Duocylinder (EntityTopic, 14)

From Hi.gher. Space

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.


  • 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


The net of a duocylinder is two touching cylinders which have the length equal 2πr.

(image) (image)


Blue disk-first: (image) Red disk-first: (image) Face-first: (image)


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: pyrochoronaerochorongeochoronxylochoronhydrochoroncosmochoron
Powertopes: triangular octagoltriatesquare octagoltriatehexagonal octagoltriateoctagonal octagoltriate
Circular: glomecubinderduocylinderspherindersphonecylindronediconeconinder
Torii: tigertorispherespheritorustorinderditorus

13. 31
14. 22
15. 211
List of tapertopes

5a. (II)II
5b. ((II)II)
6a. (II)(II)
6b. ((II)(II))
7a. (III)I
7b. ((III)I)
List of toratopes

27. (<(II)I>I)
Biconic crind
28. [(II)(II)]
29. <(II)(II)>
Duocircular tegum
List of bracketopes