In a recent article, I have explained the Euler’s method for solving ordinary differential equations using as a motivation the fictionalized version in the film Hidden Figures of the scientific contribution of Katherine Goble and her two colleagues to the NASA space program. As an example of application, I have also shown a program written in the *awk *programming language for calculating the orbits of planets of the solar system. However, my interest in astrodynamics come back to my juvenile age, when still going to high school, my parents decided to gift me a more sophisticated microcomputer than my previous one (the celebrated Commodore VIC 20). So I became a programmer of a Philips MSX VG 8010 that I still jealously own in its original box. So, powered by the versatile Federico Faggin’s Zilog Z80 processor with a clock 3.58 MHz, with an impressive (for a previous owner of a VIC20 with a mere 3.583 kB!) memory of 32 kB RAM , 16kB of video RAM and a dedicated tape-record device as storage system, I started to write more sophisticated in MSX Basic. At that time, I was eagerly following the department “Ricreazioni al Computer” by the famous computer scientist A. K. Dewdney on the magazine “Le Scienze”, the Italian edition of Scientific American. The new microcomputer allowed me to experiment with the fascinating computational topics that Dewdney was offering every month. One of these topics was dedicated to the simulation of stars using the algorithm based on the Euler integration of the Newton equation. Following the instruction of Dewdney, I managed to write a small program in MSX basic and this was the starting of my interest in computational astronomy.

# Author: Danilo Roccatano

# L’Integrazione Numerica di Equazioni Differenziali: 50 anni fa l’ uomo ha messo piede sulla Luna

Nel giorno in cui iniziai a scrivere questo articolo ricorreva l’anniversario della prima esplorazione da parte dei cosmonauti americani Neil Armstrong, Michael Collins, Buzz Aldrin della nostra Luna. Anche se non ho una memoria diretta di questi eventi, le missioni delle progetto Apollo evocano in tutti noi una forte emozione rappresentando un momento unico ed epico nella storia della conquista dello spazio. Nessun altro uomo ha messo di nuovo piede sulla Luna dopo l’ultima missione (Apollo 17) nel 1972, per cui i recenti annunci della NASA di nuove esplorazioni umane del nostro satellite rende l’anniversario ancora più eccitante.

L’ immagine di copertina di questo articolo è stata creata da mio figlio Leonardo per una sua ricerca scolastica sulle missioni Apollo ed è un collage d’immagini ottenute usando i programmi Google Earth e Sketchup. Nella figura si confrontano le dimensioni del razzo Saturno V con quelle della meravigliosa cattedrale di Lincoln in Gran Bretagna per dare un’idea dell’enorme grandezza del vettore spaziale. Il pensiero che l’uomo sia riuscito nel giro pochissimi anni in uno sforzo tecnologico e scientifico immenso a costruire questa cattedrale volante della tecnologia mi ha suscitato una forte curiosità a cui non ho saputo trattenermi e ho così iniziato a spiluccare la miriade di documenti disponibili sul sito della NASA sul programma Apollo nella ricerca di qualche aspetto interessante e curioso di questa impresa tecnologica senza eguali.

Continue reading# The Logistic Map and the Feigenbaum Constants: a Retro Programming Inspired Excursion

“… Mitchell Feigenbaum was an unusual case. He had exactly one published article to his name, and he was working on nothing that seemed to have any particular promise. His hair was a ragged mane, sweeping back from his wide brow in the style of busts of German composers… At the age of twenty he had already become a savant among savants, an ad hoc consultant [at Los Alamos National Laboratory, USA] whom scientist would go to see about any expecially intractable problem.”

James Gleick, Chaos: the amazing science of the unpredectable.

This year, on June 30th 2019, **Mitchell J. Feigenbaum** died at the age of 74. Feigenbaum was an American mathematician that became famous with the discovery of the universal constants that bear his name. In the middle of the ’80, reading Le Scienze magazine (the Italian edition of Scientific American) I got to know of the contribution to the chaos theory of this charismatic mathematician. In particular, I was delighted by reading the Douglas Hofstadter’s article in the rubric “Temi Metamagici” ( Methamagical themes) (Scientific American, November 1981). The article explained the emergence of the chaos in the iteration map of the logistic equation, the same equation deeply studied by Feigenbaum. The full story about the Mitchell Feigenbaum and his discovery of his universal constants is delightly narrated in the beautiful book *Chaos:the amazing science of the unpredectable* by J. Gleick [1]. Here it is just another small extract:

Continue reading“… in the summer of 1975, at a gathering in Aspen, Colorado, he heard Steve Smale

[another key mathematicial in the developing of the chaos theory, NDA]talk about some of the mathematical qualities of the same quadratic difference equation [the same studied by Robert May, NDA]. Smale seemed to think that there were some interesting open questions about the exact point at which the mapping changes from periodic to chaotic. As always, Smale had a sharp instinct for questions worth exploring. Feigenbaum decided to look into it once more.”[1]

# Numerical Integration of Differential Equations. Part I.: Katherine Johnson and the Euler’s Method.

This article was inspired by the beautiful 2016 movie *Hidden Figures* (based on the book of the same name by M. L. Shetterley) which tell the dramatic story of three talented black women scientist that worked as “human computers” for NASA in 1961 for the Mercury project.

In the movie, the mathematician Creola Katherine Johnson (or Globe) (interpreted by Taraji P. Henson), had a brilliant intuition on how to numerically solve the complex problem to find the transfer trajectory for the reentry into the Earth atmosphere of the Friendship 7 capsule with the astronaut John Glenn on board. In the particular scene, she was standing together with other engineers and the director of the Langley Research Center (a fictional character interpreted by Kevin Coster) in front of the vast blackboard looking to graph and equations when she says that the solution might be in the “old math” and she runs to take an old book from a bookshelf with the description of the Euler method [*UPDATE, May 2022: I just come across another excellent youtube video by the Tibees about the mathematical work of K. Johnson at NASA, and at the end of the video she reveal that the book shown in the movie is the classic H Margenau & G. M Murphy, The Mathematics of Physics and Chemistry. Published by Van Nostrand, 1956*]. The scene is also nicely described in the youtube video lesson by Prof. Alan Garfinkel of the UCLA. A detailed description of the numerical solution based on the original derivation of K. Johnson is in the Wolfram blog website.

Katherine Globe was using for these complex calculation her brilliant brain with the support of a mechanical calculator (the Friden STW-10, in the movie, this machine is visible in different scenes). In a scene of the film, she revealed that her typical computing performance was of 10000 calculations per day and probably for calculations, she was not referring to single arithmetic operations! These exceptional mathematical skills have given a significative contribution at the beginning of the American space program, but it became insufficient to handle the more complex mathematics necessary to land the man on the Moon, and the other fantastic NASA achievements.

Continue reading# The Calculation of the Lattice Energy: The Born-Haber Cycle

My blog in italian on this topics is very popular and for this reason I decided to add an English translation (when I have some free time, I will also translate the text in the Figure and Table). So be tune and more will come!

The stability of a crystal lattice at constant T and P conditions is linked to the Gibbs free energy of lattice formation by the relations

If is more negative for the formation of the structure than for the structure, the transition will be spontaneous and the solid will have that structure.

Continue reading# Modelling Patterns and Forms in Nature II: (Easter 2019) Eggs

*When you start with a portrait and search for a pure form, **a clear volume, through successive eliminations, you arrive **inevitably at the egg. Likewise, starting with the egg and **following the same process in reverse, one finishes with **the portrait.*

PABLO PICASSO

Easter is coming and what better time to talk about eggs!

During my recent mathematical explorations of natural shapes and forms, my attention has been catched by the shape of birds eggs. In the interesting book by J. Adams, *A Mathematical walk in Nature* [1], you can find a short review on the different mathematical modelling approach to describe the shape of an egg. Among them, the geometrical one by Baker [2] is revealed one of the most versatile as it can very accurately reproduce the shapes of a large variety of bird eggs [2]. More recently, the model was used to perform a systematic and comparative study of the shape of bird eggs. This study, published on Science magazine [3], a two-dimensional morphological space defined by the parameters of the Baker’s equation, has been used to show the diversity of the shape of 1400 species of birds. Combining these information with a mechanical model and phylogenetics information, the authors have shown that egg shape correlates with flight ability on broad taxonomic scales. They concluded that adaptations for flight may have been critical drivers of egg-shape variation in birds [3].

# I Primi 150 Anni della Tavola Periodica degli Elementi

Il 6 Marzo del 1869 il chimico russo *Dmitri Ivanovich Mendeleyev* presento’ alla Societa’ di Chimica Russa, una comunicazione dal titolo *La dipendenza delle proprieta’ degli elementi chimica dal peso atomico*. In questa storica comunicazione, Mendeleev pesento’ una tabella in cui organizzava gli elementi chimici allora noti. Questa tabella segno’ anche la fama del suo autore poiche’ fu la prima versione della moderna tavola periodica degli elementi chimici.

Mendeleyev, preparando una seconda edizione del suo libro di chimica, stava cercando un modo per classificare gli elementi chimici allora conosciuti (53 ovvero meno della meta’ di quelli che conosciamo oggi) per fare chiarezza sulle loro proprieta’. In una nota, Mendeleyev racconta che l’ispirazione gli sia venuta in sogno (non e’ la prima volta che Orfeo suggerisce a chimici le loro grandi scoperte scientifici!) [2]:

*I saw in a dream a table where all the elements fell into place as required. Awakening, I immediately wrote it down on a piece of paper.*

# Calculus in a Nutshell: the Definite Integral of Monovariate Functions

The definite integral is the key tool in calculus for defining and calculating quantities important to mathematics and science, such as areas, volumes, lengths of curved paths, probabilities, and the weights of various objects, just to mention a few.

The idea behind the integral is that we can effectively compute such quantities by breaking them into small pieces and then summing the contributions from each piece.

Continue reading# The Lissajous-Bowditch Curves

Try to glue a small mirror to an end of a bent piece of wire fixed to a stable platform and let the laser beam of a laser pointer reflect on it. Entangled spires of an ephemeral dragon of light will perform a hypnotic dance on the wall of your room. This voluptuous dance is the results of two mutually perpendicular harmonic oscillations produced by the oscillations of the elastic wire.

The curved patterns are called *Lissajous-Bowditch* figures and named after the French physicist Jules Antoine Lissajous who did a detailed study of them (published in his *Mémoire sur l’étude optique des mouvements vibratoires, 1857*). The American mathematician Nathaniel Bowditch (1773 – 1838) conducted earlier and independent studies on the same curves and for this reason, the figures are also called Lissajous-Bowditch curves. Lissajous invented different mechanical devices consisting of two mirrors attached to two oriented diapasons (or other oscillators) by double reflecting a collimated ray of light on a screen, produce these figures upon oscillations of the diapasons. The diapason can be substituted with elastic wires, speakers, pendulum or electronic circuits. I the last case, the light is the electron beam of a cathodic tube (or its digital equivalent) of an oscilloscope. This blog is about these curves and shows demonstrations and applications.

# The Magic Imaginary Numbers

Complex numbers may appear a difficult subject given the name. However, there is nothing of really complicated about complex numbers. However, they definitively add a pinch of \em magic \em in the mathematics manipulations that you can do with them!

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