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| Genu = |
| Genu = |
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| Face_type = 12个[[正五边形]]{5} |
| Face_type = 12个[[正五边形]]{5} |
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| Vertice_type = (5<sup>5</sup>)/2<ref>{{Cite Web | url = http://www.software3d.com/GreatDod.php | title = Great Dodecahedron | website = software3d.com | author = Robert Webb | accessdate = 2017-07-25 |
| Vertice_type = (5<sup>5</sup>)/2<ref>{{Cite Web | url = http://www.software3d.com/GreatDod.php | title = Great Dodecahedron | website = software3d.com | author = Robert Webb | accessdate = 2017-07-25 }}</ref> |
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| Schläfli = {5,<sup>5</sup>/<sub>2</sub>} |
| Schläfli = {5,<sup>5</sup>/<sub>2</sub>} |
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| Wythoff = <span class="sfrac nowrap" style="display:inline-block; vertical-align:-0.5em; font-size:85%; text-align:center;"><span style="display:block; line-height:1em; margin:0 0.1em;">5</span><span style="display:block; line-height:1em; margin:0 0.1em; border-top:1px solid;">2</span></span> | 2 5 |
| Wythoff = <span class="sfrac nowrap" style="display:inline-block; vertical-align:-0.5em; font-size:85%; text-align:center;"><span style="display:block; line-height:1em; margin:0 0.1em;">5</span><span style="display:block; line-height:1em; margin:0 0.1em; border-top:1px solid;">2</span></span> | 2 5 |
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| net_image = |
| net_image = |
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在[[几何学]]中,'''大十二面体'''<ref name="mathworld"/>又称为'''第二星形正十二面体'''<ref name="Polyhedron Models, Wenninger 1989"/><ref name="mathworld, Dodecahedron Stellations"/>,是一个由6对互相[[平行]]的[[正五边形]]组成的非凸正多面体,同时也是一种[[星形正多面体]]<ref name="richeson2008euler">{{Cite book | title=Euler's Gem: The Polyhedron Formula and the Birth of Topology | author=Richeson, D.S. | isbn=9780691126777 | lccn=2008062108 | | year=2008 | publisher=Princeton University Press | page=151 }}{{dead link|date=2018年2月 |bot=InternetArchiveBot |fix-attempted=yes }}</ref>,其外形有如内有星形图案的正二十面体或每面内凹三角锥的[[正二十面体]]<ref name = "hilton2010mathematical"/>,是三种[[星形十二面体]]之一<ref name="Polyhedron Models, Wenninger 1989">{{cite book|author = Wenninger, M. J.| title = Polyhedron Models | publisher = New York: Cambridge University Press| page = pp. 35 and 38-40 | year = 1989}}</ref><ref name="mathworld, Dodecahedron Stellations">{{mathworld | urlname = DodecahedronStellations| title = Dodecahedron Stellations }}</ref>。其顶点的布局与正二十面体相同,但边的链接方式不同,因此可以视为[[正二十面体]]经过{{link-en|刻面|faceting}}后的多面体<ref name="mathworld">{{mathworld | urlname = GreatDodecahedron| title = Great Dodecahedron }}</ref>,[[对偶多面体]]为[[小星形十二面体]]。这个多面体被认为是由{{link-en|路易·龐索|Louis Poinsot}}在1810年发现<ref>{{link-en|路易·龐索|Louis Poinsot|Louis Poinsot}}, Memoire sur les polygones et polyèdres. ''J. de l'École Polytechnique'' '''9''', pp. 16–48, 1810.</ref><ref name="senechal2013shaping">{{Cite book|title=Shaping Space: Exploring Polyhedra in Nature, Art, and the Geometrical Imagination|author=Senechal, M.|isbn=9780387927145|lccn=2013932331|series=EBSCOhost ebooks online||year=2013|publisher=Springer New York|page=60|access-date=2017-09-07 |
在[[几何学]]中,'''大十二面体'''<ref name="mathworld"/>又称为'''第二星形正十二面体'''<ref name="Polyhedron Models, Wenninger 1989"/><ref name="mathworld, Dodecahedron Stellations"/>,是一个由6对互相[[平行]]的[[正五边形]]组成的非凸正多面体,同时也是一种[[星形正多面体]]<ref name="richeson2008euler">{{Cite book | title=Euler's Gem: The Polyhedron Formula and the Birth of Topology | author=Richeson, D.S. | isbn=9780691126777 | lccn=2008062108 | | year=2008 | publisher=Princeton University Press | page=151 }}{{dead link|date=2018年2月 |bot=InternetArchiveBot |fix-attempted=yes }}</ref>,其外形有如内有星形图案的正二十面体或每面内凹三角锥的[[正二十面体]]<ref name = "hilton2010mathematical"/>,是三种[[星形十二面体]]之一<ref name="Polyhedron Models, Wenninger 1989">{{cite book|author = Wenninger, M. J.| title = Polyhedron Models | publisher = New York: Cambridge University Press| page = pp. 35 and 38-40 | year = 1989}}</ref><ref name="mathworld, Dodecahedron Stellations">{{mathworld | urlname = DodecahedronStellations| title = Dodecahedron Stellations }}</ref>。其顶点的布局与正二十面体相同,但边的链接方式不同,因此可以视为[[正二十面体]]经过{{link-en|刻面|faceting}}后的多面体<ref name="mathworld">{{mathworld | urlname = GreatDodecahedron| title = Great Dodecahedron }}</ref>,[[对偶多面体]]为[[小星形十二面体]]。这个多面体被认为是由{{link-en|路易·龐索|Louis Poinsot}}在1810年发现<ref>{{link-en|路易·龐索|Louis Poinsot|Louis Poinsot}}, Memoire sur les polygones et polyèdres. ''J. de l'École Polytechnique'' '''9''', pp. 16–48, 1810.</ref><ref name="senechal2013shaping">{{Cite book|title=Shaping Space: Exploring Polyhedra in Nature, Art, and the Geometrical Imagination|author=Senechal, M.|isbn=9780387927145|lccn=2013932331|series=EBSCOhost ebooks online||year=2013|publisher=Springer New York|page=60|access-date=2017-09-07}}</ref>,虽然在{{link-en|温佐·雅姆尼策尔|Wenzel Jamnitzer}}于1568年出版的著作《Perspectiva Corporum Regularium》中有一幅形狀非常类似大十二面体的图画<ref>{{Cite book|author=Wenzel Jamnitzer|title=Perspectiva Corporum Regularium||year=1568}}</ref>。1983年时,温尼尔在他的书《[[温尼尔多面体模型列表|多面体模型]]》中列出许多星形多面体模型,其中也收录了此种形狀,并給予编号W<sub>21</sub><ref>{{cite book | author = {{link-en|马格努斯·J·温尼尔|Magnus J. Wenninger|Wenninger, Magnus}} | title = Polyhedron Models | publisher = Cambridge University Press | year = 1974 | isbn = 0-521-09859-9 }}</ref>。 |
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== 性质 == |
== 性质 == |
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大十二面体是4个非凸正多面体之一,具二十面体的对称性<ref>{{Cite Web | author = Roman E. Maeder | url = https://www.mathconsult.ch/static/unipoly/35.html | title = 35: great dodecahedron | website = mathconsult.ch | accessdate = 2017-07-25 |
大十二面体是4个非凸正多面体之一,具二十面体的对称性<ref>{{Cite Web | author = Roman E. Maeder | url = https://www.mathconsult.ch/static/unipoly/35.html | title = 35: great dodecahedron | website = mathconsult.ch | accessdate = 2017-07-25 }}</ref>,由12个面<ref>{{Cite Web | url = http://www.theuniverseismental.com/great-dodecahedron | title = great-dodecahedron | website = the universeis mental | accessdate = 2017-07-25 }}</ref>、30条边和12个顶点所组成<ref name="bulatov.org">{{Cite web|url=http://bulatov.org/polyhedra/uniform/u40.html|title=Great Dodecahedron|publisher=bulatov.org|access-date=2017-07-25}}</ref><ref name = "Gijs Korthals Altes">{{Cite Web | author = Gijs Korthals Altes | url = http://www.korthalsaltes.com/model.php?name_en=great%20dodecahedron | title = Paper Great Dodecahedron | website = korthalsaltes.com | accessdate = 2017-07-25 }}</ref>,其12个面皆为正五边形面,其中12个五边形中有6对互相平行的五边形。其每个顶角都是由5个五边形以五角星的路径构成的五面角,因此在[[施莱夫利符号]]中可利用{5,5/2}来表示<ref name = "lib.msu.edu">{{Cite web | | title = Great Dodecahedron | publisher = 密西根州立大学 | date = 1999-05-25 | author = Eric W. Weisstein | accessdate = 2017-07-25 }}</ref>,意为此立体的所有顶角组成的面皆为五边形(施莱夫利符号:{5}),并且以五角星(施莱夫利符号:{5/2})的方式构成。而在{{link-en|考克斯特-迪肯符号|Coxeter-Dynkin digram|考克斯特符号}}中则利用{{CDD|node_1|5|node|5|rat|d2|node}}来表示<ref name="book_buekenhout_2013_diagram">{{Cite book |
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|1 = |
|1 = |
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|title = Diagram Geometry: Related to Classical Groups and Buildings |
|title = Diagram Geometry: Related to Classical Groups and Buildings |
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第41行: | 第41行: | ||
|year = 2013 |
|year = 2013 |
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|publisher = Springer Berlin Heidelberg |
|publisher = Springer Berlin Heidelberg |
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}}{{dead link|date=2018年2月 |bot=InternetArchiveBot |fix-attempted=yes }}</ref>。在抽象几何学中,大十二面体对应到一个亏格为4的五阶五边形正则地区图(施莱夫利符号:{5,5}),同时,其对偶多面体[[小星形十二面体]]亦对应到相同的正则地区图<ref>{{cite web | url = http://www.weddslist.com/rmdb/pages/stellation/stellation3.php | title = Stellation of Regular Maps | publisher = Regular Map database, weddslist.com | accessdate = 2021-07-30 |
}}{{dead link|date=2018年2月 |bot=InternetArchiveBot |fix-attempted=yes }}</ref>。在抽象几何学中,大十二面体对应到一个亏格为4的五阶五边形正则地区图(施莱夫利符号:{5,5}),同时,其对偶多面体[[小星形十二面体]]亦对应到相同的正则地区图<ref>{{cite web | url = http://www.weddslist.com/rmdb/pages/stellation/stellation3.php | title = Stellation of Regular Maps | publisher = Regular Map database, weddslist.com | accessdate = 2021-07-30 }}</ref>,因此这个正则地区图是一个自身对偶的几何结构。<ref name="Regular Map small stellated dodecahedron, great dodecahedron embedded" >{{cite web | url = http://www.weddslist.com/rmdb/map.php?a=R4.6 | title = S4:{5,5} | publisher=Regular Map database - map details, weddslist.com | accessdate=2021-07-30}}</ref> |
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{| class=wikitable |
{| class=wikitable |
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!{{link-en|星狀图|Stellation diagram}}||外观||星狀核||[[凸包]] |
!{{link-en|星狀图|Stellation diagram}}||外观||星狀核||[[凸包]] |
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=== 二面角 === |
=== 二面角 === |
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大十二面体是一种[[星形正多面体]]<ref>Coxeter, ''Star polytopes and the Schläfli function f(α,β,γ)'' p. 121 1. The Kepler–Poinsot polyhedra</ref>,因此大十二面体仅有一种二面角,其值为五平方根倒数的反餘弦值<ref name="dmccooey">{{cite web | url = http://dmccooey.com/polyhedra/GreatDodecahedron.html | title = Kepler-Poinsot Solids: Great Dodecahedron | publisher = dmccooey.com | access-date = 2017-07-25 |
大十二面体是一种[[星形正多面体]]<ref>Coxeter, ''Star polytopes and the Schläfli function f(α,β,γ)'' p. 121 1. The Kepler–Poinsot polyhedra</ref>,因此大十二面体仅有一种二面角,其值为五平方根倒数的反餘弦值<ref name="dmccooey">{{cite web | url = http://dmccooey.com/polyhedra/GreatDodecahedron.html | title = Kepler-Poinsot Solids: Great Dodecahedron | publisher = dmccooey.com | access-date = 2017-07-25 }}</ref>: |
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:<math>\cos^{-1} \frac{\sqrt 5}{5} \approx 63.434948823^\circ</math> |
:<math>\cos^{-1} \frac{\sqrt 5}{5} \approx 63.434948823^\circ</math> |
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=== 顶点坐标 === |
=== 顶点坐标 === |
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由于大十二面体的[[凸包]]为[[正二十面体]]<ref name="mathworld"/>,且无顶点落在凸包内,因此大十二面体的顶点坐标会与相同边长的正二十面体相同,边长为单位长、[[几何中心]]在原点,则其为:<ref name="dmccooeydata">{{Cite web|url=http://dmccooey.com/polyhedra/GreatDodecahedron.txt|title=Data of Great Dodecahedron|publisher=dmccooey.com|access-date=2017-07-25 |
由于大十二面体的[[凸包]]为[[正二十面体]]<ref name="mathworld"/>,且无顶点落在凸包内,因此大十二面体的顶点坐标会与相同边长的正二十面体相同,边长为单位长、[[几何中心]]在原点,则其为:<ref name="dmccooeydata">{{Cite web|url=http://dmccooey.com/polyhedra/GreatDodecahedron.txt|title=Data of Great Dodecahedron|publisher=dmccooey.com|access-date=2017-07-25}}</ref> |
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:<math>(0, \,\pm 1, \,\pm \frac{1+\sqrt{5}}{4})</math>、 |
:<math>(0, \,\pm 1, \,\pm \frac{1+\sqrt{5}}{4})</math>、 |
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:<math>(\pm 1, \,\pm \frac{1+\sqrt{5}}{4}, \, 0)</math>、 |
:<math>(\pm 1, \,\pm \frac{1+\sqrt{5}}{4}, \, 0)</math>、 |
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|image1=[https://pbs.twimg.com/media/DC5nel6UQAADNIv.jpg:large 动画《游戏人生》中星杯的外形]<ref name=榎宫祐20>{{cite web|author1=榎宫祐||title=NO GAME NO LIFE 游戏人生[12]|url=https://ani.gamer.com.tw/animeVideo.php?sn=7955|website=巴哈母特动画疯|accessdate=2017-09-07|language=zh}}</ref> |
|image1=[https://pbs.twimg.com/media/DC5nel6UQAADNIv.jpg:large 动画《游戏人生》中星杯的外形]<ref name=榎宫祐20>{{cite web|author1=榎宫祐||title=NO GAME NO LIFE 游戏人生[12]|url=https://ani.gamer.com.tw/animeVideo.php?sn=7955|website=巴哈母特动画疯|accessdate=2017-09-07|language=zh}}</ref> |
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}} |
}} |
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星形正多面体经常出现在艺术创作中,部分小说也有使用大十二面体进行创作,如《[[NO GAME NO LIFE 游戏人生|游戏人生]]》<ref name="榎宫祐2014"/>。除了艺术创作外,常见文化也有关于大十二面体的使用,例如部分的[[魔术方块]]之外型<ref name="Alexander's Star - Jaap's Puzzle Page">{{Cite Web | url = http://www.jaapsch.net/puzzles/alexandr.htm | title = Alexander's Star - Jaap's Puzzle Page | website = jaapsch.net | accessdate = 2017-07-25 |
星形正多面体经常出现在艺术创作中,部分小说也有使用大十二面体进行创作,如《[[NO GAME NO LIFE 游戏人生|游戏人生]]》<ref name="榎宫祐2014"/>。除了艺术创作外,常见文化也有关于大十二面体的使用,例如部分的[[魔术方块]]之外型<ref name="Alexander's Star - Jaap's Puzzle Page">{{Cite Web | url = http://www.jaapsch.net/puzzles/alexandr.htm | title = Alexander's Star - Jaap's Puzzle Page | website = jaapsch.net | accessdate = 2017-07-25 }}</ref>,以及其投影图曾作为视觉化的相关实验性教材<ref>{{Cite web | url = http://www.transum.org/Software/Thinking_Skills/Dodecahedron.asp | title = THE GREAT DODECAHEDRON | website = transum.org | accessdate = 2017-07-25 }}</ref>。 |
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=== 在常见文化中 === |
=== 在常见文化中 === |
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* 魔术方块[[亚历山大之星]]的外形也是大十二面体<ref name="Alexander's Star - Jaap's Puzzle Page"/>。 |
* 魔术方块[[亚历山大之星]]的外形也是大十二面体<ref name="Alexander's Star - Jaap's Puzzle Page"/>。 |
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=== 在小说中 === |
=== 在小说中 === |
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* 在小说《[[NO GAME NO LIFE 游戏人生|游戏人生]]》中,唯一神[[NO GAME NO LIFE 游戏人生#特图|特图]]持有的星杯外形为大十二面体<ref name="榎宫祐2014">{{Cite book | title = 《NO GAME NO LIFE 游戏人生6》 聽说游戏玩家夫妻向世界挑战了| page=第330-331頁 | id={{EAN|4710945544663}}| date = 2014-04-25 | author = 榎宫祐 |
* 在小说《[[NO GAME NO LIFE 游戏人生|游戏人生]]》中,唯一神[[NO GAME NO LIFE 游戏人生#特图|特图]]持有的星杯外形为大十二面体<ref name="榎宫祐2014">{{Cite book | title = 《NO GAME NO LIFE 游戏人生6》 聽说游戏玩家夫妻向世界挑战了| page=第330-331頁 | id={{EAN|4710945544663}}| date = 2014-04-25 | author = 榎宫祐}}</ref>。 |
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=== 在电脑科学中 === |
=== 在电脑科学中 === |
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[[File:Small_stellated_dodecahedron.png|缩略图|大十二面体的[[对偶多面体]]<ref name="dmccooeydata"/>。]] |
[[File:Small_stellated_dodecahedron.png|缩略图|大十二面体的[[对偶多面体]]<ref name="dmccooeydata"/>。]] |
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{{Main|小星形十二面体}} |
{{Main|小星形十二面体}} |
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大十二面体的对偶多面体同样是一个[[星形正多面体]],为[[小星形十二面体]]<ref name="dmccooeydata"/>,由12个五角星面组成<ref>{{citation | first=Matthias | last=Weber | title=Kepler's small stellated dodecahedron as a Riemann surface | journal=Pacific J. Math. | volume=220 | year=2005 | pages=167–182 | url=http://msp.org/pjm/2005/220-1/p09.xhtml | doi=10.2140/pjm.2005.220.167 | accessdate=2017-08-07 |
大十二面体的对偶多面体同样是一个[[星形正多面体]],为[[小星形十二面体]]<ref name="dmccooeydata"/>,由12个五角星面组成<ref>{{citation | first=Matthias | last=Weber | title=Kepler's small stellated dodecahedron as a Riemann surface | journal=Pacific J. Math. | volume=220 | year=2005 | pages=167–182 | url=http://msp.org/pjm/2005/220-1/p09.xhtml | doi=10.2140/pjm.2005.220.167 | accessdate=2017-08-07 }}</ref>。 |
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=== 康威变换的结果 === |
=== 康威变换的结果 === |
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=== 相关[[克利多胞形|三角化]]多面体 === |
=== 相关[[克利多胞形|三角化]]多面体 === |
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大十二面体可以转换成外观相同的简单多面体,此时,多面体变为由20个凹三角锥组成<ref name = "hilton2010mathematical">{{Cite book | title=A Mathematical Tapestry: Demonstrating the Beautiful Unity of Mathematics | author=Hilton, P. and Pedersen, J. and Donmoyer, S. | isbn=9781139489072 | | year=2010 | publisher=Cambridge University Press | page=171 | access-date=2017-08-08 |
大十二面体可以转换成外观相同的简单多面体,此时,多面体变为由20个凹三角锥组成<ref name = "hilton2010mathematical">{{Cite book | title=A Mathematical Tapestry: Demonstrating the Beautiful Unity of Mathematics | author=Hilton, P. and Pedersen, J. and Donmoyer, S. | isbn=9781139489072 | | year=2010 | publisher=Cambridge University Press | page=171 | access-date=2017-08-08 }}</ref>,这时,其拓樸结构则与[[三角化二十面体]]相同,皆是在正二十面体的每个三角形面接上三角锥<ref>{{citation |
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| author = {{link-en|布兰科·格林鮑姆|Branko Grünbaum|Grünbaum, Branko}} |
| author = {{link-en|布兰科·格林鮑姆|Branko Grünbaum|Grünbaum, Branko}} |
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| doi = 10.1007/BF02759726 |
| doi = 10.1007/BF02759726 |
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<div class="rellink<nowiki> </nowiki>noprint relarticle mainarticle">主条目:{{link-en|复合大十二面体小星形十二面体|Compound_of_small_stellated_dodecahedron_and_great_dodecahedron}}</div> |
<div class="rellink<nowiki> </nowiki>noprint relarticle mainarticle">主条目:{{link-en|复合大十二面体小星形十二面体|Compound_of_small_stellated_dodecahedron_and_great_dodecahedron}}</div> |
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:[[File:Compound of great dodecahedron and small stellated dodecahedron.png|200px]] |
:[[File:Compound of great dodecahedron and small stellated dodecahedron.png|200px]] |
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大十二面体与其对偶的复合体为'''复合小星形十二面体大十二面体'''。其共有24个面、60条边和24个顶点,其[[欧拉特征数|尤拉示性数]]为-12,亏格为7<ref>{{Cite web|url=http://bulatov.org/polyhedra/uniform_compounds/uc40.html|title=compound of great dodecahedron and small stellated dodecahedron|publisher=bulatov.org|access-date=2017-07-25 |
大十二面体与其对偶的复合体为'''复合小星形十二面体大十二面体'''。其共有24个面、60条边和24个顶点,其[[欧拉特征数|尤拉示性数]]为-12,亏格为7<ref>{{Cite web|url=http://bulatov.org/polyhedra/uniform_compounds/uc40.html|title=compound of great dodecahedron and small stellated dodecahedron|publisher=bulatov.org|access-date=2017-07-25}}</ref>,而在这个立体图形中,大十二面体完全隐沒于小星形十二面体而不可见<ref>{{MathWorld |title=Great Dodecahedron-Small Stellated Dodecahedron Compound |urlname=GreatDodecahedron-SmallStellatedDodecahedronCompound}}</ref>。 |
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== 参见 == |
== 参见 == |