## Archive for April, 2016

### Sigmoid Functions: Gudermannian and Gompertz Curves

The Gudermannian is named after Christoph Gudermann (1798–1852). The Gompertz function is named after Benjamin Gompertz (1779–1865). These are two amongst several sigmoid functions. Sigmoid functions find applications in many areas, including population dynamics, artificial neural networks, cartography, control systems and probability theory. We will look at several examples in this class of functions.

### The Inexorable Rise in Life Expectancy

Life expectancy has increased in a spectacular fashion over the past 150 years. There has been a four-decade increase in average length of life over the period since 1850. World life expectancy has more than doubled, from about 25 years to around 65 years for men and 70 years for women.  [TM089, or search for “thatsmaths” at irishtimes.com].

Global (male) life expectancy in years. Image Wikimedia Commons.

### The Power Tower Fractal

We can construct a beautiful fractal set by defining an operation of iterating exponentials and applying it to the numbers in the complex plane. The operation is tetration and the fractal is called the power tower fractal or sometimes the tetration fractal. A detail of the set is shown in the figure here.

Detail of the power tower fractal.

### The Search is on for Planet Nine

Then felt I like some watcher of the skies
When a new planet swims into his ken.

John Keats: On First Looking into Chapman’s Homer

Pluto, discovered in 1930, orbits in the Kuiper Belt, a ring of asteroids and icy debris beyond Neptune. About ten years ago, it was reclassified as a “dwarf planet”, so now we have eight planets. Is there a ninth anywhere? A recent paper in the Astronomical Journal presents evidence of a large planet in an orbit far beyond Neptune. If confirmed, this would be a dramatic confirmation of our understanding of solar system dynamics [TM088, or search for “thatsmaths” at irishtimes.com].

Orbits of six trans-Neptunian objects aligned in one direction, suggesting a perturbing body, Planet Nine (?). Caltech/Robert Hurt.