Crind (EntityTopic, 10)
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- | {{Shape| | + | <[#ontology [kind topic] [cats 3D Curved Bracketope]]> |
- | + | {{STS Shape | |
+ | | image=<[#embed [hash GG1HEVCTKEEHT280S5G8MEPCG2] [width 180]]> | ||
+ | | dim=3 | ||
+ | | elements=4, 4, 2 | ||
+ | | genus=0 | ||
+ | | ssc=([xy]z) | ||
+ | | ssc2=G4oM1 | ||
+ | | extra={{STS Bracketope | ||
+ | | index=9 | ||
+ | | notation=([II]I) | ||
+ | }}}} | ||
+ | |||
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'']]. | ||
- | + | Its maximal and minimal [[compression]]s are an irregular [[octahedron]] and a [[line segment]] respectively. | |
+ | |||
+ | == 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: | ||
- | <blockquote>'' | + | <blockquote>[!x,!y] ⇒ circle |
- | <br | + | [!z] ⇒ square |
- | {{ | + | </blockquote> |
+ | |||
+ | *The [[radial slice]]s ''θ'' of a crind are: | ||
+ | <blockquote>[x:xy,x:xz] ⇒ ellipse with major radius ''r''sin(45° + (''θ'' % 90°)√2 and minor radius ''r''<br> | ||
+ | [y:xy,y:yz,z:xz,z:yz] ⇒ "circle with ends cut" of unknown proportions</blockquote> | ||
+ | |||
+ | {{Trishapes}} | ||
+ | {{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 + y ≤ x + z = r
-- or --
x + z ≤ x + 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
- The hypervolumes of a crind are given by:
total edge length = 4πsqrt(2)r
surface area = Unknown
volume = πr3
- The planar cross-sections (n) of a crind are:
[!x,!y] ⇒ circle [!z] ⇒ square
- The radial slices θ of a crind are:
[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: | tetrahedron • cube • octahedron • dodecahedron • icosahedron |
Direct truncates: | tetrahedral truncate • cubic truncate • octahedral truncate • dodecahedral truncate • icosahedral truncate |
Mesotruncates: | stauromesohedron • stauroperihedron • stauropantohedron • rhodomesohedron • rhodoperihedron • rhodopantohedron |
Snubs: | snub staurohedron • snub rhodohedron |
Curved: | sphere • torus • cylinder • cone • frustum • crind |
8. <(II)I> Bicone | 9. ([II]I) Crind | 10. [IIII] Geochoron |
List of bracketopes |