Crind (EntityTopic, 10)

From Hi.gher. Space

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{{Shape|Crind|http://img46.imageshack.us/img46/779/crind1pt.png|3|4, 4, 2|0|N/A|N/A|[[Line (object)|E]][[Circle|L]][[Cylinder|E]]&E[[Square|E]]L|N/A|[[Square]], edge 1|N/A|N/A|N/A|([xy]z)|11}}
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<[#ontology [kind topic] [cats 3D Curved Bracketope]]>
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== Geometry ==
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{{STS Shape
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| image=<[#embed [hash GG1HEVCTKEEHT280S5G8MEPCG2] [width 180]]>
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| dim=3
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| elements=4, 4, 2
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| genus=0
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| ssc=([xy]z)
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| ssc2=G4oM1
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| extra={{STS Bracketope
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| index=9
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| notation=([II]I)
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}}}}
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A '''crind''' is the [[intersect]]ion of two perpendicular [[cylinder]]s. Due to momentum it will behave similarly to a [[duocylinder]] if left to roll on a [[surface]]. However, unlike a duocylinder, a crind can be stopped and then rolled in a different direction without needing to [[rotate]] it.
A '''crind''' is the [[intersect]]ion of two perpendicular [[cylinder]]s. Due to momentum it will behave similarly to a [[duocylinder]] if left to roll on a [[surface]]. However, unlike a duocylinder, a crind can be stopped and then rolled in a different direction without needing to [[rotate]] it.
The crind is also one of the few [[curved]] [[polyhedron|polyhedra]] that satisfies [[Euler's formula|Euler's ''F'' + ''V'' = ''E'' + ''2'']].
The crind is also one of the few [[curved]] [[polyhedron|polyhedra]] that satisfies [[Euler's formula|Euler's ''F'' + ''V'' = ''E'' + ''2'']].
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=== Equations ===
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Its maximal and minimal [[compression]]s are an irregular [[octahedron]] and a [[line segment]] respectively.
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== Equations ==
*Assumption: Crind is centered at the origin.
*Assumption: Crind is centered at the origin.
*Variables:
*Variables:
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*The [[planar]] [[cross-section]]s (''n'') of a crind are:
*The [[planar]] [[cross-section]]s (''n'') of a crind are:
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<blockquote>''Unknown''</blockquote>
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<blockquote>[!x,!y] ⇒ circle
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[!z] ⇒ square
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</blockquote>
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*The [[radial slice]]s ''θ'' of a crind are:
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<blockquote>[x:xy,x:xz] ⇒ ellipse with major radius ''r''sin(45° + (''θ'' % 90°)√2 and minor radius ''r''<br>
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[y:xy,y:yz,z:xz,z:yz] ⇒ "circle with ends cut" of unknown proportions</blockquote>
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{{Polyhedra}}
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{{Trishapes}}
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{{Bracketope Nav|10|11|12|<(xy)z><br>Bicone|([xy]z)<br>Crind|(<xy>z)<br>Narrow crind|hedra}}
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{{Bracketope Nav|8|9|10|<(II)I><br>Bicone|([II]I)<br>Crind|[IIII]<br>Geochoron|hedra}}

Latest revision as of 02:40, 26 March 2017


A crind is the intersection of two perpendicular cylinders. Due to momentum it will behave similarly to a duocylinder if left to roll on a surface. However, unlike a duocylinder, a crind can be stopped and then rolled in a different direction without needing to rotate it.

The crind is also one of the few curved polyhedra that satisfies Euler's F + V = E + 2.

Its maximal and minimal compressions are an irregular octahedron and a line segment respectively.

Equations

  • Assumption: Crind is centered at the origin.
  • Variables:
r ⇒ radius of crind
  • All points (x, y, z) that lie on the surface of a crind will satisfy the following equations:
x + yx + z = r
   -- or --
x + zx + y = r
  • All points (x, y, z) that lie on the edges of a crind will satisfy the following equation:
x + y = x + z = r
total edge length = 4πsqrt(2)r
surface area = Unknown
volume = πr3
[!x,!y] ⇒ circle [!z] ⇒ square
[x:xy,x:xz] ⇒ ellipse with major radius rsin(45° + (θ % 90°)√2 and minor radius r
[y:xy,y:yz,z:xz,z:yz] ⇒ "circle with ends cut" of unknown proportions


Notable Trishapes
Regular: tetrahedroncubeoctahedrondodecahedronicosahedron
Direct truncates: tetrahedral truncatecubic truncateoctahedral truncatedodecahedral truncateicosahedral truncate
Mesotruncates: stauromesohedronstauroperihedronstauropantohedronrhodomesohedronrhodoperihedronrhodopantohedron
Snubs: snub staurohedronsnub rhodohedron
Curved: spheretoruscylinderconefrustumcrind


8. <(II)I>
Bicone
9. ([II]I)
Crind
10. [IIII]
Geochoron
List of bracketopes