Cylinder Area Paradox

​
height h
10
radius r
5
bands m
5
arcs n
7
area of the cylinder ≈ 314.20
area of the triangles ≈ 312.90
Let
S
be the surface of a cylinder of height
h
and radius
r
. (
S
does not include the flat circular ends of the cylinder.) This Demonstration constructs a set of triangles that tend uniformly to
S
—yet their total area does not tend to the area of
S
!
Divide
S
into
m
subcylinders (or bands) of height
h/m
. Construct
2n
congruent isosceles triangles in each band with vertices at the vertices of a regular
2n
-gon inscribed in the circles at the top and bottom of each band, offset by
π/n
.
For any point
p
in
3

(except the axis of the cylinder), let
s(p)
be the axial projection of
p
onto
S
. As
m,n∞
, to say that the triangles approximate
S
uniformly means that for any point
t
on a triangle and any
ϵ>0
(independent of
t
), there is a
K∈
such that for all
m,n>K
,
|t-s(t)|<ϵ
.
The sum of the areas of the triangles is
A(m,n)=2rnsin
π
n
2
h
+
2
(mr)
2
1-cos
π
n
.
Depending on how the limit is taken,
lim
m,n∞
A(m,n)
can differ. If first
n∞
with
m
held fixed and then
m∞
, the limit is
2πrh
, the expected area of the cylinder. If first
m∞
with
n
held fixed and then
n∞
, the limit is infinity. If
m∞
and
n∞
together so that
m/
2
n
is some positive constant
c
, the limit can be chosen to be any number greater than
2πrh
.
Therefore
A(m,n)
does not have a limit.
The surface is known as Schwarz's lantern, Schwarz's polyhedron, or Schwarz's cylinder.

Details

For details, see Freida Zames' Surface Area and the Cylinder Area Paradox.

External Links

Surface Area (Wolfram MathWorld)
Schwarz's Polyhedron (Wolfram MathWorld)

Permanent Citation

George Beck, Izidor Hafner
​
​"Cylinder Area Paradox"​
​http://demonstrations.wolfram.com/CylinderAreaParadox/​
​Wolfram Demonstrations Project​
​Published: November 4, 2008