Since the term was coined in 1655 by British mathematician John Wallis, ‘infinity’ has been a mathematical concept that has fascinated mathematicians, philosophers, and humans alike. This abstract idea represents something limitless, larger than the largest number you could probably imagine right now!
In mathematics, specifically calculus, infinity has a special significance, pushing the boundaries of what we perceive as ‘measurable’. An interesting example of this concept is Gabriel’s horn - a shape with infinite area but a finite volume - revealing a paradox, at the heart of this boundless phenomenon. So how exactly is this possible? If a shape has an infinite surface area, shouldn’t its volume also be infinite? First, we must see what Gabriel’s horn is and how this mathematical construct is formed.
What is Gabriel’s Horn?
Figure 1: Revolution of y=1x on the x-axis
Gabriel's horn is formed as the result of revolving the function y=1x (x1) around the x-axis. Figure 1 shows us this, but how exactly can we quantify this ‘paradox.’ In other words, how can we prove that this shape has an infinite area and a finite volume?
In reality, the process is quite simple. We can first start with the volume; the formula for finding the volume of a curve revolved around the x-axis is:
We can use this formula but before we must figure out what each variable of the formula is. Hence, we can let:
Now we can substitute everything into the formula setting up our volume equation!
With the equation complete, all that is left is to simply integrate and substitute our bounds which gives us:
Our volume for our horn is , so it definitely is finite!
Checking for our surface area now, we can use this formula (slightly more advanced, but watch as it trickles down and gets simplified):
First, since the formula requires the first derivative of our equation:
For very large values of x, the above will approach 0, therefore, substituting this along with all our original terms, our formula for the surface area of the horn becomes:
Now all that's left is simply integrating and applying our bounds to solve for the surface area!
Interestingly, we were indeed able to prove using some simple calculus formulae, the paradox of Gabriel’s horn, and how it defies the laws of our universe to possess an infinite surface area with a finite volume. Mathematically, this can be explained due to the asymptotic nature of the graph as x approaches infinity, demonstrating the nuanced nature of equations in the world of numbers!
Can you think of any other graphs with odd properties?
Works Cited
Kaufman, R. (2023). Gabriel’s Horn and the Painter’s Paradox in Perspective. Ohio Journal of School Mathematics, [online] 94, pp.1–5. Available at: https://ohiomathjournal.org/index.php/OJSM/article/view/9483 [Accessed 9 Oct. 2024].
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