五角化六十面體 - 维基百科，自由的百科全书

（按這裡觀看旋轉模型）

verse-and-dimensions的wikiaBowers acronym
sapedit

60
150

V3.3.3.3.5
V34.5[1]:97

12個5階頂點[1]:97

I, [5,3]+, (532)

 扭棱十二面體（對偶多面體） （展開圖）

性質

 五角化六十面體的旋轉透視圖 五角化六十面體的另一個手性鏡像的旋轉透視圖

二面角

${\displaystyle \arccos {\left(-{\frac {2\left(x+{\frac {2}{x}}\right)\left(1+15\varphi \right)+\left(15+16\varphi \right)}{209}}\right)}\approx }$2.67347322717678${\displaystyle \approx }$153.178732558°

面的組成

${\displaystyle {\frac {1}{x}}:{\frac {x\left(2+7\varphi \right)+\left(5\varphi -3\right)+{\frac {2\left(8-3\varphi \right)}{x}}}{31}}\approx }$0.582899534744982414 : 1.019988247022845898

${\displaystyle l={\frac {1+\xi }{2-\xi ^{2}}}\approx 1.749\,852\,566\,74}$.

幾何

${\displaystyle {\frac {\sqrt {3\left(x\varphi +1+\varphi +{\frac {1}{x}}\right)}}{2}}\approx }$2.1172098986

${\displaystyle {\frac {\sqrt {x^{2}\left(1009+1067\varphi \right)+x\left(1168+2259\varphi \right)+\left(1097+941\varphi \right)}}{62}}\approx }$2.220000699

體積與表面積

${\displaystyle t={\frac {{\sqrt[{3}]{44+12\varphi (9+{\sqrt {81\varphi -15}})}}+{\sqrt[{3}]{44+12\varphi (9-{\sqrt {81\varphi -15}})}}-4}{12}}\approx 0.471\,575\,629\,622}$ .

${\displaystyle A={\frac {30b^{2}\cdot (2+3t)\cdot {\sqrt {1-t^{2}}}}{1-2t^{2}}}\approx 162.698\,964\,198b^{2}}$.

${\displaystyle V={\frac {5b^{3}(1+t)(2+3t)}{(1-2t^{2})\cdot {\sqrt {1-2t}}}}\approx 189.789\,852\,067b^{3}}$.

${\displaystyle \Psi ={\frac {\pi ^{\frac {1}{3}}(6V)^{\frac {2}{3}}}{A}}\approx 0.98163}$

參考文獻

1. Williams, Robert. The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. 1979. ISBN 0-486-23729-X.
2. Weisstein, Eric W. (编). Pentagonal Hexecontahedron. at MathWorld--A Wolfram Web Resource. Wolfram Research, Inc. （英语）.
3. ^ Alan Holden. Shapes, Space, and Symmetry. New York: Columbia University Press. 1971.
4. ^ Conway, J.H. and Burgiel, H. and Goodman-Strauss, C. The Symmetries of Things. AK Peters/CRC Recreational Mathematics Series. CRC Press. 2016 [2022-07-25]. ISBN 9781439864890. LCCN 2007046446. （原始内容存档于2022-07-26）. (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 287, pentagonal icosikaitetrahedron)
5. ^ Coxeter, H. S. M., Kaleidoscopes: Selected Writings, John Wiley and Sons: 282, 1995, ISBN 9780471010036.
6. David I. McCooey. Catalan Solids: Pentagonal Hexecontahedron (dextro). [2022-07-24]. （原始内容存档于2022-07-27）.
7. David I. McCooey. Catalan Solids: Pentagonal Hexecontahedron (laevo). [2022-07-24]. （原始内容存档于2022-07-27）.
8. ^ Pentagonal Hexecontahedron. polyhedra.org. [2008-09-24]. （原始内容存档于2008-07-14）.
9. ^ Livio Zefiro and Maria Rosa Ardigo. Description of the Forms Belonging to the 235 and m35 Icosahedral Point Groups Starting from the Pairs of Dual Polyhedra: Icosahedron-Dodecahedron and Archimedean Polyhedra-Catalan Polyhedra. [2022-07-25]. （原始内容存档于2021-05-06）.
10. ^ Pentagonal Hexecontahedron - Geometry Calculator. rechneronline.de. [2020-05-26]. （原始内容存档于2022-05-23）.
11. ^ Fair Dice. mathpuzzle.com. [2022-07-25]. （原始内容存档于2022-04-26）.