THE GABRIEL'S HORN (PAINTER'S PARADOX)

                                        

Have you ever heard about something called the Gabriel’s Horn? (See the figure above so you can get a rough idea of what it could look like) Well, according to the Holy Bible, it is the horn the Archangel Gabriel will blow on the Judgment Day. Anyway, why we are talking about it here? Well, that’s because this is a mysterious ‘shape’ that exists in nature.

As it can be seen above, the shape can be generated by rotating the graph of y=1/x  about the x- axis. Now, let us think for a moment that such an instrument is there with us and we are planning to paint it with some newly bought paint. So go ahead and see what happens!

From the above diagram, the curve in red is given by

Let us rotate the curve from x=1 to x=k.

We like to find the surface area formed by this rotation.

We begin with the “differential Pythagoras Theorem”:

Where ds is the differential arc length.

Then,

= 1+→→

By rotating ds around x-axis by 4 right angles, we get a belt with differential surface area,

Hence the surface area can be found by integrating the above:

The volume of revolution about the x-axis is given by,

Putting k→∞, we get,

So we can see that the surface area is infinite.

But,

Which means the Volume is Finite. 

So what does it mean?

Nothing but we cannot paint the Gabriel’s horn completely, no matter how hard we try, but we can fill the instrument with a finite amount of paint - apparently a paradox!

But if viewed in another way, when the instrument is filled with paint, we can see that its inner surface is automatically painted with the paint, so literally the horn’s outer surface cannot be painted while the inner surface could be – another practical paradox!

(Remember, in the above cases we assumed that the layer of paint coated on the outer surface of the horn is of constant thickness, but if we apply the paint on the surface in such a way that the layer gradually fades, the instrument can really be painted.)

So isn’t this a painful conclusion? Just grab a cup of coffee and think about it again in your free time. Oh and thankfully we don’t have such weird objects in nature!

 

ASHAN FERNANDO

2011/2012

Acknowledgements; www.qc.edu.hk