# 正十六胞體堆砌

{3,3}
{3}

## 性質

### D4網格

=

4

4

= =

The D*
4

4

4
）可以透過所有四個D4網格的聯集來構成，但其與D4網格相同，同時他也是2個超立方體堆砌放置在對方的對偶位置的聯集，也就是四維空間中立方晶系結構[6]

= =

D*
4

## 不同對稱性的結構

[3,4,3], 1152階
24： 正十六胞體

[3,3,4], 384階
16+8： 正十六胞體
${\displaystyle {\tilde {D}}_{4}}$ = {3,31,1,1}
= h{4,3,31,1}
=
[31,1,1], 192階
8+8+8： 正十六胞體

## 相關多胞體與堆砌

 图像 施萊夫利符號 超立方體堆砌 正十六胞體堆砌 正二十四胞體堆砌 {4,3,3,4} {3,3,4,3} {3,4,3,3}
D5堆砌體

[31,1,3,31,1] ${\displaystyle {\tilde {D}}_{5}}$
<[31,1,3,31,1]>
↔ [31,1,3,3,4]

${\displaystyle {\tilde {D}}_{5}}$×21 = ${\displaystyle {\tilde {B}}_{5}}$ , , ,

, , ,

[[31,1,3,31,1]] ${\displaystyle {\tilde {D}}_{5}}$×22 ,
<2[31,1,3,31,1]>
↔ [4,3,3,3,4]

${\displaystyle {\tilde {D}}_{5}}$×41 = ${\displaystyle {\tilde {C}}_{5}}$ , , , , ,
[<2[31,1,3,31,1]>]
↔ [[4,3,3,3,4]]

${\displaystyle {\tilde {D}}_{5}}$×8 = ${\displaystyle {\tilde {C}}_{5}}$×2 , ,

## 註釋

1. ^ 當n<8時為2n-1；n=8時為240；n>8時為2n(n-1)

## 參考文獻

1. Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8
• pp. 154–156: Partial truncation or alternation, represented by h prefix: h{4,4} = {4,4}; h{4,3,4} = {31,1,4}, h{4,3,3,4} = {3,3,4,3}, ...
2. 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 [1]
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
3. George Olshevsky, Uniform Panoploid Tetracombs, Manuscript (2006) (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)
4. Richard Klitzing, 4D, Euclidean tesselations x3o3o4o3o - hext - O104
1. ^ The Lattice F4. math.rwth-aachen.de.
2. The Lattice D4. math.rwth-aachen.de.
3. ^ O. R. Musin. The problem of the twenty-five spheres. Russ. Math. Surv. 2003, 58: 794–795. doi:10.1070/RM2003v058n04ABEH000651.
4. ^ Conway JH, Sloane NJH. Sphere Packings, Lattices and Groups 3rd. 1998. ISBN 0-387-98585-9.
5. ^ Conway（1998）[4], p. 119
6. ^ Conway and Sloane, Sphere packings, lattices, and groups, 7.4 The dual lattice D3*, p.120
7. ^ Conway and Sloane, Sphere packings, lattices, and groups, p. 120
8. ^ Conway and Sloane, Sphere packings, lattices, and groups, p. 466