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[[Image:Cuboctahedron.png|thumb|A rectified cube is a [[cuboctahedron]] – edges reduced to vertices, and vertices expanded into new faces]]
All plants should be grown in nutrient-rich soil that is toxin free. It is rich in Vitamin C, foliate, calcium, iron and the intrinsic fatty acids that our body demands. Somersizing is not far off from Paleo to begin with, and the foods that are not allowed on Paleo are extremely limited with Somersizing. Simply by selecting to stick to the Paleo diet program, we are increasing our probability of having a healthy, disease-free life. Conversely raw plants naturally have digestive enzymes. <br><br>A good rule of thumb when going to the supermarket is to shop around the edges of the building where all the fresh food is kept. A very easy and favorite recipe is to make  chicken stake with  eggs and carrot. In all allergies a professional doctor should be consulted for treatment and diet. Further, a diet low in carbohydrates will ensure there is a less chance of excess carbohydrates getting converted into fats and stored in the liver. The Paleo Diet Cookbook demonstrates how to utilize these identical natural foods so your family can eat the healthful foods our bodies were created for. <br><br>Especially with Paleo, where many ingredients can be quite specific, planning your meals and recipes is key to enjoing the diet while leading a relaxing and healthy day to day life. The Paleo diet is predominantly made up of high-protein meals. Both fruit and veggies contain a number of vitamins. Today, it appears to be a staple of the American diet. Diaz is qualified to handle any oral health need, including Invisalign, cosmetic dentistry, laser dentistry and dental implants in Pembroke Pines. <br><br>Still, he said that's better than his previously sedentary lifestyle of couch-surfing and boozing. There are three levels in Loren Cordain's book, The Paleo Diet. Desserts are usually made from fruits such as apples and cranberries. Eating alone brings its own unique kind of comfort. Results from most diets take a great deal of time, but not with this plan. <br><br>These tips will help you initially navigate your new eating habits. This diet,  also known as the caveman diet, stone age diet, or hunter-gatherer diet, is based on  the habitually food consumption patterns of humans during the Paleolithic era. It is infinitely customizable to meet YOUR specific needs and goals. If better taste is required then start from rabbit or goat meat. Are you looking for more information regarding Paleo Diet Grocery List. <br><br>The Dukan Diet starts with a short, sharp Attack stage where you consume nothing but protein - only meat, fish, eggs and (different of Atkins) no-fat or low-fat dairy products. Lobster, Mussels, Oysters, Scallops and shrimps (prawns). There has been a dramatic increase in the use of paleo cookbooks recently,just do a google search and see the amount of results returned. Like most other people with acne, I also tried the typical over the counter cleansers with salicylic acid and benzoyl peroxide, only to left with an irritated face and continuing breakouts. They avoid foods, like wheat and other grains that are strictly agrarian.<br><br>Should you have any issues relating to wherever in addition to how you can utilize [http://gritsandgroceries.info/sitemap/ paleo beef stew recipe], you are able to call us on our web site.
[[Image:Dual Cube-Octahedron.svg|thumb|A ''birectified'' cube is an octahedron – faces are reduced to points and new faces are centered on the original vertices.]]
[[Image:Rectified cubic honeycomb.jpg|thumb|A [[rectified cubic honeycomb]] – edges reduced to vertices, and vertices expanded into new cells.]]
In [[Euclidean geometry]], '''rectification''' is the process of truncating a [[polytope]] by marking the midpoints of all its edges, and cutting off its vertices at those points. The resulting polytope will be bounded by the [[vertex figure]]s and the rectified facets of the original polytope.
 
== Example of rectification as a final truncation to an edge ==
Rectification is the final point of a truncation process. For example on a cube this sequence shows four steps of a continuum of truncations between the regular and rectified form:
 
[[Image:Cube truncation sequence.svg|520px]]
 
 
== Higher degree rectifications ==
Higher degree rectification can be performed on higher-dimensional regular polytopes. The highest degree of rectification creates the [[dual polytope]]. A rectification truncates edges to points. A birectification truncates faces to points. A trirectification truncates cells to points, and so on.
 
== Example of birectification as a final truncation to a face ==
This sequence shows a ''birectified cube'' as the final sequence from a cube to the dual where the original faces are truncated down to a single point:
:[[Image:Birectified cube sequence.png|480px]]
 
== In polygons ==
 
The dual of a polygon is the same as its rectified form. New vertices are placed at the center of the edges of the original polygon.
 
== In polyhedra and plane tilings ==
{{See|quasiregular polyhedron}}
Each [[platonic solid]] and its [[dual polyhedron|dual]] have the same rectified polyhedron. (This is not true of polytopes in higher dimensions.)
 
The rectified polyhedron turns out to be expressible as the intersection of the original platonic solid with an appropriated scaled concentric version of its dual.  For this reason, its name is a combination of the names of the original and the dual:
 
# The rectified [[tetrahedron]], whose dual is the tetrahedron, is the ''tetratetrahedron'', better known as the [[octahedron]].  
# The rectified [[octahedron]], whose dual is the [[cube]], is the [[cuboctahedron]].
# The rectified [[icosahedron]], whose dual is the [[dodecahedron]], is the [[icosidodecahedron]].
# A rectified [[square tiling]] is a [[square tiling]].
# A rectified [[triangular tiling]] or [[hexagonal tiling]] is a [[trihexagonal tiling]].
 
Examples
{| class="wikitable"
!Family
!Parent
!Rectification
!Dual
|-
!{{CDD|node|p|node|q|node}}<BR>[p,q]
!{{CDD|node_1|p|node|q|node}}
!{{CDD|node|p|node_1|q|node}}
!{{CDD|node|p|node|q|node_1}}
|- align=center
![3,3]
|[[Image:Uniform polyhedron-33-t0.png|75px]]<BR>[[Tetrahedron]]
|[[Image:Uniform polyhedron-33-t1.png|75px]]<BR>[[Tetratetrahedron|Octahedron]]
|[[Image:Uniform polyhedron-33-t2.png|75px]]<BR>[[Tetrahedron]]
|- align=center
![4,3]
|[[Image:Uniform polyhedron-43-t0.png|75px]]<BR>[[Cube]]
|[[Image:Uniform polyhedron-43-t1.png|75px]]<BR>[[Cuboctahedron]]
|[[Image:Uniform polyhedron-43-t2.png|75px]]<BR>[[Octahedron]]
|- align=center
![5,3]
|[[Image:Uniform polyhedron-53-t0.png|75px]]<BR>[[Dodecahedron]]
|[[Image:Uniform polyhedron-53-t1.png|75px]]<BR>[[Icosidodecahedron]]
|[[Image:Uniform polyhedron-53-t2.png|75px]]<BR>[[Icosahedron]]
|- align=center
![6,3]
|[[Image:Uniform tiling 63-t0.png|75px]]<BR>[[Hexagonal tiling]]
|[[Image:Uniform tiling 63-t1.png|75px]]<BR>[[Trihexagonal tiling]]
|[[Image:Uniform tiling 63-t2.png|75px]]<BR>[[Triangular tiling]]
|- align=center
![7,3]
|[[Image:Uniform tiling 73-t0.png|75px]]<BR>[[Order-3 heptagonal tiling]]
|[[Image:Uniform tiling 73-t1.png|75px]]<BR>[[Triheptagonal tiling]]
|[[Image:Uniform tiling 73-t2.png|75px]]<BR>[[Order-7 triangular tiling]]
|- align=center
![4,4]
|[[Image:Uniform tiling 44-t0.png|75px]]<BR>[[Square tiling]]
|[[Image:Uniform tiling 44-t1.png|75px]]<BR>[[Square tiling]]
|[[Image:Uniform tiling 44-t2.png|75px]]<BR>[[Square tiling]]
|- align=center
![5,4]
|[[Image:Uniform tiling 54-t0.png|75px]]<BR>[[Order-4 pentagonal tiling]]
|[[Image:Uniform tiling 54-t1.png|75px]]<BR>[[tetrapentagonal tiling]]
|[[Image:Uniform tiling 54-t2.png|75px]]<BR>[[Order-5 square tiling]]
|}
 
=== In nonregular polyhedra ===
If a polyhedron is not regular, the edge midpoints surrounding a vertex may not be coplanar. However, a form of rectification is still possible in this case: every polyhedron has a [[polyhedral graph]] as its [[n-skeleton|1-skeleton]], and from that graph one may form the [[medial graph]] by placing a vertex at each edge midpoint of the original graph, and connecting two of these new vertices by an edge whenever they belong to consecutive edges along a common face. The resulting medial graph remains polyhedral, so by [[Steinitz's theorem]] it can be represented as a polyhedron.
 
The [[Conway polyhedron notation]] equivalent to rectification is '''ambo''', represented by '''a'''. Applying twice '''aa''', (rectifying a rectification) is Conway's [[expansion (geometry)|'''expand''']] operation, '''e''', which is the same as Johnson's [[Cantellation (geometry)|cantellation]] operation, t<sub>0,2</sub> generated from regular polyhedral and tilings.
 
== In polychora and 3d honeycomb tessellations ==
 
Each [[Convex_regular_4-polytope|convex regular polychoron]] has a rectified form as a [[uniform polychoron]].
 
A regular polychoron {p,q,r} has cells {p,q}. Its rectification will have two cell types, a rectified {p,q} polyhedron left from the original cells and {q,r} polyhedron as new cells formed by each truncated vertex.
 
A rectified {p,q,r} is not the same as a rectified {r,q,p}, however. A further truncation, called [[bitruncation (geometry)|bitruncation]], is symmetric between a polychoron and its dual. See [[Uniform_polychoron#Geometric_derivations]].
 
Examples
{| class="wikitable"
!Family
!Parent
!Rectification
!Birectification<BR>(Dual rectification)
!Trirectification<BR>(Dual)
|-
!{{CDD|node|p|node|q|node|r|node}}<BR>[p,q,r]
!{{CDD|node_1|p|node|q|node|r|node}}
!{{CDD|node|p|node_1|q|node|r|node}}
!{{CDD|node|p|node|q|node_1|r|node}}
!{{CDD|node|p|node|q|node|r|node_1}}
|- align=center
![3,3,3]
|[[Image:Schlegel wireframe 5-cell.png|120px]]<BR>[[5-cell]]
|[[Image:Schlegel half-solid rectified 5-cell.png|120px]]<BR>[[rectified 5-cell]]
|[[Image:Schlegel half-solid rectified 5-cell.png|120px]]<BR>[[rectified 5-cell]]
|[[Image:Schlegel wireframe 5-cell.png|120px]]<BR>[[5-cell]]
|- align=center
![4,3,3]
|[[Image:Schlegel wireframe 8-cell.png|150px]]<BR>[[tesseract]]
|[[Image:Schlegel half-solid rectified 8-cell.png|150px]]<BR>[[rectified tesseract]]
|[[Image:Schlegel half-solid rectified 16-cell.png|150px]]<BR>Rectified 16-cell<BR>([[24-cell]])
|[[Image:Schlegel wireframe 16-cell.png|150px]]<BR>[[16-cell]]
|- align=center
![3,4,3]
|[[Image:Schlegel wireframe 24-cell.png|150px]]<BR>[[24-cell]]
|[[File:Schlegel half-solid cantellated 16-cell.png|150px]]<BR>[[rectified 24-cell]]
|[[File:Schlegel half-solid cantellated 16-cell.png|150px]]<BR>[[rectified 24-cell]]
|[[Image:Schlegel wireframe 24-cell.png|150px]]<BR>[[24-cell]]
|- align=center
![5,3,3]
|[[Image:Schlegel wireframe 120-cell.png|150px]]<BR>[[120-cell]]
|[[File:Rectified 120-cell schlegel halfsolid.png|150px]]<BR>[[rectified 120-cell]]
|[[File:Rectified_600-cell_schlegel_halfsolid.png|150px]]<BR>[[rectified 600-cell]]
|[[Image:Schlegel wireframe 600-cell vertex-centered.png|150px]]<BR>[[600-cell]]
|- align=center
![4,3,4]
|[[Image:Partial cubic honeycomb.png|150px]]<BR>[[Cubic honeycomb]]
|[[Image:Rectified cubic honeycomb.jpg|150px]]<BR>[[Rectified cubic honeycomb]]
|[[Image:Rectified cubic honeycomb.jpg|150px]]<BR>[[Rectified cubic honeycomb]]
|[[Image:Partial cubic honeycomb.png|150px]]<BR>[[Cubic honeycomb]]
|- align=center
![5,3,4]
|[[Image:Hyperbolic_orthogonal_dodecahedral_honeycomb.png|150px]]<BR>[[Order-4 dodecahedral honeycomb|Order-4 dodecahedral]]
|[[File:Rectified_order_4_dodecahedral_honeycomb.png|150px]]<BR>[[Rectified order-4 dodecahedral honeycomb|Rectified order-4 dodecahedral]]
|(No image)<BR>[[Rectified order-5 cubic honeycomb|Rectified order-5 cubic]]
|[[Image:Hyperb gcubic hc.png|150px]]<BR>[[Order-5 cubic honeycomb|Order-5 cubic]]
|}
 
== Degrees of rectification ==
 
A first rectification truncates edges down to points. If a polytope is [[Regular polytope|regular]], this form is represented by an extended [[Schläfli symbol]] notation t<sub>1</sub>{p,q,...}.
 
A second rectification, or '''birectification''', truncates [[Face (geometry)|faces]] down to points. If regular it has notation t<sub>2</sub>{p,q,...}. For [[polyhedron|polyhedra]], a birectification creates a [[dual polyhedron]].
 
Higher degree rectifications can be constructed for higher dimensional polytopes. In general an n-rectification truncates ''n-faces'' to points.
 
If an n-polytope is (n-1)-rectified, its [[Facet (mathematics)|facets]] are reduced to points and the polytope becomes its [[Dual polytope|dual]].
 
=== Notations and facets ===
 
There are different equivalent notations for each degree of rectification. These tables show the names by dimension and the two type of [[Facet (mathematics)|facet]]s for each.
 
==== Regular [[polygon]]s ====
 
[[Facet (mathematics)|Facet]]s are edges, represented as {2}.
 
{| class="wikitable"
!rowspan=2|name<BR>{p}
!rowspan=2|[[Coxeter-Dynkin diagram|Coxeter-Dynkin]]
!rowspan=2|t-notation<BR>[[Schläfli symbol]]
!colspan=3|Vertical [[Schläfli symbol]]
|-
!Name
!Facet-1
!Facet-2
|- align=center
|Parent
|{{CDD|node_1|p|node}}
|t<sub>0</sub>{p}
| {p}
| {2}
|
|- align=center
|Rectified
|{{CDD|node|p|node_1}}
|t<sub>1</sub>{p}
| {p}
|
| {2}
|}
 
==== Regular [[Uniform polyhedron|polyhedra]] and [[List of uniform tilings|tiling]]s ====
 
[[Facet (mathematics)|Facet]]s are regular polygons.
{| class="wikitable"
!rowspan=2|name<BR>{p,q}
!rowspan=2|[[Coxeter-Dynkin diagram|Coxeter-Dynkin]]
!rowspan=2|t-notation<BR>[[Schläfli symbol]]
!colspan=3|Vertical [[Schläfli symbol]]
|-
!Name
!Facet-1
!Facet-2
|- align=center
|Parent
|{{CDD|node_1|p|node|q|node}}
|t<sub>0</sub>{p,q}
| {p,q}
| {p}
|
|- align=center
|Rectified
|{{CDD|node|p|node_1|q|node}}
|t<sub>1</sub>{p,q}
| <math>\begin{Bmatrix} p \\ q  \end{Bmatrix}</math> = r{p,q}
| {p}
| {q}
|- align=center
|Birectified
|{{CDD|node|p|node|q|node_1}}
|t<sub>2</sub>{p,q}
| {q,p}
|
| {q}
|}
 
==== Regular [[Uniform polychoron|polychora]] and [[honeycomb]]s ====
[[Facet (mathematics)|Facet]]s are regular or rectified polyhedra.
{| class="wikitable"
!rowspan=2|name<BR>{p,q,r}
!rowspan=2|[[Coxeter-Dynkin diagram|Coxeter-Dynkin]]
!rowspan=2|t-notation<BR>[[Schläfli symbol]]
!colspan=3|Extended [[Schläfli symbol]]
|-
!Name
!Facet-1
!Facet-2
|- align=center
|Parent
|{{CDD|node_1|p|node|q|node|r|node}}
|t<sub>0</sub>{p,q,r}
| {p,q,r}
| {p,q}
|
|- align=center
|Rectified
|{{CDD|node|p|node_1|q|node|r|node}}
|t<sub>1</sub>{p,q,r}
| <math>\begin{Bmatrix} p \ \ \\ q , r \end{Bmatrix}</math> = r{p,q,r}
| <math>\begin{Bmatrix} p \\ q \end{Bmatrix}</math> = r{p,q}
| {q,r}
|- align=center
|Birectified<BR>(Dual rectified)
|{{CDD|node|p|node|q|node_1|r|node}}
|t<sub>2</sub>{p,q,r}
| <math>\begin{Bmatrix} q , p \\ r \ \ \end{Bmatrix}</math> = r{r,q,p}
| {q,r}
|<math>\begin{Bmatrix} q \\ r \end{Bmatrix}</math> = r{q,r}
|- align=center
|Trirectified<BR>(Dual)
|{{CDD|node|p|node|q|node|r|node_1}}
|t<sub>3</sub>{p,q,r}
| {r,q,p}
|
| {r,q}
|}
 
==== Regular [[polyteron]]s and 4-space [[Honeycomb (geometry)|honeycombs]] ====
[[Facet (mathematics)|Facet]]s are regular or rectified polychora.
{| class="wikitable"
!rowspan=2|name<BR>{p,q,r,s}
!rowspan=2|[[Coxeter-Dynkin diagram|Coxeter-Dynkin]]
!rowspan=2|t-notation<BR>[[Schläfli symbol]]
!colspan=3|Extended [[Schläfli symbol]]
|-
!Name
!Facet-1
!Facet-2
|- align=center
|Parent
|{{CDD|node_1|p|node|q|node|r|node|s|node}}
|t<sub>0</sub>{p,q,r,s}
| {p,q,r,s}
| {p,q,r}
|
|- align=center
|Rectified
|{{CDD|node|p|node_1|q|node|r|node|s|node}}
|t<sub>1</sub>{p,q,r,s}
| <math>\begin{Bmatrix} p \ \ \ \ \ \\ q , r , s \end{Bmatrix}</math> = r{p,q,r,s}
|<math>\begin{Bmatrix} p \ \ \\ q , r \end{Bmatrix}</math> = r{p,q,r}
|{q,r,s}
|- align=center
|Birectified<BR>(Birectified dual)
|{{CDD|node|p|node|q|node_1|r|node|s|node}}
|t<sub>2</sub>{p,q,r,s}
| <math>\begin{Bmatrix} q , p \\ r , s \end{Bmatrix}</math> = 2r{p,q,r,s}
|<math>\begin{Bmatrix} q , p \\  r \ \ \end{Bmatrix}</math> = r{r,q,p}
|<math>\begin{Bmatrix} q \ \ \\ r , s \end{Bmatrix}</math>  = r{q,r,s}
|- align=center
|Trirectified<BR>(Rectified dual)
|{{CDD|node|p|node|q|node|r|node_1|s|node}}
|t<sub>3</sub>{p,q,r,s}
| <math>\begin{Bmatrix} r , q , p \\ s \ \ \ \ \ \end{Bmatrix}</math> = r{s,r,q,p}
|{r,q,p}
|<math>\begin{Bmatrix} r , q \\ s \ \ \end{Bmatrix}</math> = r{s,r,q}
|- align=center
|Quadrirectified<BR>(Dual)
|{{CDD|node|p|node|q|node|r|node|s|node_1}}
|t<sub>4</sub>{p,q,r,s}
| {s,r,q,p}
|
| {s,r,q}
|}
 
==See also==
* [[Dual polytope]]
* [[Quasiregular polyhedron]]
* [[List of regular polytopes]]
* [[Truncation (geometry)]]
* [[Conway polyhedron notation]]
 
== References ==
* [[Coxeter|Coxeter, H.S.M.]] ''[[Regular Polytopes (book)|Regular Polytopes]]'', (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 (pp.145-154 Chapter 8: Truncation)
* [[Norman Johnson (mathematician)|Norman Johnson]] ''Uniform Polytopes'', Manuscript (1991)
** [[Norman Johnson (mathematician)|N.W. Johnson]]: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. Dissertation, University of Toronto, 1966
 
== External links ==
* {{Mathworld | urlname=Rectification | title=Rectification }}
* {{GlossaryForHyperspace | anchor=Rectification | title=Rectification }}
 
{{Polyhedron_operators}}
 
[[Category:Polyhedra]]
[[Category:Polychora]]
[[Category:Polytopes]]

Latest revision as of 14:14, 9 January 2015

All plants should be grown in nutrient-rich soil that is toxin free. It is rich in Vitamin C, foliate, calcium, iron and the intrinsic fatty acids that our body demands. Somersizing is not far off from Paleo to begin with, and the foods that are not allowed on Paleo are extremely limited with Somersizing. Simply by selecting to stick to the Paleo diet program, we are increasing our probability of having a healthy, disease-free life. Conversely raw plants naturally have digestive enzymes.

A good rule of thumb when going to the supermarket is to shop around the edges of the building where all the fresh food is kept. A very easy and favorite recipe is to make chicken stake with eggs and carrot. In all allergies a professional doctor should be consulted for treatment and diet. Further, a diet low in carbohydrates will ensure there is a less chance of excess carbohydrates getting converted into fats and stored in the liver. The Paleo Diet Cookbook demonstrates how to utilize these identical natural foods so your family can eat the healthful foods our bodies were created for.

Especially with Paleo, where many ingredients can be quite specific, planning your meals and recipes is key to enjoing the diet while leading a relaxing and healthy day to day life. The Paleo diet is predominantly made up of high-protein meals. Both fruit and veggies contain a number of vitamins. Today, it appears to be a staple of the American diet. Diaz is qualified to handle any oral health need, including Invisalign, cosmetic dentistry, laser dentistry and dental implants in Pembroke Pines.

Still, he said that's better than his previously sedentary lifestyle of couch-surfing and boozing. There are three levels in Loren Cordain's book, The Paleo Diet. Desserts are usually made from fruits such as apples and cranberries. Eating alone brings its own unique kind of comfort. Results from most diets take a great deal of time, but not with this plan.

These tips will help you initially navigate your new eating habits. This diet, also known as the caveman diet, stone age diet, or hunter-gatherer diet, is based on the habitually food consumption patterns of humans during the Paleolithic era. It is infinitely customizable to meet YOUR specific needs and goals. If better taste is required then start from rabbit or goat meat. Are you looking for more information regarding Paleo Diet Grocery List.

The Dukan Diet starts with a short, sharp Attack stage where you consume nothing but protein - only meat, fish, eggs and (different of Atkins) no-fat or low-fat dairy products. Lobster, Mussels, Oysters, Scallops and shrimps (prawns). There has been a dramatic increase in the use of paleo cookbooks recently,just do a google search and see the amount of results returned. Like most other people with acne, I also tried the typical over the counter cleansers with salicylic acid and benzoyl peroxide, only to left with an irritated face and continuing breakouts. They avoid foods, like wheat and other grains that are strictly agrarian.

Should you have any issues relating to wherever in addition to how you can utilize paleo beef stew recipe, you are able to call us on our web site.

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