Cylinder (EntityTopic, 14)

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A cylinder is a special case of a prism where the base is a circle.

Equations

  • Variables:
r ⇒ radius of cylinder
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: tetrahedroncubeoctahedrondodecahedronicosahedron
Direct truncates: tetrahedral truncatecubic truncateoctahedral truncatedodecahedral truncateicosahedral truncate
Mesotruncates: stauromesohedronstauroperihedronstauropantohedronrhodomesohedronrhodoperihedronrhodopantohedron
Snubs: snub staurohedronsnub rhodohedron
Curved: spheretoruscylinderconefrustumcrind


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


2a. III
Cube
2b. (III)
Sphere
3a. (II)I
Cylinder
3b. ((II)I)
Torus
4a. IIII
Tesseract
4b. (IIII)
Glome
List of toratopes


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