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Changes: Penrose C4 Cartwheel

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{{Geomag-construction|PageTitle=Penrose C4 Cartwheel|Title=Penrose C4 Cartwheel|Filename=Penrose C4 model.jpg|Caption=CAD model of a Penrose C4 cartwheel
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{{Geomag-construction|PageTitle=Penrose C4 Cartwheel|Title=Penrose C4 Cartwheel|Filename=Penrose C4 model opt 2.jpg|Caption=CAD model of a Geomag Penrose C4 cartwheel
 
|Type=[[:Category:Mathematical|Mathematical]]
  
|Type=[[:Category:Mathematical|Mathematical]]
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|Triangles=|Squares=|Pentagons=|Rhombic=|Rods=<815|Spheres=361|Author=[[User:Amafirlian|Amafirlian]] 22:11, 5 August 2007 (UTC)}}
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|Triangles=126|Squares=|Pentagons=|Rhombic=|Rods=634+82|Spheres=361|Author=[[User:Amafirlian|Amafirlian]] 22:11, 5 August 2007 (UTC)}}
   
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This is a placeholder for a Penrose C4 cartwheel. It is build using the property that if you build a triangular pyramid on top of two connected rods that are spanning an angle of 70°, the top sphere is directly above the point where the sphere of a penrose dart needs to be. By using two levels one can build a complete penrose tiling. Note that the central rod (grey in the model) is effectively a support rod.
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This is a placeholder for a Penrose C4 cartwheel. It can be built using the property that if you build a triangular pyramid on top of two connected rods that are spanning an angle of 72°, the top sphere is directly above the point where the sphere of a penrose dart needs to be. So looking from directly above (and ignoring the perspective distortion) this looks like penrose dart. By using two levels one can build a complete penrose tiling. Note that the central rod (grey in the model) is effectively a support rod.
   
 
There is just one problem. When you put 2 Penrose darts built this way together, the two pyramids cause steric problems. The smallest angle between the rods is about 43°. The solution is to remove some of the rods that clash.
  
There is just one problem. When you put 2 Penrose darts built this way together, the two pyramids cause steric problems. The smallest angle between the rods is about 43°. The solution is to remove some of the rods that clash.
   
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Only the penrose darts can be filled with panels (2 triangular). After removing clashing rods only 126 remain.
 
{{clr}}
  
{{clr}}
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==Other views==
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<gallery>
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Image:Penrose C4 model.jpg|Top view (all rods).
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Image:Penrose C4 model 2.jpg|Top view with colored faces (all rods).
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Image:Penrose C4 model 3.jpg|View at an angle (all rods).
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Image:Penrose C4 model opt.jpg|Top view after removing clashing rods.
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Image:Penrose C4 model opt 2.jpg|View at an angle after removing clashing rods.
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</gallery>
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==Partial views==
  
==Partial views==
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Image:Penrose C4 partial c.jpg
  
Image:Penrose C4 partial c.jpg
 
Image:Penrose C4 partial d.jpg
  
Image:Penrose C4 partial d.jpg
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</gallery>
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==The real tiling==
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It's quite difficult to ''see'' the tiling from the Geomag representation (I'm still pondering good ways to color the tilings in Geomag). Here are a some renderings of the ''real'' thing:
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This paragraph and the picture below reproduced (permission sought)
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from Steve Dutch's site: [http://www.uwgb.edu/DutchS/symmetry/penrose.htm link]
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In some ways the pattern here, the Cartwheel, is the most important Penrose Tiling. The purple region at the center is outlined by a decagon consisting of a kite and dart edge. Every point in every tiling is contained inside an identical decagon (although the contents may differ).
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The outer portion of the pattern consists of two parts. There are ten yellow sectors and ten blue spokes. The spokes consist of "bowtie" units and the spokes can be flipped 180 degrees and still fit their adjacent sectors. That means there are 1024 possible spoke arrangements but after rotations and reflections are eliminated there are only 62 distinct patterns.
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<gallery>
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Image:PenrosTilingEJ.jpg |Image generated by Junaid Rashid
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Image:Dutch_Pencart.gif |Penrose Cartwheel by Steve Dutch
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Image:Dutch_penrose_infinite_sun.gif |Penrose Infinite Sun by Steve Dutch
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Image:Dutch_penrose_infinite_star.gif |Penrose Infinite Star by Steve Dutch
 
</gallery>
  
</gallery>
   
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*[http://www2.spsu.edu/math/tile/aperiodic/penrose/penrose2.htm Penrose kite and dart information]
  
*[http://www2.spsu.edu/math/tile/aperiodic/penrose/penrose2.htm Penrose kite and dart information]
 
*[http://www2.spsu.edu/math/tile/aperiodic/cartwheel/cartwheel1.htm Penrose Cartwheels]
  
*[http://www2.spsu.edu/math/tile/aperiodic/cartwheel/cartwheel1.htm Penrose Cartwheels]
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* [http://www.flickr.com/photos/junaidrashid/449860244 Junaid Rashid's generated picture on flickr]
  
* [http://www.flickr.com/photos/junaidrashid/449860244 Junaid Rashid's generated picture on flickr]
   

Latest revision as of 19:22, 11 August 2007

Penrose C4 Cartwheel
Penrose C4 model opt 2
CAD model of a Geomag Penrose C4 cartwheel
Type Mathematical
Panels 126 × Geomag triangle panel icon
Rods 634+82 × Geomag rod icon
Spheres 361 × Geomag ball icon
Author Amafirlian 22:11, 5 August 2007 (UTC)


This is a placeholder for a Penrose C4 cartwheel. It can be built using the property that if you build a triangular pyramid on top of two connected rods that are spanning an angle of 72°, the top sphere is directly above the point where the sphere of a penrose dart needs to be. So looking from directly above (and ignoring the perspective distortion) this looks like penrose dart. By using two levels one can build a complete penrose tiling. Note that the central rod (grey in the model) is effectively a support rod.

There is just one problem. When you put 2 Penrose darts built this way together, the two pyramids cause steric problems. The smallest angle between the rods is about 43°. The solution is to remove some of the rods that clash.

Only the penrose darts can be filled with panels (2 triangular). After removing clashing rods only 126 remain.

Other views[]


Partial views[]

The real tiling[]

It's quite difficult to see the tiling from the Geomag representation (I'm still pondering good ways to color the tilings in Geomag). Here are a some renderings of the real thing:

This paragraph and the picture below reproduced (permission sought) from Steve Dutch's site: link

In some ways the pattern here, the Cartwheel, is the most important Penrose Tiling. The purple region at the center is outlined by a decagon consisting of a kite and dart edge. Every point in every tiling is contained inside an identical decagon (although the contents may differ).

The outer portion of the pattern consists of two parts. There are ten yellow sectors and ten blue spokes. The spokes consist of "bowtie" units and the spokes can be flipped 180 degrees and still fit their adjacent sectors. That means there are 1024 possible spoke arrangements but after rotations and reflections are eliminated there are only 62 distinct patterns.

External Links[]

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