Duocylinder (EntityTopic, 14)
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- | {{Shape | + | <[#ontology [kind topic] [cats 4D Curved Rotatope] [alt [[freebase:06n2fd]] [[wikipedia:Duocylinder]]]]> |
+ | {{STS Shape | ||
| dim=4 | | dim=4 | ||
- | | elements=2, 1, 0, 0 | + | | elements=2 [[Torus|torii]], 1 duocylinder margin, 0, 0 |
- | + | ||
| ssc=[(xy)(zw)] | | ssc=[(xy)(zw)] | ||
- | | | + | | ssc2=T2<sup>2</sup> |
- | | | + | | extra={{STS Tapertope |
- | | | + | | order=2, 0 |
- | | | + | | notation=22 |
- | | | + | | index=14 |
- | }} | + | }}{{STS Toratope |
+ | | expand=[[Duocylinder|22]] | ||
+ | | notation=(II)(II) | ||
+ | | index=6a | ||
+ | }}{{STS Bracketope | ||
+ | | index=28 | ||
+ | | notation=[(II)(II)] | ||
+ | }}}} | ||
- | A '''duocylinder''' is the [[Cartesian product]] of two [[circle]]s. It is also the limit of the [[set]] of [[duoprism]]s as ''m'' and ''n'' tend to infinity. | + | A '''duocylinder''' is the [[Cartesian product]] of two [[circle]]s, and is therefore the [[square]] of the circle. It is also the limit of the [[set]] of ''m'',''n''-[[duoprism]]s as ''m'' and ''n'' tend to infinity. |
== Equations == | == Equations == | ||
Line 29: | Line 36: | ||
*Each of these bounding volumes are topologically equivalent to the inside of a 3D torus. The set of points (w,x,y,z) that satisfy either the first or the second set of equations constitute the surface of the duocylinder. | *Each of these bounding volumes are topologically equivalent to the inside of a 3D torus. The set of points (w,x,y,z) that satisfy either the first or the second set of equations constitute the surface of the duocylinder. | ||
- | *The [[hypervolume]]s of a duocylinder are given by: | + | *The [[hypervolume]]s of a duocylinder are given by: |
<blockquote>total surface area = 4π<sup>2</sup>''ab'' <br> | <blockquote>total surface area = 4π<sup>2</sup>''ab'' <br> | ||
surcell volume = 2π<sup>2</sup>''ab''(''a'' + ''b'')<br> | surcell volume = 2π<sup>2</sup>''ab''(''a'' + ''b'')<br> | ||
Line 36: | Line 43: | ||
*The [[realmic]] [[cross-section]]s (''n'') of a duocylinder are cylinders of varying heights. | *The [[realmic]] [[cross-section]]s (''n'') of a duocylinder are cylinders of varying heights. | ||
+ | |||
+ | == Net == | ||
+ | |||
+ | The net of a duocylinder is two touching cylinders which have the length equal 2πr. | ||
+ | |||
+ | <[#embed [hash P3A8JCZSJHV3G19EZSQP28E1BA]]> | ||
+ | <[#embed [hash GHQST8GAA0DWPGB61TJFT4ZKK7]]> | ||
+ | |||
+ | == Cross-sections == | ||
+ | Blue disk-first: | ||
+ | <[#embed [hash 56J0GK3126QJMFXXTZEWW6MHKC]]> | ||
+ | Red disk-first: | ||
+ | <[#embed [hash 4JX9MPMKTHPHJ6BWARW1JXMGXM]]> | ||
+ | Face-first: | ||
+ | <[#embed [hash FN1JYDRFC5JAY1TTK3THWH2KZN]]> | ||
== Projection == | == Projection == | ||
The perspective projection of a duocylinder is the following shape. The purple part is one cell, and the black part is the other cell. | The perspective projection of a duocylinder is the following shape. The purple part is one cell, and the black part is the other cell. | ||
- | <blockquote> | + | <blockquote><[#embed [hash 144YT0MJBXKCP6M5ZEWQCMT5KV]]></blockquote> |
In a parallel projection, both cells collapse to [[cylinder]]s, one [[capped]] and one uncapped, resulting in a single cylinder being observed as the projection. | In a parallel projection, both cells collapse to [[cylinder]]s, one [[capped]] and one uncapped, resulting in a single cylinder being observed as the projection. | ||
{{Tetrashapes}} | {{Tetrashapes}} | ||
- | {{ | + | {{Tapertope Nav|13|14|15|31<br>Spherinder|22<br>Duocylinder|211<br>Cubinder|chora}} |
- | {{Bracketope Nav| | + | {{Toratope Nav A|5|6|7|(II)II<br>Cubinder|((II)II)<br>Spheritorus|(II)(II)<br>Duocylinder|((II)(II))<br>Tiger|(III)I<br>Spherinder|((III)I)<br>Torisphere|chora}} |
- | + | {{Bracketope Nav|27|28|29|(<(II)I>I)<br>Biconic crind|[(II)(II)]<br>Duocylinder|<(II)(II)><br>Duocircular tegum|chora}} | |
- | + |
Latest revision as of 08:02, 27 June 2018
A duocylinder is the Cartesian product of two circles, and is therefore the square of the circle. It is also the limit of the set of m,n-duoprisms as m and n tend to infinity.
Equations
- Variables:
a ⇒ radius of the circle in the xy plane
b ⇒ radius of the circle in the zw plane
- All points (x, y, z, w) that lie on the sole 2D face of a duocylinder will satisfy the following equations:
x2 + y2 = a2
z2 + w2 = b2
- A duocylinder has two cells which meet at the 2D face. These are given respectively by the systems of equations:
- x2 + y2 = a2; z2 + w2 ≤ b2
- x2 + y2 ≤ a2; z2 + w2 = b2
- Each of these bounding volumes are topologically equivalent to the inside of a 3D torus. The set of points (w,x,y,z) that satisfy either the first or the second set of equations constitute the surface of the duocylinder.
- The hypervolumes of a duocylinder are given by:
total surface area = 4π2ab
surcell volume = 2π2ab(a + b)
bulk = π2a2b2
- The realmic cross-sections (n) of a duocylinder are cylinders of varying heights.
Net
The net of a duocylinder is two touching cylinders which have the length equal 2πr.
Cross-sections
Blue disk-first: Red disk-first: Face-first:
Projection
The perspective projection of a duocylinder is the following shape. The purple part is one cell, and the black part is the other cell.
In a parallel projection, both cells collapse to cylinders, one capped and one uncapped, resulting in a single cylinder being observed as the projection.
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 |
13. 31 Spherinder | 14. 22 Duocylinder | 15. 211 Cubinder |
List of tapertopes |
5a. (II)II Cubinder | 5b. ((II)II) Spheritorus | 6a. (II)(II) Duocylinder | 6b. ((II)(II)) Tiger | 7a. (III)I Spherinder | 7b. ((III)I) Torisphere |
List of toratopes |
27. (<(II)I>I) Biconic crind | 28. [(II)(II)] Duocylinder | 29. <(II)(II)> Duocircular tegum |
List of bracketopes |