## Posts Tagged 'Geometry'

### Malfatti’s Circles

Sometimes the “obvious” answer to a mathematical problem is not the correct one. The case of Malfatti’s circles is an example of this. In an equilateral triangle of unit side length, we must draw three non-overlapping circles such that the total area of the circles is maximal.

The solution seems obvious: draw three identical circles, each one tangent to two sides and to the other two circles (above figure, left). This is certainly the most symmetric arrangement possible. However, it turns out not to be the optimal solution. There is another arrangement (above figure, right) for which the three circles have greater total area.

### 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.

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

Continue reading ‘Drawing Multi-focal Ellipses: The Gardener’s Method’

### Locating the HQ with Multi-focal Ellipses

Motivation

Ireland 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.

### It’s as Easy as Pi

Every circle has the property that the distance around it is just over three times the distance across. This has been known since the earliest times  [see TM120 or search for “thatsmaths” at irishtimes.com].

The constant ratio of the circumference to the diameter, denoted by the Greek letter pi, is familiar to every school-child. You might expect to find a proof in Euclid’s Elements of Geometry, he could not prove it, and he made no mention of the ratio (see last week’s post).

### 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?

Slicing a disk to estimate pi (Image Wikimedia).

### Quadrivium: The Noble Fourfold Way

According to Plato, a core of mathematical knowledge – later known as the Quadrivium – was essential for an understanding of the Universe. The curriculum was outlined in Plato’s Republic. The name Quadrivium means four ways, but this term was not used until the time of Boethius in the 6th century AD [see TM119 or search for “thatsmaths” at irishtimes.com].

Image from here.

It is said that an inscription over the entrance to Plato’s Academy read “Let None But Geometers Enter Here”. This indicated that the Quadrivium was a prerequisite for the study of philosophy in ancient Greece.

### Hearing Harmony, Seeing Symmetry

Musical notes that are simply related to each other have a pleasing effect when sounded together. Each tone has a characteristic rate of oscillation, or frequency. For example, Middle C on the piano oscillates 264 times per second or has a frequency of 264 Hz (Hertz). If the frequencies of two notes have a ratio of two small whole numbers, the notes are harmonically related and sound pleasant when played together.

Beats from two notes close in pitch.