截角八面體堆砌

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截角八面體堆砌
Truncated octahedra.jpg
類型 均勻堆砌
維度 3
(4.6.6) Truncated octahedron.png
{4} Kvadrato.svg
{6} Regular hexagon.svg
棱圖 {3} Triangle.Isosceles.svg
等腰三角形
顶点图 Bitruncated cubic honeycomb verf2.png
鍥形四面體
施萊夫利符號 2t{4,3,4}
t1,2{4,3,4}
考克斯特記號英语Coxeter–Dynkin_diagram

CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png or CDel branch 11.pngCDel 4a4b.pngCDel nodes.png

CDel nodes 11.pngCDel split2.pngCDel node 1.pngCDel 4.pngCDel node.png = CDel node h0.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel split2.pngCDel node 1.png = CDel node h0.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node h0.png
對稱群 , [4,3,4]
空間群 Im3m (229)
考克斯特群 [[4,3,4]]
纖維流形 8o:2
對偶多胞體 鍥形四面體堆砌英语Disphenoid tetrahedral honeycomb
特性 顶点正英语vertex-transitive

幾何學中,截角八面體堆砌三維空間內28個半正密鋪之一,由截角八面體獨立堆積而成,雖然他每個都全等、每皆等長,但其不能稱為正密鋪,因為雖然它只由一種,截角八面體組成,但是該胞不是正多面體,因此並非所有“面”皆全等,因此截角八面體堆砌只能稱為半正堆砌。

性質[编辑]

圖中每一個正方體都是一個體心立方晶格原子,他們可以擴充為截角八面體堆砌

在截角八面體堆砌中,每個頂點周圍皆有有四個截角八面體,且全由截角八面體組成,因此其胞可遞英语cell-transitive。它也存在邊可遞英语edge-transitive的特性,由於其具有2個六邊形和一個正方形,且截角八面體堆砌的每個頂點都是4個截角八面體的公共頂點,因此每條邊和頂點也存在點可遞英语vertex-transitive的特性。

截角八面體堆砌可以被視為體心立方晶格沃羅諾伊圖開爾文男爵推測截角八面體堆砌若將其面和邊彎曲且保留原來的布局將會變為最佳的肥皂泡沫理想結構。

Bitruncated cubic tiling.png HC-A4.png

命名[编辑]

康威稱截角八面體堆砌為truncated octahedrille[1],在他的建築學和反射的細分列表,與其對偶合稱oblate tetrahedrille又稱為鍥形四面體堆砌。雖然正四面體不能單獨填充整個空間,但截角八面體堆砌的對偶具有相同的鍥形四面體胞與等腰三角形的面。

此外由於截角八面體堆砌的每個頂點都是4個截角八面體的公共頂點,因此也可稱為四階截角八面體堆砌。

對稱性與表面塗色[编辑]

五種半正表面塗色
空間群 Im3m (229) Pm3m (221) Fm3m (225) F43m (216) Fd3m (227)
纖維流形 8o:2 4:2 2:2 1o:2 2+:2
考克斯特群 ×2
[[4,3,4]]
=[4[3[4]]]
CDel node.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c1.pngCDel 4.pngCDel node.png = CDel branch c1.pngCDel 3ab.pngCDel branch c1.png

[4,3,4]
=[2[3[4]]]
CDel node.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c2.pngCDel 4.pngCDel node.png = CDel branch c1-2.pngCDel 3ab.pngCDel branch c2-1.png

[4,31,1]
=<[3[4]]>
CDel nodeab c1-2.pngCDel split2.pngCDel node c3.pngCDel 4.pngCDel node.png = CDel node c3.pngCDel split1.pngCDel nodeab c1-2.pngCDel split2.pngCDel node c3.png

[3[4]]
 
CDel node c3.pngCDel split1.pngCDel nodeab c1-2.pngCDel split2.pngCDel node c4.png
×2
[[3[4]]]
=[[3[4]]]
CDel branch c1.pngCDel 3ab.pngCDel branch c2.png
考克斯特符號英语Coxeter diagram CDel branch 11.pngCDel 4a4b.pngCDel nodes.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png CDel nodes 11.pngCDel split2.pngCDel node 1.pngCDel 4.pngCDel node.png CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel split2.pngCDel node 1.png CDel branch 11.pngCDel 3ab.pngCDel branch 11.png
截角八面體 1
Uniform polyhedron-43-t12.svg
1:1
Uniform polyhedron-43-t12.svg:Uniform polyhedron-43-t12.svg
2:1:1
Uniform polyhedron-43-t12.svg:Uniform polyhedron-43-t12.svg:Uniform polyhedron-33-t012.png
1:1:1:1
Uniform polyhedron-33-t012.png:Uniform polyhedron-33-t012.png:Uniform polyhedron-33-t012.png:Uniform polyhedron-33-t012.png
1:1
Uniform polyhedron-33-t012.png:Uniform polyhedron-33-t012.png
頂點圖 Bitruncated cubic honeycomb verf2.png Bitruncated cubic honeycomb verf.png Cantitruncated alternate cubic honeycomb verf.png Omnitruncated 3-simplex honeycomb verf.png Omnitruncated 3-simplex honeycomb verf2.png
頂點

對稱性
[2+,4]
(order 8)
[2]
(order 4)
[ ]
(order 2)
[ ]+
(order 1)
[2]+
(order 2)
圖像
表面依胞
上色
Bitruncated Cubic Honeycomb1.svg Bitruncated Cubic Honeycomb.svg Bitruncated cubic honeycomb3.png Bitruncated cubic honeycomb2.png Bitruncated Cubic Honeycomb1.svg

相關多面體和鑲嵌[编辑]

六角四片四角孔扭歪多面體日语六角四片四角孔ねじれ正多面体是一個正扭歪無限面體 {6,4|4},此形狀中包含此堆砌狀的六邊形。

考克斯特群[4,3,4]、CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png產生15個排列均勻的鑲嵌中,9個具有獨特的的幾何形狀,包括交錯立方体堆砌、擴展立方堆砌是幾何上相同的立方體堆砌。

空間群 纖維流形 擴展
對稱群
擴展
标记
蜂巢體
(堆砌)
Pm3m
(221)
4:2 [4,3,4] CDel node c1.pngCDel 4.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 4.pngCDel node c4.png ×1 CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png 1, CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png 2, CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png 3, CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png 4,
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png 5, CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png 6
Fm3m
(225)
2:2 [1+,4,3,4]
↔ [4,31,1]
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node c1.pngCDel 4.pngCDel node c2.png
CDel nodes 10ru.pngCDel split2.pngCDel node c1.pngCDel 4.pngCDel node c2.png
Half CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png 7, CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png 11, CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png 12, CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.png 13
I43m
(217)
4o:2 [[(4,3,4,2+)]] CDel branch.pngCDel 4a4b.pngCDel nodes hh.png Half × 2 CDel branch.pngCDel 4a4b.pngCDel nodes hh.png (7),
Fd3m
(227)
2+:2 [[1+,4,3,4,1+]]
↔ [[3[4]]]
CDel branch.pngCDel 4a4b.pngCDel nodes h1h1.png
CDel branch 11.pngCDel 3ab.pngCDel branch.png
Quarter × 2 CDel branch.pngCDel 4a4b.pngCDel nodes h1h1.png 10,
Im3m
(229)
8o:2 [[4,3,4]] CDel branch c2.pngCDel 4a4b.pngCDel nodeab c1.png ×2

CDel branch.pngCDel 4a4b.pngCDel nodes 11.png (1), CDel branch 11.pngCDel 4a4b.pngCDel nodes.png 8, CDel branch 11.pngCDel 4a4b.pngCDel nodes 11.png 9

考克斯特群[4,31,1], CDel node.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodes.png, 考克斯特群產生 9個排列均勻的鑲嵌中,其中4個具有獨特的的幾何形狀,包括交替立方体堆砌。

空間群 纖維流形 擴展
對稱群
擴展
标记
蜂巢體
(堆砌)
Fm3m
(225)
2:2 [4,31,1]
↔ [4,3,4,1+]
CDel node c1.pngCDel 4.pngCDel node c2.pngCDel split1.pngCDel nodes 10lu.png
CDel node c1.pngCDel 4.pngCDel node c2.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h1.png
×1 CDel node.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodes 10lu.png 1, CDel node 1.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodes 10lu.png 2, CDel node.pngCDel 4.pngCDel node 1.pngCDel split1.pngCDel nodes 10lu.png 3, CDel node 1.pngCDel 4.pngCDel node 1.pngCDel split1.pngCDel nodes 10lu.png 4
Fm3m
(225)
2:2 <[1+,4,31,1]>
↔ <[3[4]]>
CDel node h1.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodeab c1.png
CDel node 1.pngCDel split1.pngCDel nodeab c1.pngCDel split2.pngCDel node.png
×2 CDel node h1.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodes.png (1), CDel node h1.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodes 11.png (3)
Pm3m
(221)
4:2 <[4,31,1]> CDel node c3.pngCDel 4.pngCDel node c2.pngCDel split1.pngCDel nodeab c1.png ×2

CDel node 1.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodes.png 5, CDel node.pngCDel 4.pngCDel node 1.pngCDel split1.pngCDel nodes.png 6, CDel node 1.pngCDel 4.pngCDel node 1.pngCDel split1.pngCDel nodes.png 7, CDel node.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodes 11.png (6), CDel node 1.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodes 11.png 9, CDel node.pngCDel 4.pngCDel node 1.pngCDel split1.pngCDel nodes 11.png 10, CDel node 1.pngCDel 4.pngCDel node 1.pngCDel split1.pngCDel nodes 11.png 11

立方體堆砌是考克斯特群中的五個結構特別的均勻堆砌[2]之一,其對稱性可以乘以環在考克斯特-迪肯符號的對稱性:

空間群 纖維流形 方形
對稱群
擴展
對稱群
擴展
标记
擴展
蜂巢體
(堆砌)
F43m
(216)
1o:2 a1 [3[4]] CDel node.pngCDel split1.pngCDel nodes.pngCDel split2.pngCDel node.png ×1 (None)
Fd3m
(227)
2+:2 p2 [[3[4]]] CDel branch 11.pngCDel 3ab.pngCDel branch.png
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h1.png
×2 CDel branch 11.pngCDel 3ab.pngCDel branch.png 3
Fm3m
(225)
2:2 d2 <[3[4]]>
↔ [4,3,31,1]
CDel node c3.pngCDel split1.pngCDel nodeab c1-2.pngCDel split2.pngCDel node c3.png
CDel node.pngCDel 4.pngCDel node c3.pngCDel split1.pngCDel nodeab c1-2.png
×2 CDel node.pngCDel split1.pngCDel nodes 10luru.pngCDel split2.pngCDel node.png 1,CDel node 1.pngCDel split1.pngCDel nodes 10luru.pngCDel split2.pngCDel node 1.png 2
Pm3m
(221)
4:2 d4 [2[3[4]]]
↔ [4,3,4]
CDel node c1.pngCDel split1.pngCDel nodeab c2.pngCDel split2.pngCDel node c1.png
CDel node.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c2.pngCDel 4.pngCDel node.png
×4 CDel node.pngCDel split1.pngCDel nodes 11.pngCDel split2.pngCDel node.png 4
Im3m
(229)
8o:2 r8 [4[3[4]]]
↔ [[4,3,4]]
CDel branch c1.pngCDel 3ab.pngCDel branch c1.png
CDel branch c1.pngCDel 4a4b.pngCDel nodes.png
×8 CDel branch 11.pngCDel 3ab.pngCDel branch 11.png 5, CDel branch hh.pngCDel 3ab.pngCDel branch hh.png (*)

交錯形式[编辑]

截角八面體堆砌可以交錯,從截角八面體的空隙中創建不規則四面體單元建立正二十面體。有三個相關的結構對應三種考克斯特—迪肯符號:CDel node.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.pngCDel node.pngCDel 4.pngCDel node h.pngCDel split1.pngCDel nodes hh.pngCDel node h.pngCDel split1.pngCDel nodes hh.pngCDel split2.pngCDel node h.png,且有對稱性[4,3+,4]、[4,(31,1)+]和[3[4]]+。第一個和最後一個的對稱性為[[4,3+,4]] and [[3[4]]]+的一倍。

截角八面體堆砌可以當作在α-rhombihedral晶體內的硼原子位置,在二十面體的中心是面心立方晶格的位置。[3]

五種半正表面塗色
空間群 I3 (204) Pm3 (200) Fm3 (202) Fd3 (203) F23 (196)
纖維流形 8−o 4 2 2o+ 1o
考克斯特群 [[4,3+,4]] [4,3+,4] [4,(31,1)+] [[3[4]]]+ [3[4]]+
考克斯特符號英语Coxeter diagram CDel branch hh.pngCDel 4a4b.pngCDel nodes.png CDel node.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.png CDel node.pngCDel 4.pngCDel node h.pngCDel split1.pngCDel nodes hh.png CDel branch hh.pngCDel 3ab.pngCDel branch hh.png CDel node h.pngCDel split1.pngCDel nodes hh.pngCDel split2.pngCDel node h.png
四分之一
四分之一

折疊投影[编辑]

考克斯特群
考克斯特
記號
英语Coxeter–Dynkin diagram#Geometric folding
CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel split2.pngCDel node 1.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.png
形狀 Bitruncated Cubic Honeycomb flat.png
過截角正方體堆砌
Uniform tiling 44-t012.png
截角四邊形鑲嵌

參見[编辑]

参考文獻[编辑]

  1. ^ John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, (2008) The Symmetries of Things, ISBN 978-1-56881-220-5 (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, Architectonic and Catoptric tessellations, p 292-298, includes all the nonprismatic forms)
  2. ^ [1], A000029 6-1 cases, skipping one with zero marks
  3. ^ Williams, 1979, p 199, Figure 5-38.
  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, (2008) The Symmetries of Things, ISBN 978-1-56881-220-5 (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, Architectonic and Catoptric tessellations, p 292-298, includes all the nonprismatic forms)
  • George Olshevsky, Uniform Panoploid Tetracombs, Manuscript (2006) (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)
  • Branko Grünbaum, Uniform tilings of 3-space. Geombinatorics 4(1994), 49 - 56.
  • Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [2]
    • (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10] (1.9 Uniform space-fillings)
  • A. Andreini, Sulle reti di poliedri regolari e semiregolari e sulle corrispondenti reti correlative (On the regular and semiregular nets of polyhedra and on the corresponding correlative nets), Mem. Società Italiana della Scienze, Ser.3, 14 (1905) 75–129.
  • Richard Klitzing, 3D Euclidean Honeycombs, o4x3x4o - batch - O16
  • Uniform Honeycombs in 3-Space: 05-Batch
  • Williams, Robert. The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. 1979. ISBN 0-486-23729-X. 
  • MathWorldSpace-filling polyhedron的资料,作者:埃里克·韦斯坦因