Spheritorus (EntityTopic, 11)

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{{Shape|Toracubinder|''No image''|4|1, ?, ?, 0|1|N/A|N/A|[[Line (object)|E]][[Circle|L]][[Cylinder|E]]Q|(211) ((x,y),z,w)|N/A|N/A|N/A}}
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<[#ontology [kind topic] [cats 4D Curved Toratope]]>
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{{STS Shape
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| dim=4
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| elements=1, ?, ?, 0
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| genus=0
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| ssc2=T((2)2)
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| extra={{STS Toratope
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| expand=[[Cylspherinder|32]]
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| notation=((II)II)
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| index=5b
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}}}}
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===Geometry===
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Of all the [[four-dimensional torii]], the '''spheritorus''', previously known as the '''toracubinder''', is the closest analog to the three-dimensional [[torus]]. It is formed by taking an uncapped [[spherinder]] and connecting its ends in a loop. Its [[toratopic dual]] is the [[torisphere]]. It has two possible cross-sections in coordinate planes through the origin: the [[torus]], and two disjoint [[sphere]]s.  
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The '''toracubinder''' is a special case of a [[surcell of revolution]] where the base is a [[cylinder]].  
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===Equations===
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== Equations ==
*Variables:
*Variables:
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<blockquote>''R'' ⇒ major radius of the toracubinder<br>
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<blockquote>''R'' ⇒ major radius of the spheritorus<br>
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''r'' ⇒ minor radius of the toracubinder<br>
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''r'' ⇒ minor radius of the spheritorus<br>
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''h'' ⇒ height of the toracubinder</blockquote>
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''h'' ⇒ height of the spheritorus</blockquote>
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*All points (''x'', ''y'', ''z'', ''w'') that lie on the [[surcell]] of a toracubinder will satisfy the following equation:  
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*All points (''x'', ''y'', ''z'', ''w'') that lie on the [[surcell]] of a spheritorus will satisfy the following equation:  
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<blockquote>(sqrt(x<sup>2</sup>+y<sup>2</sup>)-R)<sup>2</sup> + z<sup>2</sup> + w<sup>2</sup> = r<sup>2</sup></blockquote>
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<blockquote>(√(''x''<sup>2</sup> + ''y''<sup>2</sup>) − ''R'')<sup>2</sup> + ''z''<sup>2</sup> + ''w''<sup>2</sup> = ''r''<sup>2</sup></blockquote>
*The parametric equations are:
*The parametric equations are:
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w = r sin a </blockquote>
w = r sin a </blockquote>
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*The [[hypervolume]]s of a toracubinder are given by:
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*The [[hypervolume]]s of a spheritorus are given by:
<blockquote>total edge length = ''Unknown''<br>
<blockquote>total edge length = ''Unknown''<br>
total surface area = ''Unknown''<br>
total surface area = ''Unknown''<br>
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bulk = 2π<sup>2</sup>''Rr''<sup>2</sup>''h''</blockquote>
bulk = 2π<sup>2</sup>''Rr''<sup>2</sup>''h''</blockquote>
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*The [[realmic]] [[cross-section]]s (''n'') of a toracubinder are:
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*The [[realmic]] [[cross-section]]s (''n'') of a spheritorus are:
<blockquote>''Unknown''</blockquote>
<blockquote>''Unknown''</blockquote>
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<br clear="all"><br>
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{{Polychora}}
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== Cross-sections ==
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{{Rotopes}}
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[[User:Polyhedron Dude|Jonathan Bowers aka Polyhedron Dude]] created these two excellent cross-section renderings:<br/>
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<[#embed [hash ZW0YGY7T1RAMWQP9SZH1JG565J] [width 676]]><br/>
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<[#embed [hash FM78R1H3E99EGCA10DWAJMX0M5] [width 676]]>
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{{Tetrashapes}}
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{{Toratope Nav B|4|5|6|IIII<br>Tesseract|(IIII)<br>Glome|(II)II<br>Cubinder|((II)II)<br>Spheritorus|(II)(II)<br>Duocylinder|((II)(II))<br>Tiger|chora}}

Latest revision as of 20:47, 11 February 2014


Of all the four-dimensional torii, the spheritorus, previously known as the toracubinder, is the closest analog to the three-dimensional torus. It is formed by taking an uncapped spherinder and connecting its ends in a loop. Its toratopic dual is the torisphere. It has two possible cross-sections in coordinate planes through the origin: the torus, and two disjoint spheres.

Equations

  • Variables:
R ⇒ major radius of the spheritorus
r ⇒ minor radius of the spheritorus
h ⇒ height of the spheritorus
  • All points (x, y, z, w) that lie on the surcell of a spheritorus will satisfy the following equation:
(√(x2 + y2) − R)2 + z2 + w2 = r2
  • The parametric equations are:
x = r cos a cos b cos c + R cos c
y = r cos a cos b sin c + R sin c
z = r cos a sin b
w = r sin a
total edge length = Unknown
total surface area = Unknown
surcell volume = 4π2Rr(r+h)
bulk = 2π2Rr2h
Unknown

Cross-sections

Jonathan Bowers aka Polyhedron Dude created these two excellent cross-section renderings:
(image)
(image)


Notable Tetrashapes
Regular: pyrochoron • aerochoron • geochoron • xylochoron • hydrochoron • cosmochoron
Powertopes: triangular octagoltriate • square octagoltriate • hexagonal octagoltriate • octagonal octagoltriate
Circular: glome • cubinder • duocylinder • spherinder • sphone • cylindrone • dicone • coninder
Torii: tiger • torisphere • spheritorus • torinder • ditorus


4a. IIII
Tesseract 		4b. (IIII)
Glome 		5a. (II)II
Cubinder 		5b. ((II)II)
Spheritorus 		6a. (II)(II)
Duocylinder 		6b. ((II)(II))
Tiger
List of toratopes
Retrieved from "http://hi.gher.space/wiki/Spheritorus"