# Cylinder (EntityTopic, 14)

A cylinder is a special case of a prism where the base is a circle. it has two circles at the ends connected by a surface, called a hose.

## Equations

• Variables:
h ⇒ height of cylinder
• All points (x, y, z) that lie on the surface of a cylinder will satisfy the following equations:
x2 + y2 = r2
abs(z) ≤ h/2
-- or --
x2 + y2 < r2
abs(z) = h/2
• All points (x, y, z) that lie on the edges of a cylinder will satisfy the following equations:
x2 + y2 = r2
abs(z) = h/2
total edge length = 4πr
surface area = 2πr(r+h)
volume = πr2h
[!x,!y] ⇒ rectangle with width (2rcos(πn/2)), height (h)
[!z] ⇒ circle of radius (r)

## Homology groups

All homology groups are zero unless stated. Here X is the shape in the given frame, and nZ is the direct sum of n copies of the group of integers Z.

1-frame (two circles)
H0X = 2ℤ, H1X = 2ℤ
2-frame (2 disks and a tube)
H0X = ℤ, H1X = 0, H2X = ℤ
3-frame (solid cylinder)
H0X = ℤ

## Cylindrogram

A cylindrogram is the surface of revolution of a parallelogram, just as a cylinder is the surface of revolution of a rectangle. It can also be thought of as a cylinder with a cone removed from one end and placed on the other. As such, this shape has the same volume as a cylinder with the same radius and height.

 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

 5. 3Sphere 6. 21Cylinder 7. 111Cube List of tapertopes

 2a. IIICube 2b. (III)Sphere 3a. (II)ICylinder 3b. ((II)I)Torus 4a. IIIITesseract 4b. (IIII)Glome List of toratopes

 6. (III)Sphere 7. [(II)I]Cylinder 8. <(II)I>Bicone List of bracketopes