"… I seem […] only like a boy playing on the sea-shore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me". – Isaac Newton.
Gentilissimi/e Lettori e Lettrici, Dear Reader, Sehr geehrte Leserinnen und Leser,
Grazie mille per aver fatto tappa durante le vostre peregrinazioni cibernautiche nel mio sito web e per dedicare un po’ del vostro tempo nel leggere i miei articoli. Spero che li avete trovati tanto interessanti e utili da continuate a tornare a leggermi. Voglio anticipare alcune delle prossime pubblicazioni. Tra breve usciranno nuovi titoli:
The Logistic Map and the Feigenbaum Constants: a Retro Programming Inspired Excursion.
L’integrazione numerica di equazioni differenziali, parte II: 50 anni fa l’uomo ha messo piede sulla Luna
Retro Programming: Acid-base Titration.
Retro Programming: Plant evolution.
Per il momento auguro a tutti voi di trascorrere con le vostri cari un felice Natale e di avere un nuovo anno pieno di buone notizie.
On the 7 October 2019, Prof Dr Herman Johan Christiaan Berendsen passed away just shortly after his 85 birthday. Prof Berendsen is considered the founder of the molecular dynamics simulation of biological system: the area of theoretical research that also shaped my scientific career. He was working at the University of Groningen in the picturesque Northern part of the Netherlands. It was there that I meet him the first time as it allowed me to conduct research in his lab during the last year of my doctorate researches training at the University of Rome “La Sapienza”. After I completed my doctorate, Herman gave me the opportunity to continue working in his group with a postdoc position within the “Protein Folding” EU Training network. This happens just two years before his retirement and therefore I was also one of his last postdocs. After retirement, Herman dedicated himself to write two books that distillate all his experience in the area of molecular simulation  and in the education . He stated in a project on the social scientific network Researchgate that “I am retired and work occasionally on methods for multiscale simulations.”
This year marks the 150th anniversary of the periodic table of the elements (TPE) which currently has 118 entries, the latest arrival (the Tennessine) was discovered 10 years ago (2009), and I feel obliged as a chemist to give some a small informative contribution to celebrate this important event.
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.
“… 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 . Here it is just another small extract:
“… 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.”
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 Katherine Goble (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. 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. Globe 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.
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.
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.
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 , 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  is revealed one of the most versatile as it can very accurately reproduce the shapes of a large variety of bird eggs . 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 , 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 .
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!) :
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.
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.