學上，三不互扣環（Borromean rings[a]）係三條喺三維空間入面嘅簡單閉曲線。呢三條曲線喺拓撲學上係互相扣埋嘅，冇辦法將佢哋彼此分開；但係，只要將其中任何一條切斷或者拎走，剩低嗰兩條就會即刻變成兩個冇打結、亦冇互相扣住嘅環。

最常見嘅畫法，係將三個圓畫喺同一個平面上，個樣有啲似文氏圖（Venn diagram）。三個圓喺交叉位會輪流一條喺另一條上面、另一條喺下面。其他三條曲線只要喺拓撲學上同呢幅圖入面嘅三條曲線等價，就可以叫做一組博羅米安環。

「博羅米安環」呢個名，係嚟自意大利嘅博羅米奧家族（House of Borromeo）。呢個家族以前將呢種圓環圖案用喺佢哋嘅家族徽章上面。不過，類似博羅米安環嘅圖案其實喺好多唔同文化都有出現，例如北歐人同日本。喺基督教象徵入面，佢亦曾經用嚟代表三位一體；去到現代商業，Ballantine 啤酒亦曾經用博羅米安環做標誌，所以博羅米安環亦有「Ballantine rings」呢個叫法。

現實世界入面，亦有人用互相連住嘅 DNA 或者其他分子，整出真正嘅博羅米安環。物理學上亦有一啲同佢好似嘅現象，例如 Efimov 態同博羅米安原子核（Borromean nuclei）。呢兩種情況都有三個成分互相綁住，但奇怪嘅地方係：任何兩個成分單獨攞出嚟，都唔會互相綁住。

從幾何學角度嚟講，博羅米安環可以用互相扣住嘅橢圓整出嚟；亦可以利用正二十面體嘅頂點，整出由互相連住嘅黃金矩形組成嘅博羅米安環。不過，如果要求三條環都一定要係真正嘅圓，就冇可能喺三維空間入面整到博羅米安環。

另外，有人猜想，只要用任何一條唔係圓形嘅簡單閉曲線，再用佢嘅複製品，都可以整出博羅米安環。

喺結理論（knot theory）入面，可以透過數 Fox n-著色（Fox n-colorings）去證明博羅米安環係互相連結嘅。作為一種鏈結（link），博羅米安環屬於 **Brunnian、交替（alternating）、代數（algebraic）同雙曲（hyperbolic）**鏈結。

喺算術拓撲學（arithmetic topology）入面，有一啲素數三元組亦有同博羅米安環相似嘅「連結」性質。

定義同記號

[編輯]

數學出版物如果要定義博羅米安環，通常都會用鏈結圖（link diagram）嚟表示，即係喺一個平面上畫幾條曲線，再喺佢哋交叉嘅位置加上記號，表示每一個交叉位邊一條曲線、或者曲線嘅邊一部分，係經過另一條嘅上面或者下面。

有咗呢種圖之後，可以將佢變成三維空間入面嘅曲線系統：先將個平面嵌入三維空間，再按照幅圖嘅標示，喺每個交叉位將畫咗嘅曲線拉高或者壓低，令佢經過嵌入咗嘅平面上面或者下面。

最常用嘅博羅米安環圖，係由三個大小相同嘅圓組成。三個圓嘅圓心分別位於一個等邊三角形嘅三個頂點，而且三個圓要畫得夠近，令到佢哋嘅內部有一個共同交集。呢個畫法有啲似文氏圖入面嘅三個圓，亦似用嚟定義勒洛三角形（Reuleaux triangle）嘅三個圓。

沿住每一個圓逐個交叉位睇落去，佢嘅交叉會輪流喺上面同下面[2][3][4]。 亦可以用另一種等價嘅方式去描述三個圓之間嘅上下關係：每一個圓喺佢同第二個圓嘅兩個交叉位，都係行喺對方上面；而去到同第三個圓嘅兩個交叉位，就都係行喺對方下面[5]。

如果可以透過空間嘅連續變形，即係所謂嘅環境同痕變（ambient isotopy），將一個鏈結變成另一個鏈結，就話呢兩個鏈結係等價嘅。因此，「博羅米安環」亦可以指任何一組同呢個標準圖喺呢種意義下等價嘅鏈結[4]。

喺 The Knot Atlas 入面，博羅米安環嘅編號係「L6a4」。呢個記號表示佢係一個有六個交叉位、而且採用交替圖嘅鏈結；喺 Morwen Thistlethwaite 列出嘅所有最多有13個交叉位嘅素鏈結（prime links）之中，佢係五個六交叉位交替鏈結入面嘅第四個[6]。

至於 Dale Rolfsen 喺 1976 年出版嘅《Knots and Links》入面嘅結同鏈結表，亦延續咗 Alexander 同 Briggs 喺1920年代整落嘅較早列表，博羅米安環就用 **Alexander–Briggs 記號「6³₂」**表示。呢個記號即係話，佢係三個六交叉位、三個組成部分嘅鏈結入面，第二個列出嘅鏈結[6][7]。

博羅米安環嘅 Conway 記號「.1」，就係用一個簡化咗嘅方式去描述呢個鏈結嘅標準鏈結圖[8]。

攷

[編輯]
  1. ↑ Mackey & Mackay 1922 The Pronunciation of 10,000 Proper Names
  2. ↑ Gunn, Charles; Sullivan, John M. (2008), "The Borromean Rings: A video about the New IMU logo", 出自 Sarhangi, Reza; Séquin, Carlo H. (編), Bridges Leeuwarden: Mathematics, Music, Art, Architecture, Culture, London: Tarquin Publications, pp. 63–70, ISBN 978-0-9665201-9-4; see the video itself at "The Borromean Rings: A new logo for the IMU 互聯網檔案館嘅歸檔，歸檔日期2021-03-08." [w/video], International Mathematical Union
  3. ↑ Cromwell, Peter; Beltrami, Elisabetta; Rampichini, Marta (March 1998), "The Borromean rings", The mathematical tourist, The Mathematical Intelligencer, 20 (1): 53–62, doi:10.1007/bf03024401, S2CID 189888135
  4. 1 2 Aigner, Martin; Ziegler, Günter M. (2018), "Chapter 15: The Borromean Rings Don't Exist", Proofs from THE BOOK (第6版), Springer, pp. 99–106, doi:10.1007/978-3-662-57265-8_15, ISBN 978-3-662-57265-8
  5. ↑ Chamberland, Marc; Herman, Eugene A. (2015), "Rock-paper-scissors meets Borromean rings", The Mathematical Intelligencer, 37 (2): 20–25, doi:10.1007/s00283-014-9499-4, MR 3356112, S2CID 558993
  6. 1 2 Template:Knot Atlas
  7. ↑ Rolfsen, Dale (1990), Knots and Links, Mathematics Lecture Series,第7卷 (第2版), Publish or Perish, Inc., Houston, TX, p. 425, ISBN 0-914098-16-0, MR 1277811
  8. ↑ Conway, J. H. (1970), "An enumeration of knots and links, and some of their algebraic properties", Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967), Oxford: Pergamon, pp. 329–358, MR 0258014; see description of notation, pp. 332–333, and second line of table, p. 348.
  9. Abbott, Steve (July 1997), "Review of Not Knot and Supplement to Not Knot", The Mathematical Gazette, 81 (491): 340–342, doi:10.2307/3619248, JSTOR 3619248, S2CID 64589738
  10. Aravind, P. K. (1997), "Borromean entanglement of the GHZ state" (PDF), 出自 Cohen, R. S.; Horne, M.; Stachel, J. (編), Potentiality, Entanglement and Passion-at-a-Distance, Boston Studies in the Philosophy of Science, Springer, pp. 53–59, doi:10.1007/978-94-017-2732-7_4, MR 1739812
  11. Ashley, Clifford Warren (1993) [1944], The Ashley Book of Knots, Doubleday, p. 354, ISBN 978-0-385-04025-9
  12. Bai, Sheng; Wang, Weibiao (2020), "New criteria and constructions of Brunnian links", Journal of Knot Theory and Its Ramifications, 29 (13): 2043008, 27, arXiv:2006.10290, doi:10.1142/S0218216520430087, MR 4213076, S2CID 219792382
  13. Baird, Joseph L. (1970), "Unferth the þyle", Medium Ævum, 39 (1): 1–12, doi:10.2307/43631234, JSTOR 43631234, the stone bears also representations of three horns interlaced
  14. Bruns, Carson J.; Stoddart, J. Fraser (2011), "The mechanical bond: A work of art", 出自 Fabbrizzi, L. (編), Beauty in Chemistry, Topics in Current Chemistry,第323卷, Springer, pp. 19–72, doi:10.1007/128_2011_296, PMID 22183145
  15. Burgiel, H.; Franzblau, D. S.; Gutschera, K. R. (1996), "The mystery of the linked triangles", Mathematics Magazine, 69 (2): 94–102, doi:10.1080/0025570x.1996.11996399, JSTOR 2690662, MR 1394792
  16. Cantarella, Jason; Fu, Joseph H. G.; Kusner, Rob; Sullivan, John M.; Wrinkle, Nancy C. (2006), "Criticality for the Gehring link problem" (PDF), Geometry & Topology, 10 (4): 2055–2116, arXiv:math/0402212, doi:10.2140/gt.2006.10.2055, MR 2284052
  17. Cantarella, Jason; Kusner, Robert B.; Sullivan, John M. (2002), "On the minimum ropelength of knots and links" (PDF), Inventiones Mathematicae, 150 (2): 257–286, arXiv:math/0103224, Bibcode:2002InMat.150..257C, doi:10.1007/s00222-002-0234-y, MR 1933586, S2CID 730891
  18. Chichak, Kelly S.; Cantrill, Stuart J.; Pease, Anthony R.; Chiu, Sheng-Hsien; Cave, Gareth W. V.; Atwood, Jerry L.; Stoddart, J. Fraser (May 28, 2004), "Molecular Borromean rings" (PDF), Science, 304 (5675): 1308–1312, Bibcode:2004Sci...304.1308C, doi:10.1126/science.1096914, PMID 15166376, S2CID 45191675
  19. Crum Brown, Alexander (December 1885), "On a case of interlacing surfaces", Proceedings of the Royal Society of Edinburgh, 13: 382–386
  20. Didron, Adolphe Napoléon (1843), Iconographie Chrétienne (法文), Paris: Imprimerie Royale, pp. 568–569
  21. Freedman, Michael H.; Skora, Richard (1987), "Strange actions of groups on spheres", Journal of Differential Geometry, 25: 75–98, doi:10.4310/jdg/1214440725; see in particular Lemma 3.2, p. 89
  22. Freeman, Jim (2015), "Gathering clues from Margot's extraordinary objects", Tewkesbury Historical Society Bulletin, 24
  23. "African Borromean Rings", Mathematics and Knots, Centre for the Popularisation of Maths, University of Wales, 2002, 喺2021-02-12搵到
  24. Gardner, Martin (September 1961), "Surfaces with edges linked in the same way as the three rings of a well-known design", Mathematical Games, Scientific American, reprinted as Gardner, Martin (1991), "Knots and Borromean Rings", The Unexpected Hanging and Other Mathematical Diversions, University of Chicago Press, pp. 24–33; see also Gardner, Martin (September 1978), "The Toroids of Dr. Klonefake", Asimov's Science Fiction,第2卷第5號, p. 29
  25. Glick, Ned (September 1999), "The 3-ring symbol of Ballantine Beer", The mathematical tourist, The Mathematical Intelligencer, 21 (4): 15–16, doi:10.1007/bf03025332, S2CID 123311380
  26. Hilden, Hugh M.; Lozano, María Teresa; Montesinos, José María (1983), "The Whitehead link, the Borromean rings and the knot 946 are universal", Seminario Matemático de Barcelona, 34 (1): 19–28, MR 0747855
  27. Howards, Hugh Nelson (2013), "Forming the Borromean rings out of arbitrary polygonal unknots", Journal of Knot Theory and Its Ramifications, 22 (14): 1350083, 15, arXiv:1406.3370, doi:10.1142/S0218216513500831, MR 3190121, S2CID 119674622
  28. Jablan, Slavik V. (1999), "Are Borromean links so rare?", Proceedings of the 2nd International Katachi U Symmetry Symposium, Part 1 (Tsukuba, 1999), Forma, 14 (4): 269–277, MR 1770213
  29. Kraemer, T.; Mark, M.; Waldburger, P.; Danzl, J. G.; Chin, C.; Engeser, B.; Lange, A. D.; Pilch, K.; Jaakkola, A.; Nägerl, H.-C.; Grimm, R. (2006), "Evidence for Efimov quantum states in an ultracold gas of caesium atoms", Nature, 440 (7082): 315–318, arXiv:cond-mat/0512394, Bibcode:2006Natur.440..315K, doi:10.1038/nature04626, PMID 16541068, S2CID 4379828
  30. Kumar, Vijith; Pilati, Tullio; Terraneo, Giancarlo; Meyer, Franck; Metrangolo, Pierangelo; Resnati, Giuseppe (2017), "Halogen bonded Borromean networks by design: topology invariance and metric tuning in a library of multi-component systems", Chemical Science, 8 (3): 1801–1810, doi:10.1039/C6SC04478F, PMC 5477818, PMID 28694953
  31. Lakshminarayan, Arul (May 2007), "Borromean triangles and prime knots in an ancient temple", Resonance, 12 (5): 41–47, doi:10.1007/s12045-007-0049-7, S2CID 120259064
  32. Lindström, Bernt; Zetterström, Hans-Olov (1991), "Borromean circles are impossible", American Mathematical Monthly, 98 (4): 340–341, doi:10.2307/2323803, JSTOR 2323803. Note however that Gunn & Sullivan (2008) write that this reference "seems to incorrectly deal only with the case that the three-dimensional configuration has a projection homeomorphic to" the conventional three-circle drawing of the link.
  33. Mao, C.; Sun, W.; Seeman, N. C. (1997), "Assembly of Borromean rings from DNA", Nature, 386 (6621): 137–138, Bibcode:1997Natur.386..137M, doi:10.1038/386137b0, PMID 9062186, S2CID 4321733
  34. Morishita, Masanori (2010), "Analogies between knots and primes, 3-manifolds and number rings", Sugaku Expositions, 23 (1): 1–30, arXiv:0904.3399, MR 2605747
  35. Moskowitz, Clara (December 16, 2009), "Strange physical theory proved after nearly 40 years", Live Science
  36. Nanyes, Ollie (October 1993), "An elementary proof that the Borromean rings are non-splittable", American Mathematical Monthly, 100 (8): 786–789, doi:10.2307/2324788, JSTOR 2324788
  37. Natarajan, Ganapati; Mathew, Ammu; Negishi, Yuichi; Whetten, Robert L.; Pradeep, Thalappil (2015-12-02), "A unified framework for understanding the structure and modifications of atomically precise monolayer protected gold clusters", The Journal of Physical Chemistry C, 119 (49): 27768–27785, doi:10.1021/acs.jpcc.5b08193, ISSN 1932-7447
  38. "Mandala", Principia Discordia (第4版), March 1970, p. 43
  39. Ragland-Sullivan, Ellie; Milovanovic, Dragan (2004), "Introduction: Topologically Speaking", Lacan: Topologically Speaking, Other Press, ISBN 978-1-892746-76-4
  40. Ratcliffe, John G. (2006), "The Borromean rings complement", Foundations of Hyperbolic Manifolds, Graduate Texts in Mathematics,第149卷 (第2版), Springer, pp. 459–461, ISBN 978-0-387-33197-3, MR 2249478
  41. Riley, Robert (1979), "An elliptical path from parabolic representations to hyperbolic structures", 出自 Fenn, Roger (編), Topology of Low-Dimensional Manifolds: Proceedings of the Second Sussex Conference, 1977, Lecture Notes in Mathematics,第722卷, Springer, pp. 99–133, doi:10.1007/BFb0063194, ISBN 978-3-540-09506-4, MR 0547459
  42. Saiber, Arielle; Mbirika, aBa (2013), "The Three Giri of Paradiso 33" (PDF), Dante Studies (131): 237–272, JSTOR 43490498
  43. Schoeck, Richard J. (Spring 1968), "Mathematics and the languages of literary criticism", The Journal of Aesthetics and Art Criticism, 26 (3): 367–376, doi:10.2307/429121, JSTOR 429121
  44. Tanaka, K. (2010), "Observation of a Large Reaction Cross Section in the Drip-Line Nucleus 22C", Physical Review Letters, 104 (6), Bibcode:2010PhRvL.104f2701T, doi:10.1103/PhysRevLett.104.062701, PMID 20366816, S2CID 7951719 {{citation}}: Unknown parameter |article-number= ignored (help)
  45. Thistlethwaite, Morwen B. (1991), "On the algebraic part of an alternating link", Pacific Journal of Mathematics, 151 (2): 317–333, doi:10.2140/pjm.1991.151.317, MR 1132393
  46. William Thurston (March 2002), "7. Computation of volume", The Geometry and Topology of Three-Manifolds, p. 165, 原著 (PDF)喺2020-07-27歸檔, 喺2012-01-17搵到
  47. Tverberg, Helge (2010), "On Borromean rings" (PDF), The Mathematical Scientist, 35 (1): 57–60, MR 2668444, 原著 (PDF)喺2021-03-16歸檔, 喺2021-03-16搵到
  48. Uberti, R.; Janse van Rensburg, E. J.; Orlandini, E.; Tesi, M. C.; Whittington, S. G. (1998), "Minimal links in the cubic lattice", 出自 Whittington, Stuart G.; Sumners, Witt De; Lodge, Timothy (編), Topology and Geometry in Polymer Science, IMA Volumes in Mathematics and its Applications,第103卷, New York: Springer, pp. 89–100, doi:10.1007/978-1-4612-1712-1_9, MR 1655039; see Table 2, p. 97
  49. Veliks, Janis; Seifert, Helen M.; Frantz, Derik K.; Klosterman, Jeremy K.; Tseng, Jui-Chang; Linden, Anthony; Siegel, Jay S. (2016), "Towards the molecular Borromean link with three unequal rings: double-threaded ruthenium(ii) ring-in-ring complexes", Organic Chemistry Frontiers, 3 (6): 667–672, doi:10.1039/c6qo00025h
  50. Vogel, Denis (2005), Masseyprodukte in der Galoiskohomologie von Zahlkörpern [Massey products in the Galois cohomology of number fields], Mathematisches Institut, Georg-August-Universität Göttingen: Seminars Winter Term 2004/2005, Göttingen: Universitätsdrucke Göttingen, pp. 93–98, doi:10.11588/heidok.00004418, MR 2206880
引用錯誤 喺 <references> 入面定義、名稱為 "notknot" 嘅 <ref> 標籤喺之前嘅正文冇被使用。引用錯誤 喺 <references> 入面定義、名稱為 "ghz" 嘅 <ref> 標籤喺之前嘅正文冇被使用。引用錯誤 喺 <references> 入面定義、名稱為 "ashley" 嘅 <ref> 標籤喺之前嘅正文冇被使用。引用錯誤 喺 <references> 入面定義、名稱為 "brunnian" 嘅 <ref> 標籤喺之前嘅正文冇被使用。引用錯誤 喺 <references> 入面定義、名稱為 "unferth" 嘅 <ref> 標籤喺之前嘅正文冇被使用。引用錯誤 喺 <references> 入面定義、名稱為 "mechbond" 嘅 <ref> 標籤喺之前嘅正文冇被使用。引用錯誤 喺 <references> 入面定義、名稱為 "triangles" 嘅 <ref> 標籤喺之前嘅正文冇被使用。引用錯誤 喺 <references> 入面定義、名稱為 "criticality" 嘅 <ref> 標籤喺之前嘅正文冇被使用。引用錯誤 喺 <references> 入面定義、名稱為 "4arc-rope" 嘅 <ref> 標籤喺之前嘅正文冇被使用。引用錯誤 喺 <references> 入面定義、名稱為 "stoddart" 嘅 <ref> 標籤喺之前嘅正文冇被使用。引用錯誤 喺 <references> 入面定義、名稱為 "crumbrown" 嘅 <ref> 標籤喺之前嘅正文冇被使用。引用錯誤 喺 <references> 入面定義、名稱為 "chartres" 嘅 <ref> 標籤喺之前嘅正文冇被使用。引用錯誤 喺 <references> 入面定義、名稱為 "strange" 嘅 <ref> 標籤喺之前嘅正文冇被使用。引用錯誤 喺 <references> 入面定義、名稱為 "tripod1" 嘅 <ref> 標籤喺之前嘅正文冇被使用。引用錯誤 喺 <references> 入面定義、名稱為 "tripod2" 嘅 <ref> 標籤喺之前嘅正文冇被使用。引用錯誤 喺 <references> 入面定義、名稱為 "gardner" 嘅 <ref> 標籤喺之前嘅正文冇被使用。引用錯誤 喺 <references> 入面定義、名稱為 "glick" 嘅 <ref> 標籤喺之前嘅正文冇被使用。引用錯誤 喺 <references> 入面定義、名稱為 "universal" 嘅 <ref> 標籤喺之前嘅正文冇被使用。引用錯誤 喺 <references> 入面定義、名稱為 "polygonal" 嘅 <ref> 標籤喺之前嘅正文冇被使用。引用錯誤 喺 <references> 入面定義、名稱為 "rare" 嘅 <ref> 標籤喺之前嘅正文冇被使用。引用錯誤 喺 <references> 入面定義、名稱為 "efimov" 嘅 <ref> 標籤喺之前嘅正文冇被使用。引用錯誤 喺 <references> 入面定義、名稱為 "halogen" 嘅 <ref> 標籤喺之前嘅正文冇被使用。引用錯誤 喺 <references> 入面定義、名稱為 "enneagram" 嘅 <ref> 標籤喺之前嘅正文冇被使用。引用錯誤 喺 <references> 入面定義、名稱為 "impossible" 嘅 <ref> 標籤喺之前嘅正文冇被使用。引用錯誤 喺 <references> 入面定義、名稱為 "dna" 嘅 <ref> 標籤喺之前嘅正文冇被使用。引用錯誤 喺 <references> 入面定義、名稱為 "analogies" 嘅 <ref> 標籤喺之前嘅正文冇被使用。引用錯誤 喺 <references> 入面定義、名稱為 "physical" 嘅 <ref> 標籤喺之前嘅正文冇被使用。引用錯誤 喺 <references> 入面定義、名稱為 "nanyes" 嘅 <ref> 標籤喺之前嘅正文冇被使用。引用錯誤 喺 <references> 入面定義、名稱為 "thiolate" 嘅 <ref> 標籤喺之前嘅正文冇被使用。引用錯誤 喺 <references> 入面定義、名稱為 "discord" 嘅 <ref> 標籤喺之前嘅正文冇被使用。引用錯誤 喺 <references> 入面定義、名稱為 "lacan" 嘅 <ref> 標籤喺之前嘅正文冇被使用。引用錯誤 喺 <references> 入面定義、名稱為 "ratcliffe" 嘅 <ref> 標籤喺之前嘅正文冇被使用。引用錯誤 喺 <references> 入面定義、名稱為 "riley" 嘅 <ref> 標籤喺之前嘅正文冇被使用。引用錯誤 喺 <references> 入面定義、名稱為 "dante" 嘅 <ref> 標籤喺之前嘅正文冇被使用。引用錯誤 喺 <references> 入面定義、名稱為 "litcrit" 嘅 <ref> 標籤喺之前嘅正文冇被使用。引用錯誤 喺 <references> 入面定義、名稱為 "nucleus" 嘅 <ref> 標籤喺之前嘅正文冇被使用。引用錯誤 喺 <references> 入面定義、名稱為 "algebraic" 嘅 <ref> 標籤喺之前嘅正文冇被使用。引用錯誤 喺 <references> 入面定義、名稱為 "volume" 嘅 <ref> 標籤喺之前嘅正文冇被使用。引用錯誤 喺 <references> 入面定義、名稱為 "inversive" 嘅 <ref> 標籤喺之前嘅正文冇被使用。引用錯誤 喺 <references> 入面定義、名稱為 "lattice" 嘅 <ref> 標籤喺之前嘅正文冇被使用。引用錯誤 喺 <references> 入面定義、名稱為 "unequal" 嘅 <ref> 標籤喺之前嘅正文冇被使用。引用錯誤 喺 <references> 入面定義、名稱為 "massey" 嘅 <ref> 標籤喺之前嘅正文冇被使用。

拎出去

[編輯]

Template:Knot theory

  1. ↑ Pronounced /bɒroʊˈmiːən/[1]
引用錯誤 存在組名為 "lower-alpha" 嘅 <ref> 標籤，但搵唔到對應嘅 標籤。
由「https://zh-yue.wikipedia.org/w/index.php?title=三不互扣環&oldid=2456275」收