# 小斜方截半二十面体

(按這裡觀看旋轉模型)

verse-and-dimensions的wikiaBowers acronym
srid

rr{5,3}

3 5 | 2

62
120

20個{3}
30個{4}
12個{5}

-

 3.4.5.4（頂點圖） 鳶形六十面體(對偶多面體) (展開圖)

## 性質

### 幾何關聯

 小斜方截半二十面體 小十二面截半二十面體 小斜方十二面體 小星形截角十二面體

### 其他特性

 小斜方截半二十面體的上下分层着色

### 頂點坐標

(±1, ±1, ±φ3),
φ2, ±φ, ±2φ),
(±(2+φ), 0, ±φ2),

R = ${\displaystyle {\frac {\sqrt {8\phi +7}}{2}}}$= ${\displaystyle {\frac {\sqrt {11+4{\sqrt {5}}}}{2}}}$≈ 2.233

### 體積與表面積

{\displaystyle {\begin{aligned}A&=\left\{30+{\sqrt {30\left[10+3{\sqrt {5}}+{\sqrt {15(5+2{\sqrt {5}})}}\right]}}\right\}a^{2}\\&\approx 59.3059828a^{2}\\V&={\frac {1}{3}}(60+29{\sqrt {5}})a^{3}\approx 41.6153238a^{3}\\\end{aligned}}}[8]

### 二面角

${\displaystyle \arccos {\left(-{\frac {\sqrt {10\left(5+{\sqrt {5}}\right)}}{10}}\right)}\approx 148.282525589^{\circ }}$[8]

${\displaystyle \arccos {\left(-{\frac {{\sqrt {3}}+{\sqrt {15}}}{6}}\right)}\approx 159.094842552^{\circ }}$[8]

## 正交投影

[2] [2] [2] [2] [6] [10]

## 球面鑲嵌

 正交投影 球極平面投影 以正五邊形為中心 以正三角形為中心 以正方形為中心

## 相關多面體與鑲嵌

Zometool英语Zometool積木組英语construction set的其中一個頂點連接零件是使用黃金比例矩形版本的小斜方截半二十面體[10]

{5,3} t0,1{5,3} t1{5,3} t0,1{3,5} {3,5} t0,2{5,3} t0,1,2{5,3} s{5,3}

V5.5.5 V3.10.10 V3.5.3.5 V5.6.6 V3.3.3.3.3 V3.4.5.4 V4.6.10 V3.3.3.3.5

### 變異對稱性

*n32
[n,3]

*232
[2,3]
*332
[3,3]
*432
[4,3]
*532
[5,3]
*632
[6,3]
*732
[7,3]
*832
[8,3]...
*∞32
[∞,3]

3.4.2.4 3.4.3.4 3.4.4.4 3.4.5.4 3.4.6.4

### 詹森多面體

 J5 80 81 83

## 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. (Section 3-9)
2. Cromwell, P. Polyhedra. United Kingdom: Cambridge. 1997: 79–86 Archimedean solids. ISBN 0-521-55432-2.
3. The Big Bang Theory Series 8 Episode 2 - The Junior Professor Solution: features this solid as the answer to an impromptu science quiz the main four characters have in Leonard and Sheldon's apartment, and is also illustrated in Chuck Lorre's Vanity Card #461 at the end of that episode.
1. ^ Cromwell, P. Polyhedra, CUP hbk (1997), pbk. (1999). Ch.2 p.79-86 Archimedean solids
2. ^ Rhombicosidodecahedron. bulatov.com. [2018-08-27]. （原始内容存档于2017-10-11）.
3. ^ Weisstein, Eric W. (编). DeltoidalHexecontahedron and Hamiltonian path. at MathWorld--A Wolfram Web Resource. Wolfram Research, Inc. （英语）.
4. ^ Harmonies Of The World by Johannes Kepler, Translated into English with an introduction and notes by E. J. Aiton, A. M. Duncan, "J. V. Field, 1997, ISBN 0-87169-209-0 (page 123)
5. ^ Weisstein, Eric W. (编). Small Rhombicosidodecahedron. at MathWorld--A Wolfram Web Resource. Wolfram Research, Inc. （英语）.
6. ^ The Rhombicosidodecahedron. The Geometry Center. [2018-08-27]. （原始内容存档于2016-12-04）.
7. ^ Weisstein, Eric W. (编). Icosahedral group. at MathWorld--A Wolfram Web Resource. Wolfram Research, Inc. （英语）.
8. Archimedean Solids: Rhombicosidodecahedron. dmccooey.com. [2018-08-27]. （原始内容存档于2018-01-29）.
9. ^
10. ^ Weisstein, Eric W. (编). Zome. at MathWorld--A Wolfram Web Resource. Wolfram Research, Inc. （英语）.