### Life’s a Drag Crisis

The character of fluid flow depends on a dimensionless quantity, the Reynolds number. Named for Belfast-born scientist Osborne Reynolds, it determines whether the flow is laminar (smooth) or turbulent (rough). Normally the drag force increases with speed.

The Reynolds number is defined as Re = VL/ν where V is the flow speed, L the length scale and ν the viscosity coefficient. The transition from laminar to turbulent flow occurs at a critical value of Re which depends on details of the system, such as surface roughness.

### Mathematics Solving Crimes

What use is maths? Why should we learn it? A forensic scientist could answer that virtually all the mathematics we learn at school is used to solve crimes. Forensic science considers physical evidence relating to criminal activity and practitioners need competence in mathematics as well as in the physical, chemical and biological sciences [TM080: search for “thatsmaths” at irishtimes.com ].

Trigonometry, the measurement of triangles, is used in the analysis of blood spatter. The shape indicates the direction from which the blood has come. The most probable scenario resulting in blood spatter on walls and floor can be reconstructed using trigonometric analysis. Such analysis can also determine whether the blood originated from a single source or from multiple sources.

### Numbering the Family Tree

The availability of large historical data sets online has spurred interest in genealogy and family history. Anyone who has assembled information knows how important it is to organize it systematically. A simple family tree showing the direct ancestors of Wanda One is shown here:

This has just three generations but, as more people are added, the chart expands in an unwieldy fashion. However, it is possible to represent the information in simple text form, thanks to a clever numbering system first devised by Michaël Eytzinger, an Austrian historian who published the system in 1590.

The system uses what are called Ahnentafel numbers (German for pedigree numbers). We will call them A-numbers for short. The base person or Subject (Wanda in the chart above) is Number 1. Her father is given double this number and her mother double-plus-one. Thus, her father and mother become Number 2 and Number 3. The parents of each of these are numbered in the same pattern: father’s father is double father’s number and his mother is double-plus one. In this way, all the generations are numbered. The following Ahnentafel Report shows the same information as the chart above.

For just three generations, there is no real simplification, but larger genealogical charts can be represented compactly using the system. In effect, an Ahnentafel report is a method of representing a binary tree in text form.

The Ahnentafel numbers have some interesting properties. The system allows one to derive an ancestor’s number without access to the full chart. Similarly, knowledge of the A-number determines the relationship to the Subject of the pedigree. Apart from Number 1, who may be a man or woman, odd numbers correspond to females and even numbers to males. Any pair (2n, 2n+1) is a set of parents. If the Subject is Generation 1, the generation number corresponding to A-number N is [ log2N + 1 ] where [ x ] denotes the integer part of x. The N-th generation is spanned by the numbers from 2N-1 to 2N-1. e.g, for N = 4 this is from 8 to 15. The patrilineal line (fathers only) has A-numbers 2N or the sequence {2, 4, 8, 16, … } while the matrilineal sequence is 2N+1 -1 or {3, 7, 15, 31, … }.

Binary form

The direct ancestor chart is a simple binary graph. Each node has one link to the left and two to the right. More properties of the A-numbers are revealed when we express them as binary numbers. Let us write the numbers for the first three generations in both decimal and binary form:

We see that, ignoring leading zeros, the generation number is equal to the number of binary digits or bits in the A-number. Apart from Number 1, males all end in zero and females in one. Indeed, we can directly translate the binary form into plain text. Replace the initial “1” by “Wanda’s” and, thereafter, every “0” by “father’s” and every “1” by “mother’s” (dropping “’s” for the final bit). For example

1 1 0 1         becomes         Wanda’s mother’s father’s mother.

Indeed 1101, or decimal 13 is the paternal grandmother of Wanda’s mother. Taking a more extreme example, decimal 100 becomes binary 1100100, so the A-number 100 is for Wanda’s mother’s father’s father’s mother’s father’s father. So the individual numbered decimal 100 is in the seventh generation, or six generations back from Wanda.

Let us convert an A-number explicitly to a relationship. Suppose the A-number is 45, a female. Since 45 = 44 + 1 she is the mother of 22, who is the father of 11 = 10 + 1, who is the mother of 5 = 4 + 1, who is the mother of 2, who, finally, is the father of 1. Thus, A-number 45 is the subject’s father’s mother’s mother’s father’s mother. And the binary representation must be 101101. Indeed 4510 = ( 32 + 8 + 4 + 1 ) = 25 + 23 + 22 + 20 = 1011012.

Other numbering systems have been devised for use in genealogy. Popular ones include Henry numbers, the Register system (a combination of arabic and Roman numerals) and the NGSQ system (National Genealogical Society Quarterly). There are several others.

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Peter Lynch’s book about walking around the coastal counties of Ireland is now available as an ebook (at a very low price!). For more information and photographs go to RRI.

### Melencolia: An Enigma for Half a Millennium

Albrecht Dürer, master painter and engraver of the German Renaissance, made his Melencolia I in 1514, just over five centuries ago. It is one of the most brilliant engravings of all time, and amongst the most intensively debated works of art [TM079; or search for “thatsmaths” at irishtimes.com ].

The winged figure, Melancholy, sits in a mood of lassitude and brooding dejection, weighed down by intellectual cares. Her head rests on her left hand while her right hand holds a mathematical compass, one of many symbols and motifs in the work that reflect Dürer’s interest in mathematics.

### Mowing the Lawn in Spirals

Like a circle in a spiral / Like a wheel within a wheel / Never ending or beginning / On an ever-spinning reel.    The Windmills Of Your Mind

Broadly speaking, a spiral curve originates at a central point and gets further away (or closer) as it revolves around the point. Spirals abound in nature, being found at all scales from the whorls at our finger-tips to vast rotating spiral galaxies. The seeds in a sunflower are arranged in spiral segments. In the technical world, the grooves of a gramophone record and the coils of a watch balance-spring are spiral in form.

Left: Archimedean spiral. Centre: Fermat spiral. Right: Hyperbolic spiral.

### A Few Wild Functions

Sine Function: ${\mathbf{y=\sin x}}$

The function ${y=\sin x}$ is beautifully behaved, oscillating regularly along the entire real line ${\mathbb{R}}$ (it is also well-behaved for complex ${x}$ but we won’t consider that here).

The sine function, the essence of good behaviour.

### It’s a Small – Networked – World

Networks are everywhere in the modern world. They may be physical constructs, like the transport system or power grid, or more abstract entities like family trees or the World Wide Web. A network is a collection of nodes linked together, like cities connected by roads or people genetically related to each other. Such a system of nodes and links is what mathematicians call a graph [TM078; or search for “thatsmaths” at irishtimes.com ].

Detail of a Twitter communications network.