查看“︁黃金菱形”︁的源代码

[[Image:GoldenRhombus.svg|240px|thumb|黃金菱形]]
在[[幾何學]]中，'''黃金菱形'''是指兩對角線長度的比值呈[[黃金比例]]的菱形<ref name=senechal>{{citation
 | last = Senechal | first = Marjorie
 | editor1-last = Davis | editor1-first = Chandler
 | editor2-last = Ellers | editor2-first = Erich W.
 | contribution = Donald and the golden rhombohedra
 | isbn = 0-8218-3722-2
 | mr = 2209027
 | pages = 159–177
 | publisher = American Mathematical Society, Providence, RI
 | title = The Coxeter Legacy
 | year = 2006}}</ref>：
:<math>{D\over d} = \varphi = {{1+\sqrt5}\over2} \approx 1.618~034</math>
其中<math>D</math>為長對角線的長度、<math>d</math>為短對角線的長度。而由[[黃金矩形]]中取到的[[伐里農平行四邊形]]皆為黃金菱形。<ref name=senechal/>
有數種知名的多面體皆由黃金菱形組成，例如{{link-en|比林斯基十二面體|Bilinski_dodecahedron}}<ref>{{cite journal | author=Branko Grünbaum | title=The Bilinski Dodecahedron and Assorted Parallelohedra, Zonohedra, Monohedra, Isozonohedra, and Otherhedra | year=2010 | volume=32 | issue=4 | pages=5–15 | url=https://dlib.lib.washington.edu/researchworks/bitstream/handle/1773/15593/Bilinski_dodecahedron.pdf | deadurl=yes | archiveurl=https://web.archive.org/web/20150402132516/https://dlib.lib.washington.edu/researchworks/bitstream/handle/1773/15593/Bilinski_dodecahedron.pdf | archivedate=2015-04-02 | df= | access-date=2020-08-04 }}</ref><ref>H.S.M Coxeter, "Regular polytopes", Dover publications,  1973.</ref>、[[菱形三十面體]]<ref name="RhombicTriacontahedron.Mathworld">{{cite mathworld|urlname=RhombicTriacontahedron|title=Rhombic Triacontahedron}}</ref>等。特別地，有另一種菱形也與[[黃金比例]]相關聯，即{{link-en|潘洛斯鑲嵌|Penrose_tiling}}中的菱形，但不同之處在於，{{link-en|潘洛斯鑲嵌|Penrose_tiling}}中的菱形是邊長與對角線的比為黃金比例，而黃金菱形則是指兩對角線比值為黃金比例的菱形。<ref>{{citation
 | last = Livio | first = Mario
 | location = New York
 | page = 206
 | publisher = Broadway Books
 | title = The Golden Ratio: The Story of Phi, the World's Most Astonishing Number
 | year = 2002}}</ref>

== 性質 ==
{{Main|菱形}}
黃金菱形是菱形中的一個特例，其基本性質與菱形相同。以下討論黃金菱形的特別性質。
=== 內角 ===
黃金菱形的內角為<ref name=ogawa>{{citation
 | last = Ogawa | first = Tohru
 | date = January 1987
 | doi = 10.4028/www.scientific.net/msf.22-24.187
 | journal = Materials Science Forum
 | pages = 187–200
 | title = Symmetry of three-dimensional quasicrystals
 | volume = 22-24}}. See in particular table 1, p. 188.</ref>：
*銳角：<math>\alpha=2\arctan{1\over\varphi}</math> ;
:<math>\alpha=\arctan{{2\over\varphi}\over{1-({1\over\varphi})^2}}=\arctan{{2\over\varphi}\over{1\over\varphi}}=\arctan2\approx63.43495^\circ.</math>
:
*鈍角：<math>\beta=2\arctan\varphi=\pi-\arctan2\approx116.56505^\circ,</math>
:這個角度值與[[正十二面體]]相同<ref>{{citation
 | last = Gevay | first = G.
 | date = June 1993
 | doi = 10.1080/01411599308210255
 | issue = 1-3
 | journal = Phase Transitions
 | pages = 47–50
 | title = Non-metallic quasicrystals: Hypothesis or reality?
 | volume = 44}}</ref>
=== 邊長與對角線長 ===
由於菱形也是一種[[平行四邊形]]<ref>{{Cite book | author=Zalman Usiskin and Jennifer Griffin | title="The Classification of Quadrilaterals. A Study of Definition" | url=https://books.google.com/books?id=ff0nDwAAQBAJ&printsec=frontcover#v=onepage&q=rhombus&f=false | archive-url=https://web.archive.org/web/20200226195300/https://books.google.com/books?id=ff0nDwAAQBAJ&printsec=frontcover#v=onepage&q=rhombus&f=false | archive-date=2020-02-26 | publisher=Information Age Publishing | year=2008 | page=pp. 55-56 | access-date=2020-08-15 | dead-url=no }}</ref>，因此黃金菱形的邊長與對角線長可以用[[平行四邊形恆等式]]得出<ref>{{cite mathworld|id=Rhombus|title=Rhombus}}</ref>：

黃金菱形的邊長<math>a</math>與對角線長<math>d</math>具有以下關係：
*<math>a={1\over2}\sqrt{d^2+(\varphi d)^2}={1\over2}\sqrt{1+\varphi^2}~d={{\sqrt{2+\varphi}}\over2}~d={1\over4}\sqrt{10+2\sqrt5}~d\approx0.95106~d~.~</math> 
:因此，可以用<math>a</math>來表示長對角線<math>D</math>與短對角線<math>d</math>：<ref name=ogawa/>
*<math>d={2a\over\sqrt{2+\varphi}}=2\sqrt{{3-\varphi}\over5}~a=\sqrt{2-{2\over\sqrt5}}~a\approx1.05146~a~.</math>
*<math>D={2\varphi a\over\sqrt{2+\varphi}}=2\sqrt{{2+\varphi}\over5}~a=\sqrt{2+{2\over\sqrt5}}~a\approx1.70130~a~.</math>
=== 面積 ===
已知短對角線長<math>d</math>時，黃金菱形的的面積為<ref name=Mathworld/>：
:<math>A = {{(\varphi d)\cdot d}\over2} = {{\varphi}\over2}~d^2 = {{1+\sqrt5}\over4}~d^2 \approx 0.80902~d^2~.</math>
已知邊長為<math>a</math>時，黃金菱形的的面積為<ref name=ogawa/><ref name=Mathworld>{{cite mathworld|urlname=GoldenRhombus |title=Golden Rhombus}}</ref>：
:<math>A = (\sin(\arctan2))~a^2 = {2\over\sqrt5}~a^2 \approx 0.89443~a^2~.</math>

==在多面體中==
黃金菱形出現在許多高對稱性的多面體中，例如[[菱形三十面體]]（[[截半二十面体]]的對偶多面體）<ref name="RhombicTriacontahedron.Mathworld">{{cite mathworld|urlname=RhombicTriacontahedron|title=Rhombic Triacontahedron}}</ref>、[[菱形六十面體]]（[[菱形三十面體]]的星形化體）<ref>{{cite mathworld|title=Rhombic hexecontahedron|urlname=RhombicHexecontahedron}} </ref>。黃金菱形也構成了許多知名的多面體，例如{{link-wd|Q98398187}}、{{link-en|比林斯基十二面體|Bilinski dodecahedron}}和[[菱形二十面體]]等。由全部皆由黃金菱形組成的凸多面體僅有兩種黃金菱形六面體、比林斯基十二面體、菱形二十面體以及菱形三十面體五種。而不考慮[[凹凸性 (幾何)|凹凸性]]（即允許非凸多面體），則有無限多種多面體可以包含黃金菱形<ref>{{citation
 |last        = Grünbaum
 |first       = Branko
 |doi         = 10.1007/s00283-010-9138-7
 |issue       = 4
 |journal     = The Mathematical Intelligencer
 |mr          = 2747698
 |pages       = 5–15
 |title       = The Bilinski dodecahedron and assorted parallelohedra, zonohedra, monohedra, isozonohedra, and otherhedra
 |url         = https://dlib.lib.washington.edu/researchworks/bitstream/handle/1773/15593/Bilinski_dodecahedron.pdf
 |volume      = 32
 |year        = 2010
 |archiveurl  = https://web.archive.org/web/20150402132516/https://dlib.lib.washington.edu/researchworks/bitstream/handle/1773/15593/Bilinski_dodecahedron.pdf
 |archivedate = 2015-04-02
 |accessdate  = 2020-08-04
 |dead-url    = no
}}.</ref>。

<gallery>
File:Acute_golden_rhombohedron.png|銳角{{link-wd|Q98398187}}
File:Flat_golden_rhombohedron.png|鈍角黃金菱形六面體
File:Bilinski dodecahedron.png|{{link-en|比林斯基十二面體|Bilinski dodecahedron}}
File:Rhombic icosahedron.png|[[菱形二十面體]]
File:Rhombictriacontahedron.svg|[[菱形三十面體]]
File:Rhombic hexecontahedron.png|[[菱形六十面體]]
</gallery>

==參見==
* [[黄金三角形]]

== 參考文獻 ==
{{reflist}}

== 外部連結 ==
* [https://azimpremjiuniversity.edu.in/SitePages/resources-ara-march-2017-from-a-golden-rectangle-to-a-golden-quadrilateral.aspx From a golden rectangle to a golden rhombus and other golden quadrilaterals] {{Wayback|url=https://azimpremjiuniversity.edu.in/SitePages/resources-ara-march-2017-from-a-golden-rectangle-to-a-golden-quadrilateral.aspx |date=20181023160438 }} Explores some aspects of possible golden rhombi (and golden parallelograms)

{{貴金屬比例}}

[[Category:四邊形]]
[[Category:黃金比例]]