#### TL;DR Gabriel's wedding cake describes a geometric figure w...
Three-dimensional illustration of Gabriel’s Horn. This horn shape h...
**Cavalieri's Principle:** states that if two objects are between t...
Determining the surface of the horn is a slightly more complex than...
You can learn more about Gabriel's Horn here: - [Fermat's Library...
How to transform the the "continuous horn" into a "discrete cake".
Another similar object is the infinite Gift 🎁: The infinite gift...
Gabriel's
Wedding
Cake
Julian
F.
Fleron
Julian Fleron (j_fleron@foma.wsc.mass.edu) has been
Assistant Professor of Mathematics at
Westfield State College
since completing his Ph.D.
in
several complex variables at
SUNY University at
Albany
in
1994. He has broad
mathematical passions that he strives to share with all of his
students, whether mathematics for
liberal arts students,
pre-service teachers, or mathematics majors.
Family hobbies
include popular music, cooking, and restoring the family's
Victorian house.
We obtain the solid which nowadays
is
commonly, although
perhaps
inappropri-
ately,
known
as Gabriel's
hom
by
revolving
the
hyperbola y =
1/x
about
the line
y = 0, as
shown
in
Fig.
1.
(See, e.g.,
[2],
[5].) This
name
comes from the archangel
Gabriel
who,
the
Bible tells us,
used
a
hom
to
announce
news
that was some-
times heartening (e.g. the birth
of
Christ in Luke 1)
and
sometimes fatalistic
(e.g. Armageddon in Revelation
8-11).
Figure 1. Gabriel's
Horn.
This object
and
some
of
its remarkable properties
were
first discovered in
1641
by
Evangelista Torricelli.
At
this time Torricelli was a little
known
mathematician
and
physicist
who
was the successor to Galileo at Florence.
He
would
later
go
on
to
invent
the
barometer
and
make
many
other
important contributions to mathematics
and
physics. Torricelli communicated his discovery
to
Bonaventura Cavalieri
and
showed
how
he
had
computed
its volume using Cavalieri's principle for indivisibles.
Remarkably, this volume
is
finite! This result propelled Torricelli into the mathemati-
cal spotlight, gave rise to
many
related paradoxes
[3],
and
sparked
an
extensive
philosophical controversy that included Thomas Hobbes,
John
Locke, Isaac Barrow
and
others [
4].
VOL. 30, NO. 1, JANUARY 1999
35
This solid
is
a favorite in many calculus classes because its volume can
be
readily
computed via the method
of
disks:
co co
1 (
-1
In)
V=
1 7T(f(
x)
)
2
dx
= 1
7T-
2
dx
= lim
7r-
=
7T.
1 1 X X 1
The seeming paradox
of
an
infinite solid with a finite volume becomes even more
striking
when
one
considers its surface area. The standard method for computing
areas
of
surfaces
of
revolution gives
This last integral cannot
be
evaluated readily, although with the aid
of
a computer
algebra system
we
find
In lieu
of
this, typically
we
estimate
...J
x
4
+ 1 I x
3
0 I x
3
= 1 I x so
co
...J
x
4
+ 1
co
dx
n
5=1
27T
3
-=lim
(lln{x)lt)=oo.
1 X 1 X
Hence, Gabriel's
hom
is
an
infinite solid with finite volume but infinite surface area!
Although Gabriel's
hom
is
an
engaging
and
appropriate example for second
semester calculus, analysis
of
its remarkable features
is
complicated by two factors.
First, many
of
the
new
calculus curricula
do
not include areas
of
surfaces
of
revolution. Second, the beauty
of
the paradox
is
often obscured
by
an
integral
estimate that most students find spurious at best.
In
an
effort to alleviate these factors, as well as to find
an
example accessible to
less advanced students,
we
can use a discrete analogue
of
Gabriel's
hom
to
illustrate the same paradox. To construct
it
we
can replace the function y =
11x
with a step function.
Let
f(x)
=
1 for 1 2
1
-
2
1
- for n x < n + 1
n
Revolving the graph
of
f about the line y = 0
we
obtain the solid
of
revolution
shown in
Fig.
2.
Notice that,
when
stood
on
end, it appears to
be
a cake with
infinitely many layers.
36
THE
COLLEGE
MATHEMATICS
JOURNAL
/////
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N\
:
1
//////!'
IV
,v,,
,:"\n '
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11
Figure
2. Gabriel's Wedding Cake.
As
each layer
is
simply a cylinder, the volume and surface area
of
the solid can
be
readily computed. The
nth
layer has volume
1T(lln)
2
,
so
the total volume
of
the
cake
is
V=
i:
1T(2_)
2
=
1T
E
\.
n=I
n
n=I
n
This series converges. Calculus students will recognize the series as a p-series with
p =
2.
Less advanced students can see that the series converges
by
comparison:
00
1
111111
"-=1+-+-+-+-+-+-+
...
n'":I
nz
22
32
42
52
62
72
< 1 +
2.(2.
+
2.)
+
2.(2.
+..:. +..:. +
2.)
+ ...
-22
2
44
4 4 4
1 1 1
=
1
+ 2 + 4 + ... =
--1-
=
2
·
1--
2
Using Euler's remarkable result that the
sum
of
the series
is
E:=
1
(1ln
2
)
=
1r
2
16
[1],
one
can even obtain
an
exact result:
V=
1r
3
16.
The surface area
is
formed
by
the annular tops
and
the lateral sides
of
each layer.
The surface area
of
the
nth
annular top
is
7T(1
I n)
2
-
1T(l
In
+ 1)
2
,
so
the total area
of
the annular tops
is
given by the telescoping series
Notice this result
is
obvious if
one
"collapses" the layers since the resulting top layer
will
be
a complete disk
of
radius one. The surface area
of
the
nth
lateral side
is
21T(lln)(l),
so the total lateral surface area
is
VOL. 30, NO. 1, JANUARY 1999
37
This
is
the harmonic series,
among
the most important
of
all the infinite series,
which diverges.
Thus, this solid illustrates essentially the same paradox as Gabriel's hom:
an
infinite solid with finite volume
and
infinite surface area. In other words: a cake you
can eat,
but
cannot frost.
Regarding a name for this
new
solid, Gabriel's wedding cake seems appropriate
for physical
and
genealogical reasons. In aqdition,
it
seems a bit refreshing since
weddings are so unabashedly joyous,
and
the connotations
of
the
hom
have often
imposed a heavy burden
on
Gabriel.
References
1. William Dunham,
journey
Through Genius,
John
Wiley & Sons, 1990.
2.
P.
Gillett, Calculus
and
Analytic Geometry, 2nd ed.,
D.
C.
Heath, 1984.
3.
Jan
A.
van
Maanen, Alluvial deposits, conic sections,
and
improper
glasses,
or
history
of
mathematics
applied in
the
classroom, in
F.
Swetz,].
Fauvel,
0.
Bekken,
B.
Johansson,
and
V.
Katz, eds., Learn
from the Masters,
Mathematical Association
of
America, 1995.
4.
Paolo Mancosu
and
Ezio Vailati, Torricelli's infinitely long solid
and
its philosophical reception in the
seventeenth century,
Isis,
82:311
(1991) 50-70.
5.
D.
W.
Varberg
and
E.].
Purcell, Calculus, 7th ed., Prentice Hall, 1997.
38
---o---
Call for Referees
The
jounzal
is
in constant need
of
people
who
will read and evaluate
manuscripts, suggest
how
they could be improved, correct their grammati-
cal errors (if any), and, in general, give them a thorough going-over.
This
is
voluntary, unpaid,
and
anonymous labor, but referees are vital for
the
jounzal,
and
for the profession.
If
you would
be
willing to take
on
this
job-it
is
not too onerous,
one
or
two manuscripts a
year-please
let me
know (Underwood Dudley, Mathematics Department, DePauw University,
Greencastle, Indiana 46135; dudley@depauw.edu) and,
if
you want to,
indicate fields in which you would
be especially interested
THE
COLLEGE
MATHEMATICS
JOURNAL

Discussion

#### TL;DR Gabriel's wedding cake describes a geometric figure with a finite volume but infinite surface are. In the author's words imagine ***"a cake you can eat, but cannot frost"***. A few interesting properties of this cake (see image below): - Gabriel's wedding cake describes a particular geometric figure which has **infinite surface area but finite volume** - the rotation of an infinitely large section of the $xy$ plane about the x axis generates an object of finite volume was considered a **paradox because the section lying in the $xy$ plane has an infinite area, any other section parallel to it has a finite area. ** This paradox was first discovered in the 17th century and it originated a **dispute on the nature of infinity involving many of the key thinkers of the time, including Thomas Hobbes, John Wallis and Galileo Galilei.** ![](https://i.imgur.com/ZapbgL4.png) How to transform the the "continuous horn" into a "discrete cake". Another similar object is the infinite Gift 🎁: The infinite gift is an interesting object where the side of the nth box is $\frac{1}{√n}$. As $n\rightarrow +∞$, the gift has infinite surface area and length but finite volume! - [Learn more here: Infinit Gift](https://twitter.com/fermatslibrary/status/1166324490760065024) !["infinite_giff"](https://i.imgur.com/s1tvfvS.jpg) You can learn more about Gabriel's Horn here: - [Fermat's Library - Gabriel's Horn](https://fermatslibrary.com/s/gabriels-horn) - [Wolfram - Gabriel's Horn](https://mathworld.wolfram.com/GabrielsHorn.html) Three-dimensional illustration of Gabriel’s Horn. This horn shape has an interesting, counterintuitive property: **a finite volume but an infinite surface area.** ![GH](http://67.media.tumblr.com/47e8ed75408372defa0ccdf3ce155147/tumblr_n7umskeuNz1tzs5dao3_1280.gif) Create your own Gabriel Horn with Mathematica: ```mathematica x[u_, v_] := u y[u_, v_] := Cos[v]/u z[u_, v_] := Sin[v]/u Manipulate[ParametricPlot3D[{{x[u, v], y[u, v], z[u, v]}}, {u, 1, umax}, {v, 0, 2*Pi}, PlotRange -> {{0, 20}, {-1, 1}, {-1, 1}}, Mesh -> {Floor[umax], 20}, Axes -> False, Boxed -> False], {{umax, 20}, 1.1, 20}] ``` Source: [Fouriest Seriess.](http://fouriestseries.tumblr.com/post/90296337418/gabriels-horn-and-the-painters-paradox) Determining the surface of the horn is a slightly more complex than the volume. The area for a volume of revolution is given by: \begin{eqnarray} S &=& 2\pi \int_1^N y \sqrt{1+ \left(\frac{dy}{dx} \right)^2} dx\\ \end{eqnarray} In this scenario we know that $y = \frac{1}{x}$ and we can then write: \begin{eqnarray*} \frac{dy}{dx} &=& -\frac{1}{x^2} \end{eqnarray*}We can then rewrite (1): \begin{eqnarray} S &=& 2\pi \int_1^N \frac{1}{x} \sqrt{1+ \frac{1}{x^4} } dx \nonumber \\ &=& 2\pi \int_1^N \frac{\sqrt{x^4+ 1}}{x^3} dx\\ \end{eqnarray}using integration by parts we can rewrite the integra in equation (2) as: \begin{eqnarray} \int \frac{\sqrt{x^4+ 1}}{x^3} dx &=&- \frac{1}{2x^2}\sqrt{x^4+1} + \int \frac{x}{\sqrt{x^4+1}}dx \end{eqnarray}to compute the integral in the right side of equation (3) we perform the following substition: \begin{eqnarray*} u = x^2 ; \frac{du}{dx} = 2x \end{eqnarray*}we have then that: \begin{eqnarray*} \int \frac{x}{\sqrt{x^4+1}}dx &=& \frac{1}{2} \int \frac{1}{\sqrt{u^2+1}}du\\ &=& \frac{1}{2} \ln{\left(u + \sqrt{u^2+1}\right)}+k\\ \end{eqnarray*}and re-doing the substition we finally have that: \begin{eqnarray} \int \frac{x}{\sqrt{x^4+1}}dx &=& \frac{1}{2} \ln{\left(x^2 + \sqrt{x^4+1}\right)}+k \end{eqnarray}Taking (3) and (4) into account we can rewritte the surface area (2): \begin{eqnarray} S &=& 2\pi \left[ - \frac{1}{2x^2}\sqrt{x^4+1} + \frac{1}{2} \ln{\left(x^2 + \sqrt{x^4+1}\right)}\right]_1^N \nonumber\\ &=& \pi \left(- \frac{1}{N^2}\sqrt{N^4+1} + \ln{\left(N^2 + \sqrt{N^4+1}\right)} \nonumber \\ +\sqrt{2}- \ln{ \left( 1+\sqrt{2}\right)} \right) \end{eqnarray} As $N \rightarrow \infty$ the dominent term is $\ln{\left(N^2 + \sqrt{N^4+1}\right)}$ and taking the properties of the logarithm into accoun we can then write that: \begin{eqnarray*} \lim_{x\to\infty} S &\simeq& \lim_{x\to\infty} \left(\pi \ln{N} \right) &=& \infty \end{eqnarray*} Thus we determined with modern calculous that **Gabriel's horn has a finite Volume and an intfinite area!** Paradox! **Cavalieri's Principle:** states that if two objects are between two parallel planes and if every plane cross-section parallel to the given planes have equal areas then the two objects must have equal volume. Figure 1. below is a great illustration of different objects with matching cross-sectional areas. They all have the same volume. !["cvp1"](http://i.imgur.com/zl6t9kD.png) **Figure 1:** Simple illustration of different objects with the same volume. #### We will now use Cavalieri's principle and apply it to Gabriel's horn. Figure 2 shows shows the intersection of a cylinder with an hyperbola in the $xy$-plane at point $(x_0,y_0)$. We can easily compute the curved surface area of the cylinder: $$V_C = (2\pi y_0)x_0$$ and we know that $y= \frac{1}{x}$ thus: $$V_C = 2\pi$$ !["cvp2"](http://i.imgur.com/06NQP0o.png) **Figure 2:** Illustration of a horn with open lip (1,1) and it's intersection with a cylinder at point $(x_0,y_0)$. The area of the cylinder is constant as the point of contact varies. Now imagine that you can cut the cylinder along the $x$-axis and unravel it so that it becomes a horizontal rectangle at point $y_0$, see the left panel of Figure 3. We can construct a new object that has surface are $2\pi$ at point $y_0$, see right panel of Figure 3. !["cvp3"](http://i.imgur.com/0nJweyO.png) **Figure 3:** Applying Cavalieri's principle to Gabriel's horn. Using Cavalieri's principle we know that two objects between two parallel planes with the same matching cross-sectional areas have the same volume. We know that the cylinder that we just constructed is a succession of $2\pi$ planes from $y=0, y=1$. Thus the volume of the cylinder is $V_C = 2\pi*1 = 2\pi$. Now we can use Cavalieri's principle and infer the volume of Gabriel's horn $V_{GH}$. $$V_{GH} = V_C = 2\pi.$$ Thanks to Cavalieri's principle have shown that although the horn seems to have an infinite length it has a finite volume!