## Archive for November, 2019

### Archimedes and the Volume of a Sphere

One of the most remarkable and important mathematical results obtained by Archimedes was the determination of the volume of a sphere. Archimedes used a technique of sub-dividing the volume into slices of known cross-sectional area and adding up, or integrating, the volumes of the slices. This was essentially an application of a technique that was — close to two thousand years later — formulated as integral calculus.

Cone, sphere and cylinder on the same base. The volumes are in the ratios  1 : 2 : 3 [image from mathigon.org].

Continue reading ‘Archimedes and the Volume of a Sphere’

### Airport Baggage Screening with X-Ray Tomography

When you check in your baggage for a flight, it must be screened before it is allowed on the plane. Baggage screening detects threats within luggage and personal belongings by x-ray analysis as they pass along a conveyor belt. Hold-baggage and passenger screening systems are capable of detecting contraband materials, narcotics, explosives and weapons [TM175 or search for “thatsmaths” at irishtimes.com].

3D X-ray image of baggage [image from Rapiscan Systems ].

### Elliptic Trigonometry: Fun with “sun”, “cun” and “dun”

Introduction

The circular functions arise from ratios of lengths in a circle. In a similar manner, the elliptic functions can be defined by means of ratios of lengths in an ellipse. Many of the key properties of the elliptic functions follow from simple geometric properties of the ellipse.

Originally, Carl Gustav Jacobi defined the elliptic functions ${\mathop\mathrm{sn} u}$, ${\mathop\mathrm{cn} u}$, ${\mathop\mathrm{dn} u}$ using the integral

$\displaystyle u = \int_0^{\phi} \frac{\mathrm{d}\phi}{\sqrt{1-k^2\sin^2\phi}} \,.$

He called ${\phi}$ the amplitude and wrote ${\phi = \mathop\mathrm{am} u}$. It can be difficult to understand what motivated his definitions. We will define the elliptic functions ${\mathop\mathrm{sn} u}$, ${\mathop\mathrm{cn} u}$, ${\mathop\mathrm{dn} u}$ in a more intuitive way, as simple ratios associated with an ellipse.

### The Vastness of Mathematics: No One Knows it All

No one person can have mastery of the entirety of mathematics. The subject has become so vast that the best that can be achieved is a general understanding and appreciation of the main branches together with expertise in one or two areas [TM174 or search for “thatsmaths” at irishtimes.com].

The Princeton Companions to Maths and Applied Maths