Easter 2020: Modelling Natural Shapes: (Easter) Eggs

One year ago, I wrote an article about the modelling of the egg shapes, promising at one point to come back on the topics. A next step in studying eggs shapes is to look to real one or a copy of it. A happy occasion for experimenting with the model using three-dimensional graphics and 3d Printing! That is a natural indeed step: take half of the symmetric curve representing the egg shape


where T and \lambda are two parameters, and rotate it around the central axis

\begin{aligned} x'&=&x\\ y' &=&y*cos(\theta) \\ z' &=& y*sin(\theta) \end{aligned}

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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.


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 caught by the shape of birds’ eggs. In the exciting book by J. Adams, A Mathematical Walk in Nature [1], you can find a short review of the different mathematical modeling approaches to describe the shape of an egg. Among them, the geometrical one by Baker [2] is revealed as one of the most versatile as it can 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 in Science magazine [3], a two-dimensional morphological space defined by the parameters of Baker’s equation, has been used to show the diversity of the shape of 1400 species of birds. Combining this 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].

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I Primi 150 Anni della Tavola Periodica degli Elementi

Che la nobiltà dell’Uomo, acquisita in cento secoli di prove e di errori, era consistita nel farsi signore della materia, e che io mi ero iscritto a Chimica perché a questa nobiltà mi volevo mantenere fedele. Che vincere la materia è comprenderla, e comprendere la materia è necessario per comprendere l’universo e noi stessi: e che quindi il Sistema Periodico di Mendeleev, che proprio in quelle settimane imparavamo laboriosamente a dipanare, era una poesia, più alta e più solenne di tutte le poesie digerite in liceo: a pensarci bene, aveva perfino le rime! 

Primo Levi, Il sistema periodico

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.

La tavola periodic degli elementi di Mendeleev. I trattini rappresentano elelmenti sconosciuti nel 1871. (fonte della figura: wikipedia)

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.

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Frosinone and the Ciociaria

Glie musèu
Tra glie campanìle i l'Annunziàta,
'Ntòcce prima della uìa ‘Ngelòne,
Stamatìna sò fatte 'na fermàta
Pè rattaccà le strengh'a strucenòne.
Me uànne gli'òcchie doppe 'na utràta
I uède, tutt'a giòrne, nè salòne
Addò 'na raccòta sta urdenàta
De prete, crete, pièzze de matòne,
Cule de uàse, de pile, de pignàte,
Màneche de recciòle i de bicchièra
Ancòra prima de Nuè 'mpastàte.
Quante sècule,frà, che sò passàte
'Nche gli'Ome a trebbulà 'ncim'a sta Tera
I ch'è remàste ? Ddù cocce smenuzzàte.
              Irèno Da Vialìra (Poeta ciociaro)  

I am European of Italian nationality. I very proud of my background but, unfortunately, my carrier put me in an orbit that does intersect my country only during my holiday vacations. In these close encounters, my landing site is Frosinone. When not-Italian acquaintances want to know about my Heimat, most of them are puzzled about the name and location of my hometown. Usually, I help them to overcome the understandable disorientation by giving as reference Rome and telling them that my birthplace is somewhere 80 km in the South of the Caput Mundi. Last year (2018) my hometown soccer team (Frosinone Calcio, nicknamed Canarini, The Canaries, for their home colours)) moved in the first league (A) of the national soccer championship. So, let see if this success will help to raise its notoriety!

It is a long time that I would like to write about Frosinone. However, in this article, I won’t write about the success of the Canarini football team but about my hometown and the rural place in central Italy where it is located.

In the province of Frosinone, there are many historical famous towns such as Anagni that has the exquisite Anagni Cathedral with its museum, Alatri with its megalitic Acropolis, Ferentino another megalytic city with also a beutyful Romanesque Duomo, Veroli and the closeby Certosa of Trisulti that has been recently on news spot for the sadly attempt to trasform this wolderful medieval santuary of christianity and culture to a private business when instead it should be preserved and nominated instead as an humanity heritage. I will talk about these town in other articles. This just to mention some of the famous towns and cultural places in Ciociaria.

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Retro Programming II: the Amiga and the Computational Beauty of the Leaf

In my archaeological exploration of old computer files, I came across another simple but exciting Amiga Basic program I programmed in 1989. It is named “Foglie”, the Italian name for leaves. It was an attempt to explore some ideas of functional plant morphology modelling. The stimulus comes after the reading of the paper by Karl J. Niklas on issue 213 of Le Science (the Italian edition of the Scientific American magazine [1]). The article titled “Computer-simulated plant evolution” described the modelling of plants to study their interaction with the environment. It was a fascinating paper; still, simple and primitive graphics caught my imagination. Nowadays, the field of digital morphology has come to an age (just to mention one, Avatar), and we can have an idea of this progress in the level of realism in movies, video games, and TV programs. However, the organism’s form and shape have always caught my curiosity and interest. The structure of leaf nervation was an intriguing pattern related to my acquaintance with the fascinating fractals objects, another recurrent topic in the pages of scientific magazines of the period.

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Leaf Miners: Life in (almost) Two Dimensions

The pigmy moths of the family Nepticulidae are the smallest Lepidoptera. They are also called leaf miners as, in the larval stage, they feed on leaves by excavating tunnels in between the two laminas. Some species are considered pests by gardeners and farmers since they feed on fruit plants or flowers, like roses. It is on a plant of rose that grows on a side of our house that I discovered this seasonal intruder that decorate with long tracks lined with the black strip of their frass, the green leaf of our rose. I did some research and I found the name of the culprit: Stigmella anomalella. A nice name for a small pest! Reading more about this little creature, I discovered that has been found fossils dated 97 millions of year ago, this minuscule insect survived dinosaurs (to bother our roses!) so they deserve all my respect for their resilience! – By the way, the study of traces left by insects is a science and it is called Ichnoentomology [3]. Indeed, the curious sinuous trajectory of dwelling inside the leaf cathed my curiosity and I decided to analyze more in details their mining behavior. There are different websites dedicated to the Nepticulidae and on leaf miners in general [2, 3]. I have also found an interesting old book (1955) on the topics of E.M. Hering titled “Biology of the Leaf Miners” that contains interesting information about these insects [4]. Miner tracks are classified  accordingly to their shapes in the following types: 

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La programmazione in Awk II: Life in a Shell

Il gioco Life fu inventato negli anni ’70 dal prolifico matematico John H. Conway (vedi [5] per la sua biografia) ed è diventato famoso dopo la pubblicazione di Martin Gardner nella sua rubrica di matematica amatoriale sulla rivista Scientific American [1,2]. Il gioco è basato sugli automi cellulari concepiti da Konrad Zuse e Stanislaw M. Ulam all’inizio degli anni ’50, e poi adottati da John von Neumann per il suo studio sugli automi auto-replicanti [2,3]. Un automa cellulare è composto da unità (celle) interagenti disposte in una griglia quadrataIl sistema si evolve in cicli di vita in cui ogni cella cambia stato e nuove celle possono nascere e altre possono sopravvivere o, eventualmente, morire. Lo stato di ogni cella nel ciclo successivo è definito dall’interazione con le celle adiacenti in base a delle regole. L’interazione avviene con i primi vicini di ciascuna cella. Come mostrato nella Figura 1, è possibile utilizzare due tipi di intorni (cerchi) della cella centrale. Il gioco Life usa il tipo di proposto da Moore. 

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Awk Programming II: Life in a Shell

The game of Life was invented in the ’70 by the prolific mathematician John H. Conway (on the 11/4/2020 sadly J.H. Conway passed away at the age of 82 after having contracted the COVID-19, see [5] for his biography). The game becomes popular after Martin Gardner described it in his famous column in the Scientific American magazine [1,2].  The game is based on cellular automata conceived by Konrad Zuse and Stanislaw M. Ulam at beginning of the ’50 and then adopted by John von Neumann for his study on self-replicating automata [2,3]. A cellular automaton is composed of interacting units (cells) arranged in a square grid. The system evolves in life cycles where each cell change status and new cells can be born, and others can survive or eventually die. The status of each cell in the next cycle is defined by the interaction with their neighbor cells according to a given set of rules. The interaction occurs with the first neighbors of each cell. As shown in Figure 1, two type of neighbor’s cells (circles) can be used, the game of Life uses the Moore type neighborhood.

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