小雙三角十二面截半二十面體

小雙三角十二面截半二十面體
小雙三角十二面截半二十面體
類別均勻星形多面體
對偶多面體小雙三角十二角星化六十面體（英语：Small ditrigonal dodecacronic hexecontahedron）
識別
名稱小雙三角十二面截半二十面體
Small ditrigonal dodecicosidodecahedron
small dodekified icosidodecahedron
參考索引U43, C55, W82
鮑爾斯縮寫
（verse-and-dimensions的wikia：Bowers acronym）		sidditdid
數學表示法
威佐夫符號
（英语：Wythoff symbol）		5/3 3 | 5
5/2 3/2 | 5
性質
面44
邊120
頂點60
歐拉特徵數F=44, E=120, V=60 （χ=-16）
組成與佈局
面的種類20個正三角形
12個正五角星
12個正十邊形
頂點圖3.10.5/3.10
對稱性
對稱群Ih, [5,3], *532
圖像
立體圖
3.10.5/3.10
（頂點圖）

小雙三角十二角星化六十面體（英语：Small ditrigonal dodecacronic hexecontahedron）
（對偶多面體）

小雙三角十二面截半二十面體又稱小十二面化截半二十面體（small dodekified icosidodecahedron）[1]是一種星形均勻多面體，由20個正三角形、12個正五角星和12個正十邊形組成[2]，索引為U43，對偶多面體為小雙三角十二角星化六十面體（英语：Small ditrigonal dodecacronic hexecontahedron）[3]，具有二十面體群對稱性（英语：Icosahedral symmetry）。[4][2][5]。

性質

小雙三角十二面截半二十面體共由44個面、120條邊和60個頂點組成。[4]在其44個面中，有20個正三角形面、12個正五角星面和12個正十邊形面[6]，或以施萊夫利符號表示為20{3}+12{5⁄2}+12{10}[7]。在其60個頂點中，每個頂點都是2個十邊形面、1個三角形面和1個五角星面的公共頂點，並且這些面在構成頂角的多面角時，以五角星、十邊形、三角形和十邊形的順序排列，在頂點圖中可以用(5/3.10.3.10)[8][9]或(10.5/3.10.3)來表示[6][4]。

表示法

小雙三角十二面截半二十面體在考克斯特—迪肯符号（英语：Coxeter-Dynkin diagram）中可以表示為label5-3 branch 01rd split2-35 node 1 [10]（x5/3o3x5*a）[11]或label5 branch 11 split2-tp2 node （x3/2o5/2x5*a）[11]，在威佐夫記號中可以表示為5/3 3 | 5[10][12][13]。

尺寸

若小雙三角十二面截半二十面體的邊長為單位長，則其外接球半徑為：

邊長為單位長的小雙三角十二面截半二十面體，中分球半徑為：

二面角

小雙三角十二面截半二十面體共有兩種二面角，分別為十邊形面和三角形面的二面角、以及十邊形面和五角星面的二面角。[2][9]

其中，十邊形面和三角形面的二面角角度約為100.8123度：[2]

十邊形三角形

而十邊形面和五角星面的二面角為5平方根倒數的反餘弦值，[9]角度約為63.43度：[2]

十邊形五角星

相關多面體

小雙三角十二面截半二十面體與大星形截角十二面体共用相同的頂點佈局。其亦與小十二面二十面體和小二十面化截半二十面體共用相同的邊佈局。[9]


大星形截角十二面体
小二十面化截半二十面體
小雙三角十二面截半二十面體
小十二面二十面體

拓樸

從組成元素來看，小雙三角十二面截半二十面體似乎與大雙三角十二面截半二十面體、小十二面截半二十面體和大十二面截半二十面體拓樸同構。但實際上與小雙三角十二面截半二十面體拓樸同構的立體只有大雙三角十二面截半二十面體。[9]

參見

參考文獻

  1. ^ Jim McNeill. Uniform Polyhedra. orchidpalms.com. [2022-08-23]. （原始内容存档于2015-09-24）. 
  2. ^ 2.0 2.1 2.2 2.3 2.4 David I. McCooey. Self-Intersecting Quasi-Quasi-Regular Polyhedra: Small Ditrigonal Dodecicosidodecahedron. [2022-08-23]. （原始内容存档于2022-08-23）. 
  3. ^ Weisstein, Eric W. (编). Small Ditrigonal Dodecicosidodecahedron. at MathWorld--A Wolfram Web Resource. Wolfram Research, Inc. （英语）. 
  4. ^ 4.0 4.1 4.2 Maeder, Roman. 43: small ditrigonal dodecicosidodecahedron. MathConsult. [2022-08-23]. （原始内容存档于2022-08-23）. 
  5. ^ Paul Bourke. Uniform Polyhedra (80). Math Consult AG. October 2004 [2019-09-27]. （原始内容存档于2013-09-02）. 
  6. ^ 6.0 6.1 Zvi Har'El. Kaleido Data: Uniform Polyhedron #48, small ditrigonal dodecicosidodecahedron. harel.org.il. 2006-11-14 [2022-08-14]. （原始内容存档于2021-10-22）. 
  7. ^ Eric W. Weisstein. Small Ditrigonal Dodecicosidodecahedron. archive.lib.msu.edu. 1999-05-25 [2022-08-23]. （原始内容存档于2021-12-08）. 
  8. ^ Kovič, J. Classification of uniform polyhedraby their symmetry-type graphs (PDF). Int. J. Open Problems Compt. Math. 2012, 5 (4) [2022-08-23]. （原始内容存档 (PDF)于2022-08-14）. 
  9. ^ 9.0 9.1 9.2 9.3 9.4 Richard Klitzing. small ditrigonary dodekicosidodecahedron, small dodekified icosidodecahedron, sidditdid. bendwavy.org. [2022-08-23]. （原始内容存档于2021-09-24）. 
  10. ^ 10.0 10.1 Klitzing, Richard. Axial-Symmetrical Edge-Facetings of Uniform Polyhedra (PDF). tic. 2002, 2 (4): 3 [2022-08-23]. （原始内容存档 (PDF)于2022-08-14）. 
  11. ^ 11.0 11.1 Richard Klitzing. Icosahedral Symmetries uniform polyhedra, Polytopes & their Incidence Matrices. bendwavy.org. [2022-08-07]. （原始内容存档于2018-07-07）. 
  12. ^ Adrian Rossiter. small ditrigonal dodecicosidodecahedron. antiprism.com. 
  13. ^ V.Bulatov. small ditrigonal dodecicosidodecahedron. [2022-08-23]. （原始内容存档于2022-08-23）. 

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