# Modelling Forms in Nature: Easter Chocolate Eggs 2021

This year, your Highness, we will be featuring square eggs.
Peter Carl Fabergé

The Easter Math Bunny is back again, and he is talking again of … eggs!

Two years ago, I ended my blog with a question, “What about chocolate egg shapes?” It is now the time to give some answer. If you have read my previous articles, I and many scientists and artists have been caught by the shape of the bird eggs. Several models have been proposed to reproduce the silhouette of bird eggs. Baker [1] proposed a simple two-parameters mathematical model based on projection geometry that revealed to be versatile and accurate in producing the shapes of a large variety of bird eggs [1]. More recently, the model was used to perform a systematic and comparative study of the shape of 1400 bird eggs species [2].

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# The Dandelion (Taraxacum Officinalis) and OpenCV

The dandelion’s pallid tube
Astonishes the grass,
And winter instantly becomes
An Infinite Alas —

The tube uplifts a signal Bud
And then a shouting Flower, —
The Proclamation of the Suns
That septulture is o’er.

– Emily Dickinson

The yellow flowers and the delicate and beautiful florescence of Dandelion catch the attention of both romantic and curious souls. The aerial consistency of the fine silk decorated seeds that glance to the sunlight as crystalline material became the favourite subject of inspired photographer and the toy of amused children. Beside the grace of its forms, other interesting and curious secret is hidden in its phloem fluids. In fact, if you cut one of the stems of the plant a milky sticky liquid will flow out of the wound resection. This latex is going to polymerize at 30-35 oC in few minutes in a yellow-brown quite solid mass. Around the year 1982, I have annotated this observation but I could not find in my later notes further follow-ups study on the topics. It was a casual observation but I didn’t know at that time that this latex is indeed very useful. A variety of the Taraxacum (Taraxacum koksaghyz, Russian Dandelion) was used in Russian and American to produce a replacement of the natural rubber from Brazil during the WWII that was in shortage because of the war. Now days, many researches are in progress to exploit the lattice of Taraxacum and also Taraxacum brevicorniculatumas a convenient replacement of the rubber plant lattice. A recent study has shown the presence of rubber particles in the lattice of these plant in 32% proportion composed prevalently by poly(cis-1,4-isoprene) at >95% of purity (www.biomedcentral.com/1471-2091/11/11). The brownish lattice condensate that, as I reported in my note, forms after exposing for several minutes the latex to the air, is caused by the presence of the polyphenol oxidase (PPO) enzyme that produces the fast coagulation of the latex by catalysing the oxidation of polyphenols. Genetic engineer the plant, it is possible to reduce the amount of PPO in the latex making economically competitive the use of this resource for the production of latex.

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# Modelling Natural Shapes: (Easter) Eggs 2020

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

$y=T(1+x)^{\frac{\lambda}{1+\lambda}}(1-x)^{\frac{1}{1+\lambda}}$,

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) 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].

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# 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.

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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 fermàtaPè rattaccà le strengh'a strucenòne. Me uànne gli'òcchie doppe 'na utràtaI uède, tutt'a giòrne, nè salòneAddò 'na raccòta sta urdenàtaDe prete, crete, pièzze de matòne, Cule de uàse, de pile, de pignàte,Màneche de recciòle i de bicchièraAncòra prima de Nuè 'mpastàte. Quante sècule,frà, che sò passàte'Nche gli'Ome a trebbulà 'ncim'a sta TeraI 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 archeological exploration of old computer files, I came across to another simple but interesting Amiga Basic program that 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 modeling. 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” and was a description of the modeling of plants to study their interaction with the environment. It was a fascinating paper, and still simple and primitive graphics was catching my imagination. Nowadays the field of digital morphology 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, TV programs. My curiosity an interest has been always caught by the form and shapes of the organism and the structure of leafs nervation was an intriguing pattern very much 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 Mines: 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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