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[[Image:reinforced solids cube.jpg|thumb|Figure 1: Small cube of a material with reinforcing bars. The cube is cracked and the material above the crack is removed to show the reinforcement that crosses the crack.]]
{| class=wikitable width=280 align=right
 
!<math>{\tilde{A}}_2</math>
In [[solid mechanics]], a '''reinforced solid''' is a [[brittle]] material that is reinforced by [[ductile]] bars or fibres. A common application is [[reinforced concrete]]. When the concrete cracks the tensile force in a crack is not carried any more by the concrete but by the steel reinforcing bars only. The reinforced concrete will continue to carry the load provided that sufficient reinforcement is present. A typical design problem is to find the smallest amount of reinforcement that can carry the [[Stress (mechanics)|stresses]] on a small cube (Fig. 1). This can be formulated as an [[Mathematical optimization|optimization]] problem.
!<math>{\tilde{A}}_3</math>
 
==Optimization problem==
 
The reinforcement is directed in the x, y and z direction. The reinforcement ratio is defined in a cross-section of a reinforcing bar as the reinforcement area <math>A_{r}</math> over the total area <math>A</math>, which is the brittle material area plus the reinforcement area.
 
:<math>\rho_{x}</math> = <math>A_{rx}</math> / <math>A_{x}</math>
 
:<math>\rho_{y}</math> = <math>A_{ry}</math> / <math>A_{y}</math>
 
:<math>\rho_{z}</math> = <math>A_{rz}</math> / <math>A_{z}</math>
 
In case of reinforced concrete the reinforcement ratios are usually between 0.1% and 2%. The [[yield stress]] of the reinforcement is denoted by <math>f_{y}</math>. The [[Stress (mechanics)|stress tensor]] of the brittle material is
 
:<math>
\left[{\begin{matrix}
\sigma _{xx} - \rho_{x} f_{y} & \sigma _{xy} & \sigma _{xz} \\
\sigma _{xy} & \sigma _{yy} - \rho_{y} f_{y} & \sigma _{yz} \\
\sigma _{xz} & \sigma _{yz} & \sigma _{zz} - \rho_{z} f_{y} \\
\end{matrix}}\right]
</math>.
 
This can be interpreted as the stress tensor of the composite material minus the stresses carried by the reinforcement at yielding. This formulation is accurate for reinforcement ratio's smaller than 5%. It is assumed that the brittle material has no tensile strength. (In case of reinforced concrete this assumption is necessary because the concrete has small shrinkage cracks.) Therefore, the [[principal stresses]] of the brittle material need to be compression. The principal stresses of a stress tensor are its [[eigenvalues]].
 
The optimization problem is formulated as follows. Minimize <math>\rho_{x}</math> + <math>\rho_{y}</math> + <math>\rho_{z}</math> subject to all eigenvalues of the brittle material stress tensor are less than or equal to zero ([[Positive-definite matrix|negative-semidefinite]]). Additional constraints are <math>\rho_{x}</math> ≥ 0, <math>\rho_{y}</math> ≥ 0, <math>\rho_{z}</math> ≥ 0.
 
==Solution==
 
The solution to this problem can be presented in a form most suitable for hand calculations.<ref name="A"/><ref name="N"/> It can be presented in graphical form.<ref name="F"/> It can also be presented in a form most suitable for computer implementation.<ref name="H1"/><ref name="H2"/> In this article the latter method is shown.
 
There are 12 possible reinforcement solutions to this problem, which are shown in the table below. Every row contains a possible solution. The first column contains the number of a solution. The second column gives conditions for which a solution is valid. Columns 3, 4 and 5 give the formulas for calculating the reinforcement ratios.
 
{| class="wikitable"
|-
|  || Condition || <math>\rho_{x}</math> <math>f_{y} </math> || <math>\rho_{y}</math> <math>f_{y}</math>  || <math>\rho_{z}</math> <math>f_{y}</math>
|-
|-
| 1 || <math>I_{1}</math> ≤ 0, <math>I_{2}</math> ≥ 0, <math>I_{3}</math> ≤ 0 || 0 || 0 || 0
![[Triangular tiling]]
![[Tetrahedral-octahedral honeycomb]]
|-
|-
| 2 || <math>\sigma_{yy}\sigma_{zz} - \sigma^2_{yz}</math> > 0<br/><math>I_{1}(\sigma_{yy}\sigma_{zz} - \sigma^2_{yz}) - I_{3}</math> ≤ 0<br/><math>I_{2}(\sigma_{yy}\sigma_{zz} - \sigma^2_{yz}) - I_{3}(\sigma_{yy}+\sigma_{zz})</math> ≥ 0 || <math>\frac{I_{3}}{\sigma_{yy} \sigma_{zz} - \sigma^2_{yz}}</math> || 0 || 0
|[[File:Uniform_tiling_333-t1.png|120px]]<BR>With red and yellow equilateral triangles
|-
|[[File:Tetrahedral-octahedral honeycomb2.png|160px]]<BR>With cyan and yellow [[tetrahedron|tetrahedra]], and red rectified tetrahedra ([[octahedron]])
| 3 || <math>\sigma_{xx}\sigma_{zz} - \sigma^2_{xz}</math> > 0<br/><math>I_{1}(\sigma_{xx}\sigma_{zz} - \sigma^2_{xz}) - I_{3}</math> ≤ 0<br/><math>I_{2}(\sigma_{xx}\sigma_{zz} - \sigma^2_{xz}) - I_{3}(\sigma_{xx}+\sigma_{zz})</math> ≥ 0 || 0 || <math>\frac{I_{3}}{\sigma_{xx} \sigma_{zz} - \sigma^2_{xz}}</math> || 0
|-
| 4 || <math>\sigma_{xx}\sigma_{yy} - \sigma^2_{xy}</math> > 0<br/><math>I_{1}(\sigma_{xx}\sigma_{yy} - \sigma^2_{xy}) - I_{3}</math> ≤ 0<br/><math>I_{2}(\sigma_{xx}\sigma_{yy} - \sigma^2_{xy}) - I_{3}(\sigma_{xx}+\sigma_{yy})</math> ≥ 0 || 0 || 0 || <math>\frac{I_{3}}{\sigma_{xx} \sigma_{yy} - \sigma^2_{xy}}</math>
|-
| 5 || <math>\sigma_{xx}<0</math> || 0 || <math>\sigma_{yy}- \frac{\sigma^2_{xy}}{\sigma_{xx}} +|\sigma_{yz}-\frac{\sigma_{xz}\sigma_{xy}}{\sigma_{xx}}|</math> || <math>\sigma_{zz}-\frac{\sigma^2_{xz}}{\sigma_{xx}}+|\sigma_{yz}-\frac{\sigma_{xz}\sigma_{xy}}{\sigma_{xx}}|</math>
|-
| 6 || <math>\sigma_{yy}<0</math> || <math>\sigma_{xx}-\frac{\sigma^2_{xy}}{\sigma_{yy}} +|\sigma_{xz}-\frac{\sigma_{yz}\sigma_{xy}}{\sigma_{yy}}|</math> || 0 || <math>\sigma_{zz}-\frac{\sigma^2_{yz}}{\sigma_{yy}} +|\sigma_{xz}-\frac{\sigma_{yz}\sigma_{xy}}{\sigma_{yy}}|</math>
|-
| 7 || <math>\sigma_{zz}<0</math> || <math>\sigma_{xx}-\frac{\sigma^2_{xz}}{\sigma_{zz}} +|\sigma_{xy}-\frac{\sigma_{yz}\sigma_{xz}}{\sigma_{zz}}|</math> || <math>\sigma_{yy} -\frac{\sigma^2_{yz}}{\sigma_{zz}} +|\sigma_{xy} -\frac{\sigma_{xz}\sigma_{yz}}{\sigma_{zz}}|</math> || 0
|-
| 8 || <math>\sigma_{yz} + \sigma_{xz} + \sigma_{xy}</math> ≥ 0<br/><math>\sigma_{xz}\sigma_{xy} + \sigma_{yz}\sigma_{xy} + \sigma_{yz}\sigma_{xz}</math> ≥ 0<br/> || <math>\sigma_{xx} + \sigma_{xz} + \sigma_{xy}</math> || <math>\sigma_{yy} + \sigma_{yz} + \sigma_{xy}</math> || <math>\sigma_{zz} + \sigma_{yz} + \sigma_{xz}</math>
|-
| 9 || <math>- \sigma_{yz} - \sigma_{xz} + \sigma_{xy}</math> ≥ 0<br/><math>- \sigma_{xz}\sigma_{xy} - \sigma_{yz}\sigma_{xy} + \sigma_{yz}\sigma_{xz}</math> ≥ 0<br/> || <math>\sigma_{xx} - \sigma_{xz} + \sigma_{xy}</math> || <math>\sigma_{yy} - \sigma_{yz} + \sigma_{xy}</math> || <math>\sigma_{zz} - \sigma_{yz} - \sigma_{xz}</math>
|-
| 10 || <math>\sigma_{yz} - \sigma_{xz} - \sigma_{xy}</math> ≥ 0<br/><math>\sigma_{xz}\sigma_{xy} - \sigma_{yz}\sigma_{xy} - \sigma_{yz}\sigma_{xz}</math> ≥ 0<br/> || <math>\sigma_{xx} - \sigma_{xz} - \sigma_{xy}</math> || <math>\sigma_{yy} + \sigma_{yz} - \sigma_{xy}</math> || <math>\sigma_{zz} + \sigma_{yz} - \sigma_{xz}</math>
|-
| 11 || <math>- \sigma_{yz} + \sigma_{xz} - \sigma_{xy}</math> ≥ 0<br/><math>- \sigma_{xz}\sigma_{xy} + \sigma_{yz}\sigma_{xy} - \sigma_{yz}\sigma_{xz}</math> ≥ 0<br/> || <math>\sigma_{xx} + \sigma_{xz} - \sigma_{xy}</math> || <math>\sigma_{yy} - \sigma_{yz} - \sigma_{xy}</math> || <math>\sigma_{zz} - \sigma_{yz} + \sigma_{xz}</math>
|-
| 12 || <math>\sigma_{xy}\sigma_{xz}\sigma_{yz}<0</math> || <math>\sigma_{xx} - \frac{\sigma_{xz}\sigma_{xy}}{\sigma_{yz}}</math> || <math>\sigma_{yy} - \frac{\sigma_{yz}\sigma_{xy}}{\sigma_{xz}}</math> || <math>\sigma_{zz} - \frac{\sigma_{yz}\sigma_{xz}}{\sigma_{xy}}</math>
|-
|-
!{{CDD|node_1|split1|branch}}
!{{CDD|node_1|split1|nodes|split2|node}}
|}
|}
In [[geometry]], the '''simplectic honeycomb''' (or '''n-simplex honeycomb''') is a dimensional infinite series of [[Honeycomb (geometry)|honeycomb]]s, based on the <math>{\tilde{A}}_n</math> affine [[Coxeter group]] symmetry. It is given a [[Schläfli symbol]] {3<sup>[n+1]</sup>}, and is represented by a [[Coxeter-Dynkin diagram]] as a cyclic graph of ''n+1'' nodes with one node ringed. It is composed of n-[[simplex]] facets, along with all [[Rectification (geometry)|rectified]] n-simplices. The [[vertex figure]] of an ''n-simplex honeycomb'' is an [[Expansion (geometry)|expanded]] n-[[simplex]].


<math>I_{1}</math>, <math>I_{2}</math> and <math>I_{3}</math> are the [[Stress (mechanics)|stress invariants]] of the composite material stress tensor.
In 2 dimensions, the honeycomb represents the [[triangular tiling]], with Coxeter graph {{CDD|node_1|split1|branch}} filling the plane with alternately colored triangles. In 3 dimensions it represents the [[tetrahedral-octahedral honeycomb]], with Coxeter graph {{CDD|node_1|split1|nodes|split2|node}} filling space with alternately tetrahedral and octahedral cells. In 4 dimensions it is called the [[5-cell honeycomb]], with Coxeter graph {{CDD|node_1|split1|nodes|3ab|branch}}, with [[5-cell]] and [[rectified 5-cell]] facets. In 5 dimensions it is called the [[5-simplex honeycomb]], with Coxeter graph {{CDD|node_1|split1|nodes|3ab|nodes|split2|node}}, filling space by [[5-simplex]], [[rectified 5-simplex]], and [[birectified 5-simplex]] facets. In 6 dimensions it is called the [[6-simplex honeycomb]], with Coxeter graph {{CDD|node_1|split1|nodes|3ab|nodes|3ab|branch}}, filling space by [[6-simplex]], [[rectified 6-simplex]], and [[birectified 6-simplex]] facets.
 
The algorithm for obtaining the right solution is simple. Compute the reinforcement ratios of each possible solution that fulfills the conditions. Further ignore solutions with a reinforcement ratio less than zero. Compute the values of <math>\rho_{x}</math> + <math>\rho_{y}</math> + <math>\rho_{z}</math> and select the solution for which this value is smallest. The principal stresses in the brittle material can be computed as the eigenvalues of the brittle material stress tensor, for example by [[Jacobi method|Jacobi's method]].
 
The formulas can be simply checked by substituting the reinforcement ratios in the brittle material stress tensor and calculating the invariants. The first invariant needs to be less than or equal to zero. The second invariant needs to be greater than or equal to zero. These provide the conditions in column 2. For solution 2 to 12, the third invariant needs to be zero.<ref name="F"/>
 
==Examples==
 
The table below shows computed reinforcement ratios for 10 stress tensors. The applied reinforcement yield stress is <math>f_{y}</math> = 500 N/mm². The [[Density|mass density]] of the reinforcing bars is 7800&nbsp;kg/m<sup>3</sup>. In the table <math>\sigma_{m}</math> is the computed brittle material stress. <math>m_{r}</math> is the optimised amount of reinforcement.


== By dimension ==
{| class="wikitable"
{| class="wikitable"
|- style="height: 30px;"
!height=30|n
| width="50pt" | || <math>\sigma_{xx}</math> || <math>\sigma_{yy}</math> || <math>\sigma_{zz}</math> || <math>\sigma_{yz}</math> || <math>\sigma_{xz}</math> || <math>\sigma_{xy}</math> || || <math>\rho_{x}</math> || <math>\rho_{y}</math> || <math>\rho_{z}</math> || <math>\sigma_{m}</math> || <math>m_{r}</math>
!<math>{\tilde{A}}_{2+}</math>
!Tessellation
!Vertex figure
!Facets per vertex figure
!Vertices per vertex figure
!Edge figure
|-
|-
| 1 || 1 N/mm²|| 2 N/mm²|| 3 N/mm²|| -4 N/mm²|| 3 N/mm²|| -1 N/mm²|| || 1.00%|| 1.40%|| 2.00%|| -10.65 N/mm² || 343&nbsp;kg/m<sup>3</sup>
|1
|<math>{\tilde{A}}_1</math>
|[[File:Regular_apeirogon.png|80px]]<BR>[[Apeirogon]]<BR>{{CDD|node_1|infin|node}}
|{{CDD|node_1}}
|1
|2
| -
|-
|-
| 2 || -5 || 2 || 3 || 4 || 3 || 1 || || 0.00 || 1.36 || 1.88 || -10.31 || 253
|2
|<math>{\tilde{A}}_2</math>
|[[Image:Uniform tiling 333-t1.png|80px]]<BR>[[Triangular tiling]]<BR>2-simplex honeycomb<BR>{{CDD|node_1|split1|branch}}
|[[Image:Truncated triangle.png|80px]]<BR>[[Hexagon]]<BR>(Truncated triangle)<BR>{{CDD|node_1|3|node_1}}
|3 [[triangle]]s<BR>3 [[hexagon|rectified triangles]]
|6
|[[Line segment]]<BR>{{CDD|node_1}}
|-
|-
| 3 || -5 || -6 || 3 || 4 || 3 || 1 || || 0.00 || 0.00 || 1.69 || -10.15 || 132
|3
|<math>{\tilde{A}}_3</math>
|[[File:Tetrahedral-octahedral honeycomb2.png|80px]]<BR>[[Tetrahedral-octahedral honeycomb]]<BR>3-simplex honeycomb<BR>{{CDD|node_1|split1|nodes|split2|node}}
|[[Image:Uniform polyhedron-33-t02.png|80px]]<BR>[[Cuboctahedron]]<BR>(Cantellated tetrahedron)<BR>{{CDD|node_1|3|node|3|node_1}}
|4+4 [[tetrahedron]]<BR>6 [[octahedron|rectified tetrahedra]]
|12
|[[Rectangle]]<BR>{{CDD|node_1|2|node_1}}
|-
|-
| 4 || -5 || -6 || -6 || 4 || 3 || 1 || || 0.00 || 0.00 || 0.00 || -10.44 || 0
|4
|<math>{\tilde{A}}_4</math>
|[[4-simplex honeycomb]]<BR>{{CDD|node_1|split1|nodes|3ab|branch}}
|[[Image:Schlegel half-solid runcinated 5-cell.png|80px]]<BR>[[Runcinated 5-cell]]<BR>{{CDD|node_1|3|node|3|node|3|node_1}}
|5+5 [[5-cell]]s<BR>10+10 [[rectified 5-cell]]s
|20
|[[File:Runcinated_5-cell_verf.png|60px]]<BR>Triangular antiprism<BR>{{CDD|node_h|3|node_h|2|node_h}}
|-
|-
| 5 || 1 || 2 || 3 || -4 || -3 || -1 || || 0.60 || 1.00 || 2.00 || -10.58 || 281
|5
|<math>{\tilde{A}}_5</math>
|[[5-simplex honeycomb]]<BR>{{CDD|node_1|split1|nodes|3ab|nodes|split2|node}}
|[[File:5-simplex_t04.svg|80px]]<BR>[[Stericated 5-simplex]]<BR>{{CDD|node_1|3|node|3|node|3|node|3|node_1}}
|6+6 [[5-simplex]]<BR>15+15 [[rectified 5-simplex]]<BR>20 [[birectified 5-simplex]]
|30
|[[File:Stericated_hexateron_verf.png|60px]]<BR>Tetrahedral antiprism<BR>{{CDD|node_h|4|node|3|node|2|node_h}}
|-
|-
| 6 || 1 || -2 || 3 || -4 || 3 || 2 || || 0.50 || 0.13 || 1.80 || -10.17 || 190
|6
|<math>{\tilde{A}}_6</math>
|[[6-simplex honeycomb]]<BR>{{CDD|node_1|split1|nodes|3ab|nodes|3ab|branch}}
|[[File:6-simplex_t05.svg|80px]]<BR>[[Pentellated 6-simplex]]<BR>{{CDD|node_1|3|node|3|node|3|node|3|node|3|node_1}}
|7+7 [[6-simplex]]<BR>21+21 [[rectified 6-simplex]]<BR>35+35 [[birectified 6-simplex]]
|42
|4-simplex antiprism
|-
|-
| 7 || 1 || 2 || 3 || 4 || 2 || -1 || || 0.40 || 1.00 || 1.80 || -9.36 || 250
|7
|<math>{\tilde{A}}_7</math>
|[[7-simplex honeycomb]]<BR>{{CDD|node_1|split1|nodes|3ab|nodes|3ab|nodes|split2|node}}
|[[File:7-simplex_t06.svg|80px]]<BR>[[Hexicated 7-simplex]]<BR>{{CDD|node_1|3|node|3|node|3|node|3|node|3|node|3|node_1}}
|8+8 [[7-simplex]]<BR>28+28 [[rectified 7-simplex]]<BR>56+56 [[birectified 7-simplex]]<BR>70 [[trirectified 7-simplex]]
|56
|5-simplex antiprism
|-
|-
| 8 || 2 || -2 || 5 || 2 || -4 || 6 || || 2.40 || 0.40 || 1.40 || -15.21 || 328
|8
|<math>{\tilde{A}}_8</math>
|[[8-simplex honeycomb]]<BR>{{CDD|node_1|split1|nodes|3ab|nodes|3ab|nodes|3ab|branch}}
|[[File:8-simplex_t07.svg|80px]]<BR>[[Heptellated 8-simplex]]<BR>{{CDD|node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node_1}}
|9+9 [[8-simplex]]<BR>36+36 [[rectified 8-simplex]]<BR>84+84 [[birectified 8-simplex]]<BR>126+126 [[trirectified 8-simplex]]
|72
|6-simplex antiprism
|-
|-
| 9 || -3 || -7 || 0 || 2 || -4 || 6 || || 0.89 || 0.00 || 0.57 || -14.76 || 114
|9
|<math>{\tilde{A}}_9</math>
|[[9-simplex honeycomb]]<BR>{{CDD|node_1|split1|nodes|3ab|nodes|3ab|nodes|3ab|nodes|split2|node}}
|[[File:9-simplex_t08.svg|80px]]<BR>[[Octellated 9-simplex]]<BR>{{CDD|node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node_1}}
|10+10 [[9-simplex]]<BR>45+45 [[rectified 9-simplex]]<BR>120+120 [[birectified 9-simplex]]<br>210+210 [[trirectified 9-simplex]]<br>252 [[quadrirectified 9-simplex]]
|90
|7-simplex antiprism
|-
|-
| 10 || 3 || 0 || 10 || 0 || 5 || 0 || || 1.60 || 0.00 || 3.00 || -10.00 || 359
|10
|<math>{\tilde{A}}_{10}</math>
|[[10-simplex honeycomb]]<BR>{{CDD|node_1|split1|nodes|3ab|nodes|3ab|nodes|3ab|nodes|3ab|branch}}
|[[File:10-simplex_t09.svg|80px]]<BR>[[Ennecated 10-simplex]]<BR>{{CDD|node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node_1}}
|11+11 [[10-simplex]]<BR>55+55 [[rectified 10-simplex]]<BR>165+165 [[birectified 10-simplex]]<BR>330+330 [[trirectified 10-simplex]]<BR>462+462 [[quadrirectified 10-simplex]]
|110
|8-simplex antiprism
|-
|-
|11
|<math>{\tilde{A}}_{11}</math>
|11-simplex honeycomb
|...
|...
|...
|...
|}
|}


==Extension==
== Projection by folding ==


The above solution can be very useful to design reinforcement, however, it has some practical limitations. The following aspects can be included too if the problem is solved using [[convex optimization]].
The (2n-1)-simplex honeycombs and 2n-simplex honeycombs can be projected into the n-dimensional [[hypercubic honeycomb]] by a [[Coxeter–Dynkin diagram#Geometric folding|geometric folding]] operation that maps two pairs of mirrors into each other, sharing the same [[vertex arrangement]]:
*Multiple stress tensors in one point due to multiple loads on the structure instead of only one stress tensor,
*A constraint imposed to crack widths at the surface of the structure,
*Shear stress in the crack (aggregate interlock),
*Reinforcement in other directions than x, y and z,
*Reinforcing bars that already have been placed in the reinforcement design process,
*The whole structure instead of one small material cube in turn.
*Large reinforcement ratio's
*Compression reinforcement


Minimise |<math>\rho_{1}</math>| + |<math>\rho_{2}</math>| + |<math>\rho_{3}</math>|.
{|class=wikitable
|-
!<math>{\tilde{A}}_2</math>
|{{CDD|node_1|split1|branch}}
!<math>{\tilde{A}}_4</math>
|{{CDD|node_1|split1|nodes|3ab|branch}}
!<math>{\tilde{A}}_6</math>
|{{CDD|node_1|split1|nodes|3ab|nodes|3ab|branch}}
!<math>{\tilde{A}}_8</math>
|{{CDD|node_1|split1|nodes|3ab|nodes|3ab|nodes|3ab|branch}}
!<math>{\tilde{A}}_{10}</math>
|{{CDD|node_1|split1|nodes|3ab|nodes|3ab|nodes|3ab|nodes|3ab|branch}}
|...


Variables <math>\rho_{xx}</math>, <math>\rho_{yy}</math>, <math>\rho_{zz}</math>, <math>\rho_{yz}</math>, <math>\rho_{xz}</math>, <math>\rho_{xy}</math>.
|-
 
!<math>{\tilde{A}}_3</math>
Constraint Eigenvalues of <math>T_{ij}</math> ≤ 0.
|{{CDD|nodes_10r|splitcross|nodes}}
 
!<math>{\tilde{A}}_3</math>
<math>\rho_{1}</math>, <math>\rho_{2}</math> and <math>\rho_{3}</math> are the eigenvalues of the reinforcement tensor. <math>T_{ij}</math> is the brittle material stress tensor.
|{{CDD|node_1|split1|nodes|split2|node}}
!<math>{\tilde{A}}_5</math>
|{{CDD|node_1|split1|nodes|3ab|nodes|split2|node}}
!<math>{\tilde{A}}_7</math>
|{{CDD|node_1|split1|nodes|3ab|nodes|3ab|nodes|split2|node}}
!<math>{\tilde{A}}_9</math>
|{{CDD|node_1|split1|nodes|3ab|nodes|3ab|nodes|3ab|nodes|split2|node}}
|...
|-
!<math>{\tilde{C}}_1</math>
|{{CDD|node_1|infin|node}}
!<math>{\tilde{C}}_2</math>
|{{CDD|node_1|4|node|4|node}}
!<math>{\tilde{C}}_3</math>
|{{CDD|node_1|4|node|3|node|4|node}}
!<math>{\tilde{C}}_4</math>
|{{CDD|node_1|4|node|3|node|3|node|4|node}}
!<math>{\tilde{C}}_5</math>
|{{CDD|node_1|4|node|3|node|3|node|3|node|4|node}}
|...
|}


:<math>
== Kissing number ==
T_{ij} =  
\left[{\begin{matrix}
\sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\
\sigma_{xy} & \sigma_{yy} & \sigma_{yz} \\
\sigma_{xz} & \sigma_{yz} & \sigma_{zz} \\
\end{matrix}}\right]_{ij}
- f_{y} \sum_{k}


\left[{\begin{matrix}
These honeycombs, seen as tangent n-spheres located at the center of each honeycomb vertex have a fixed number of contacting spheres and correspond to the number of vertices in the [[vertex figure]]. For 2 and 3 dimensions, this represents the highest [[kissing number]] for 2 and 3 dimensions, but fall short on higher dimensions. In 2-dimensions, the triangular tiling defines a circle packing of 6 tangent spheres arranged in a regular hexagon, and for 3 dimensions there are 12 tangent spheres arranged in an [[cuboctahedron|cuboctahedral]] configuration. For 4 to 8 dimensions, the kissing numbers are [[Expanded 4-simplex|20]], [[Expanded 5-simplex|30]], [[Expanded 5-simplex|42]], [[Expanded 6-simplex|56]], and [[Expanded 7-simplex|72]] spheres, while the greatest solutions are 24, 40, 72, 126, and 240 spheres respectively.
\rho_{xx k} & \rho_{xy k} & \rho_{xz k} \\
\rho_{xy k} & \rho_{yy k} & \rho_{yz k} \\
\rho_{xz k} & \rho_{yz k} & \rho_{zz k} \\
\end{matrix}}\right]
- f_{y}
\left[{\begin{matrix}
\rho_{xx} & \rho_{xy} & \rho_{xz} \\
\rho_{xy} & \rho_{yy} & \rho_{yz} \\
\rho_{xz} & \rho_{yz} & \rho_{zz} \\
\end{matrix}}\right]
</math>.
<math>i</math> is the number of the load combination on the structure related to the ultimate limit state.
<math>j</math> is the number of the material point.
<math>k</math> is the number of the rebar that is already placed.


==See also==
== See also==
*[[Reinforced concrete]]
* [[Truncated simplectic honeycomb]]
*[[Solid mechanics]]
* [[Omnitruncated simplectic honeycomb]]
*[[Structural engineering]]


==References==
== References ==
* [[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.
* [[Norman Johnson (mathematician)|Norman Johnson]] ''Uniform Polytopes'', Manuscript (1991)
* [[Coxeter|Coxeter, H.S.M.]] ''[[Regular Polytopes (book)|Regular Polytopes]]'', (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8
* '''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 [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html]
** (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)
** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. Zeit. 200 (1988) 3-45]


<references>
{{Honeycombs}}
<ref name="A">Andreasen B.S., Nielsen M.P., Armiering af beton I det tredimesionale tilfælde, Bygningsstatiske meddelelser, Vol. 5 (1985), No. 2-3, pp. 25-79 (in Danish).</ref>
<ref name="F">Foster S.J., Marti P., Mojsilovic N., Design of Reinforced Concrete Solids Using Stress Analysis, ACI Structural Journal, Nov.-Dec. 2003, pp. 758-764.</ref>
<ref name="H1">Hoogenboom P.C.J., De Boer A., "Computation of reinforcement for solid concrete", Heron, Vol. 53 (2008), No. 4. pp. 247-271.</ref>
<ref name="H2">Hoogenboom P.C.J., De Boer A., "Computation of optimal concrete reinforcement in three dimensions", Proceedings of EURO-C 2010, Computational Modelling of Concrete Structures, pp. 639-646, Editors Bicanic et al. Publisher CRC Press, London.</ref>
<ref name="N">Nielsen M.P., Hoang L.C., Limit Analysis and Concrete Plasticity, third edition, CRC Press, 2011.</ref>
</references>


[[Category:Composite materials]]
[[Category:Honeycombs (geometry)]]
[[Category:Plasticity (physics)]]
[[Category:Polytopes]]
[[Category:Structural analysis]]
[[Category:Concrete]]

Revision as of 18:04, 18 August 2014

A~2 A~3
Triangular tiling Tetrahedral-octahedral honeycomb

With red and yellow equilateral triangles
With cyan and yellow tetrahedra, and red rectified tetrahedra (octahedron)
Template:CDD Template:CDD

In geometry, the simplectic honeycomb (or n-simplex honeycomb) is a dimensional infinite series of honeycombs, based on the A~n affine Coxeter group symmetry. It is given a Schläfli symbol {3[n+1]}, and is represented by a Coxeter-Dynkin diagram as a cyclic graph of n+1 nodes with one node ringed. It is composed of n-simplex facets, along with all rectified n-simplices. The vertex figure of an n-simplex honeycomb is an expanded n-simplex.

In 2 dimensions, the honeycomb represents the triangular tiling, with Coxeter graph Template:CDD filling the plane with alternately colored triangles. In 3 dimensions it represents the tetrahedral-octahedral honeycomb, with Coxeter graph Template:CDD filling space with alternately tetrahedral and octahedral cells. In 4 dimensions it is called the 5-cell honeycomb, with Coxeter graph Template:CDD, with 5-cell and rectified 5-cell facets. In 5 dimensions it is called the 5-simplex honeycomb, with Coxeter graph Template:CDD, filling space by 5-simplex, rectified 5-simplex, and birectified 5-simplex facets. In 6 dimensions it is called the 6-simplex honeycomb, with Coxeter graph Template:CDD, filling space by 6-simplex, rectified 6-simplex, and birectified 6-simplex facets.

By dimension

n A~2+ Tessellation Vertex figure Facets per vertex figure Vertices per vertex figure Edge figure
1 A~1
Apeirogon
Template:CDD 		Template:CDD 1 2 -
2 A~2
Triangular tiling
2-simplex honeycomb
Template:CDD
Hexagon
(Truncated triangle)
Template:CDD 		3 triangles
3 rectified triangles 		6 Line segment
Template:CDD
3 A~3
Tetrahedral-octahedral honeycomb
3-simplex honeycomb
Template:CDD
Cuboctahedron
(Cantellated tetrahedron)
Template:CDD 		4+4 tetrahedron
6 rectified tetrahedra 		12 Rectangle
Template:CDD
4 A~4 4-simplex honeycomb
Template:CDD
Runcinated 5-cell
Template:CDD 		5+5 5-cells
10+10 rectified 5-cells 		20
Triangular antiprism
Template:CDD
5 A~5 5-simplex honeycomb
Template:CDD
Stericated 5-simplex
Template:CDD 		6+6 5-simplex
15+15 rectified 5-simplex
20 birectified 5-simplex 		30
Tetrahedral antiprism
Template:CDD
6 A~6 6-simplex honeycomb
Template:CDD
Pentellated 6-simplex
Template:CDD 		7+7 6-simplex
21+21 rectified 6-simplex
35+35 birectified 6-simplex 		42 4-simplex antiprism
7 A~7 7-simplex honeycomb
Template:CDD
Hexicated 7-simplex
Template:CDD 		8+8 7-simplex
28+28 rectified 7-simplex
56+56 birectified 7-simplex
70 trirectified 7-simplex 		56 5-simplex antiprism
8 A~8 8-simplex honeycomb
Template:CDD 		File:8-simplex t07.svg
Heptellated 8-simplex
Template:CDD 		9+9 8-simplex
36+36 rectified 8-simplex
84+84 birectified 8-simplex
126+126 trirectified 8-simplex 		72 6-simplex antiprism
9 A~9 9-simplex honeycomb
Template:CDD 		File:9-simplex t08.svg
Octellated 9-simplex
Template:CDD 		10+10 9-simplex
45+45 rectified 9-simplex
120+120 birectified 9-simplex
210+210 trirectified 9-simplex
252 quadrirectified 9-simplex 		90 7-simplex antiprism
10 A~10 10-simplex honeycomb
Template:CDD 		File:10-simplex t09.svg
Ennecated 10-simplex
Template:CDD 		11+11 10-simplex
55+55 rectified 10-simplex
165+165 birectified 10-simplex
330+330 trirectified 10-simplex
462+462 quadrirectified 10-simplex 		110 8-simplex antiprism
11 A~11 11-simplex honeycomb ... ... ... ...

Projection by folding

The (2n-1)-simplex honeycombs and 2n-simplex honeycombs can be projected into the n-dimensional hypercubic honeycomb by a geometric folding operation that maps two pairs of mirrors into each other, sharing the same vertex arrangement:

A~2 Template:CDD A~4 Template:CDD A~6 Template:CDD A~8 Template:CDD A~10 Template:CDD ...
A~3 Template:CDD A~3 Template:CDD A~5 Template:CDD A~7 Template:CDD A~9 Template:CDD ...
C~1 Template:CDD C~2 Template:CDD C~3 Template:CDD C~4 Template:CDD C~5 Template:CDD ...

Kissing number

These honeycombs, seen as tangent n-spheres located at the center of each honeycomb vertex have a fixed number of contacting spheres and correspond to the number of vertices in the vertex figure. For 2 and 3 dimensions, this represents the highest kissing number for 2 and 3 dimensions, but fall short on higher dimensions. In 2-dimensions, the triangular tiling defines a circle packing of 6 tangent spheres arranged in a regular hexagon, and for 3 dimensions there are 12 tangent spheres arranged in an cuboctahedral configuration. For 4 to 8 dimensions, the kissing numbers are 20, 30, 42, 56, and 72 spheres, while the greatest solutions are 24, 40, 72, 126, and 240 spheres respectively.

See also

References

  • 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.
  • Norman Johnson Uniform Polytopes, Manuscript (1991)
  • Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8
  • 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 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)
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]

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