The Minkowski–Bouligand dimension of a clay brick
Notice bibliographique
Résumé
In the early 1990's, fractals and chaos were hot. In 1987, James Gleick had published "Chaos: Making a New Science", popularizing non-linear dynamics. Hydrologists played an important role in the development of fractal theory. Hurst had discovered that sequences of dry and wet years for the Nile showed very long memory effects. Instead of the chance of a dry year following a dry year being 50%, Hurst found that there were surprisingly many long series of dry or wet years. Seven fat years, seven lean years, as it is noted in Genesis. Scott Tyler found fractals in soils ("Fractal processes in soil water retention"). At Cornell, where we were at the time, David Turcotte described "Fractals in geology and geophysics". A few years later, Ignacio Rodríguez-Iturbe and Andrea Rinaldo would publish "Fractal River Basins: Chance and Self-Organization". In short, fractals were exciting scientific gold.A fractal is not just an obscure mathematical object but something that can actually be found everywhere in nature. Early on, a paper was published in Nature with the title "Fractal viscous fingering in clay slurries" by Van Damme, Obrecht, Levitz, Gatineau, and Laroche. They "only" did an experiment on a fractal embedded in 2D; we should be able to do one better and find the fractal dimension of the surface of cracking clay embedded in 3D. So out we went, collected some clay, mixed it with water in a cement mixer, siliconed together a shallow "aquarium", and poured in the slurry. To observe the cracking of the drying slurry, a video camera was mounted above the experiment, looking down and taking time-lapse images. To access the views from the sides, mirrors were installed at 45 degrees at each of the four sides. Lights made sure the camera captured high quality images. The whole set-up was enclosed in a frame with dark cloth to ensure that lighting was always the same.  We already had some box-counting code ready to calculate the fractal dimension of the surface, called the Minkowski–Bouligand dimension. One variable needed some extra attention, namely the boundary between the clay slurry and the glass sides. If the clay would cling to the sides, it would be difficult to understand the effects that this boundary condition had on the outcome of the experiment. Moreover, the cracks may not have become visible in the mirrors when the sides were covered with mud. So, instead, it was decided to make the sides hydrophobic with some mineral oil. This ensured that when the clay would start to shrink, it would come loose from the sides. Now, all we had to do was wait. It took only a week or so before the consolidated slurry started to shrink and to come loose from the sides. After that, the clay continued shrink for many weeks. This is how we learned that the fractal dimension of a shrinking brick of clay is (very close) to 3.0. 
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Prédiction distillée sur la base complète
Imitation des enseignantsNi prévalence calibrée, ni vérité terrain. Validation humaine à venir. Apprise à partir de 10 348 étiquettes directes de Codex et de 10 348 étiquettes directes de Gemma. Le mode candidate est l'union des têtes enseignantes seuillées; le consensus est leur intersection. Ces sorties portent le statut machine_predicted_unvalidated et ne sont ni des étiquettes humaines ni des étiquettes directes de modèles de pointe.
Scores Codex et Gemma par catégorie
| Catégorie | Codex | Gemma |
|---|---|---|
| Métarecherche | 0,001 | 0,000 |
| Méta-épidémiologie (sens strict) | 0,000 | 0,000 |
| Méta-épidémiologie (sens large) | 0,000 | 0,000 |
| Bibliométrie | 0,000 | 0,001 |
| Études des sciences et des technologies | 0,000 | 0,000 |
| Communication savante | 0,000 | 0,000 |
| Science ouverte | 0,002 | 0,004 |
| Intégrité de la recherche | 0,000 | 0,000 |
| Charge utile insuffisante (le modèle a refusé de juger) | 0,000 | 0,000 |
Scores machine (provisoires)
Les deux têtes enseignantes du modèle étudiant, lues sur ce travail. Un score ordonne la base pour la relecture; il n'affirme jamais une catégorie, et le statut de validation accompagne chaque rangée tel quel.
Scores de référence d'un modèle non mature (critères de maturité non atteints, 7 itérations). Un score ordonne; il n'affirme jamais une catégorie.
score_only:v0-immature-baseline · tel quel depuis la passe de notation : score_only signifie que le nombre peut ordonner les travaux, et qu'aucune étiquette de catégorie n'en découleClassification
machine, non validéePrédiction automatique; un appel candidat d’une seule tête enseignante, pas un consensus.
Le détail, modèle par modèle et score par score, se trouve en fin de page sous « Comment cette classification a été obtenue ».