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A New Mathematical Discovery from Neutrino Physics

Although abstract in character, mathematics has concrete origins: the greatest advances have been inspired by the natural world. Recently, a new result in linear algebra was discovered by three physicists trying to understand the behaviour of neutrinos [TM176 or search for “thatsmaths” at irishtimes.com].

Neutrino-Trails-in-Bubble-Chamber

Neutrino trails in a bubble chamber [image from Physics World]

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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.

SphConCyl-5

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

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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].

Baggage-Xray

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

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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.

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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].

PCM-PCAM-Covers

The Princeton Companions to Maths and Applied Maths

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An Attractive Spinning Toy: the Phi-TOP

It is fascinating to watch a top spinning. It seems to defy gravity: while it would topple over if not spinning, it remains in a vertical position as long as it is spinning rapidly.

There are many variations on the simple top. The gyroscope has played a vital role in navigation and in guidance and control systems. Many similar rotating toys have been devised. These include rattlebacks, tippe-tops and the Euler disk. The figure below shows four examples.

Spinning-Tops-4

(a) Simple top, (b) Rising egg, (c) Tippe-top, (d) Euler disk. [Image from website of Rod Cross.]

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Some Fundamental Theorems of Maths

Every branch of mathematics has key results that are so
important that they are dubbed fundamental theorems.

The customary view of mathematical research is that of establishing the truth of propositions or theorems by rigorous deduction from axioms and definitions. Mathematics is founded upon axioms, basic assumptions that are taken as true. Logical reasoning is then used to deduce the consequences of those axioms with each major result designated as a theorem.

As each new theorem is proved, it provides a basis for the establishment of further results. The most important and fruitful theorem in each area of maths is often named as the fundamental theorem of that area. Thus, we have the fundamental theorems of arithmetic, algebra and so on. For example, the fundamental theorem of calculus gives the relationship between differential calculus and integral calculus.

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