Archive for the 'Occasional' Category

Moessner’s Magical Method

Amongst the many joys of mathematics are the beautiful patterns that appear unexpectedly from time to time. In 1951, Alfred Moessner discovered a delightful and ingenious way of generating sequences of powers of natural numbers. It is surprisingly simple and offers great opportunities for development and generalization.

SumOfOddNumbers

It is well-known that the sum of odd numbers yields a perfect square:

1 + 3 + 5 + … + (2n – 1) = n 2

This is easily demonstrated in a geometric way. We start with a unit square, and repeatedly add an additional row and column on the “east” and “north” sides and a unit square at the “north-east” corner. This amounts to adding the next odd number and, at each stage, a new square is produced.

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Drawing Multi-focal Ellipses: The Gardener’s Method

Common-or-Garden Ellipses

In an earlier post we saw how a gardener may set out oval flower-beds using a well-known property of ellipses: the sum of the distances from any point on the ellipse to the two foci is always the same value, {2a}, the length of the major axis. The gardener puts down two stakes and loops a piece of rope around them. Using a stick, he pulls the loop taut, marking the points around a curve. This is illustrated here.

Ellipse-GardenersMethod

Gardener’s method of drawing an ellipse [Image Wikimedia].

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Locating the HQ with Multi-focal Ellipses

Motivation

IrelandProvincialCapitalsMapIreland has four provinces, the principal city in each being the provincial capital: Belfast, Cork, Dublin and Galway. The map here shows the location of these cities. Now imagine a company that needs to visit and to deliver goods frequently to all four cities. Where might they locate their HQ to minimize transport costs and travel times?

One possibility is to find the location with the smallest distance sum:

\displaystyle d(\mathbf{r}_0) = \sum_{j=1}^{4} |\mathbf{r}_0-\mathbf{p}_j|

where {\mathbf{r}_0} is the position of the HQ and {\mathbf{p}_j, j\in\{1,2,3,4\}} are the positions of the cities.

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Fractions of Fractions of Fractions

Numbers can be expressed in several different ways. We are familiar with whole numbers, fractions and decimals. But there is a wide range of other forms, and we examine one of them in this article. Every rational number {x} can be expanded as a continued fraction:

\displaystyle x = a_0 + \cfrac{1}{ a_1 + \cfrac{1}{ a_2 + \cfrac{1}{ a_3 + \dotsb + \cfrac{1}{a_n} } }} = [ a_0 ; a_1 , a_2 , a_3 , \dots , a_n ]

where all {a_n} are integers, all positive except perhaps {a_0}. If {a_n=1} we add it to {a_{n-1}}; then the expansion is unique.

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Who First Proved that C / D is Constant?

Every circle has the property that the distance around it is just over three times the distance across. This has been “common knowledge” since the earliest times. But mathematicians do not trust common knowledge; they demand proof. Who was first to prove that all circles are similar, in the sense that the ratio of circumference C to diameter D has the same value for all?

Circle-Area-Triagles

Slicing a disk to estimate pi (Image Wikimedia).

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Inertial Oscillations and Phugoid Flight

The English aviation pioneer Frederick Lanchester (1868–1946) introduced many important contributions to aerodynamics. He analysed the motion of an aircraft under various consitions of lift and drag. He introduced the term “phugoid” to describe aircraft motion in which the aircraft alternately climbs and descends, varying about straight and level flight. This is one of the basic modes of aircraft dynamics, and is clearly illustrated by the flight of gliders.

Glider-Loop-20

Glider in phugoid loop [photograph by Dave Jones on website of Dave Harrison]

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Patterns in Poetry, Music and Morse Code

Suppose we have to ascent a flight of stairs and can take only one or two steps at a time. How many different patterns of ascent are there? We start with the simplest cases. With one step there is only one way; with two, there are two: take two single steps or one double step. With three steps, there are three possibilities. We can now proceed in an inductive manner.

Staircase-01

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