Duocylinder (EntityTopic, 14)

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Revision as of 16:01, 26 March 2017

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:
1. x2 + y2 = a2; z2 + w2b2
2. x2 + y2a2; 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.
total surface area = 4π2ab
surcell volume = 2π2ab(a + b)
bulk = π2a2b2

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. 31Spherinder 14. 22Duocylinder 15. 211Cubinder List of tapertopes

 5a. (II)IICubinder 5b. ((II)II)Spheritorus 6a. (II)(II)Duocylinder 6b. ((II)(II))Tiger 7a. (III)ISpherinder 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