{"id":547,"date":"2023-09-24T13:45:52","date_gmt":"2023-09-24T13:45:52","guid":{"rendered":"https:\/\/tangowithbear.blog\/?p=547"},"modified":"2025-09-09T14:16:41","modified_gmt":"2025-09-09T14:16:41","slug":"the-architecture-of-growth","status":"publish","type":"post","link":"https:\/\/tangowithbear.blog\/?p=547","title":{"rendered":"THE ARCHITECTURE OF GROWTH"},"content":{"rendered":"\n

The Seashell’s Anatomy – A Story of adaptation, innovation, and survival<\/h1>\n\n\n\n

Tools:<\/strong> Rhinoceros, Grasshopper, C#<\/p>\n\n\n\n

Natural forms in the world around us often emerge as a result of growth and adaptation, a crucial aspect of survival.  Seashells serve as striking examples of how living organisms evolve to create unique and functional shapes by carefully balancing the proportions of their dwellings.<\/p>\n\n\n

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CALCIUM CARBONATE, SPECIALISED CELLS, AND ORGANIC MATRIX<\/strong><\/h2>\n\n\n\n

Seashell growth is a complex process that involves both biology and chemistry. Seashells are mostly made of a substance called calcium carbonate (CaCO3<\/sub>), which is common in the oceans. To understand how seashells grow, we need to know how marine creatures take calcium and carbonate from the water and use them to build their shells.<\/p>\n\n\n\n

Picture the oceans as vast reservoirs of minerals dissolved in water. Naturally, the oceans contain a substantial amount of calcium ions (Ca\u00b2\u207a), originating from the breakdown of rocks and minerals. These dissolved calcium ions serve as fundamental building blocks for marine organisms engaged in biomineralization, which is their ability to extract calcium and carbonate ions (CO3<\/sub>2-<\/sup>) from their surroundings and fuse them into solid calcium carbonate crystals.<\/p>\n\n\n\n

Marine organisms responsible for shell formation possess specialised cells, often referred to as “calcifying cells” or “shell-forming cells.” These cells oversee the uptake, transportation, and deposition of calcium ions.<\/p>\n\n\n\n

In addition to calcifying cells, organisms secrete an organic matrix comprising proteins, lipids, and other biomolecules that act as templates. This matrix guides the precise arrangement of calcium carbonate crystals. A shell typically encompasses a mantle region, which may extend beyond the shell itself. In this area, extra-pallial fluid is secreted, containing components required for the “construction” process.<\/p>\n\n\n\n

Ocean Acidification: The ocean absorbs approximately 30% of atmospheric carbon dioxide (CO2), leading to chemical reactions that raise hydrogen ion concentration, resulting in ocean acidification and reduced carbonate ions, making it challenging for calcifying organisms to build and maintain their calcium carbonate structures.<\/p>\n\n\n\n

GROWING STRUCTURES<\/strong><\/h2>\n\n\n\n

The spiral form of seashells is a testament to the mathematical precision that enables the shell to continue growing while accommodating the developing mollusk. The part of the shell where the creature resides, responsible for adding new material, is located at the open edge. It deposits more material on one side than the other, resulting in the shell’s spiral shape. The specific rates at which shell material is added at different points along the open edge are determined by the animal’s anatomy, and changing these rates can yield a wide range of shapes.<\/p>\n\n\n\n

There are essentially three rates of growth:<\/p>\n\n\n\n

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  1. The rate at which the aperture moves away from the coil’s axis.<\/li>\n\n\n\n
  2. The rate of translation, determining how far the shell coils along the z-axis.<\/li>\n\n\n\n
  3. The rate at which the aperture increases in size.<\/li>\n<\/ol>\n\n\n\n

    From a geometric perspective, the shell’s outer surface can be described as a three-dimensional structure formed by a curve, typically an ellipse known as the generating curve (generatrix), moving along a helical spiral called the structural curve (directrix). As the generating curve travels along the helical path, it gradually widens, giving rise to the shape of the shell.<\/p>\n\n\n

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    To build it in Grasshopper, I used the model described in JORGE PICADO’s paper [1], which gives more details on the mathematics of seashell forms.<\/p>\n\n\n\n

    The path (directrix)<\/strong><\/h2>\n\n\n\n

    The spiral pattern in shells is often referred to as a logarithmic or equiangular spiral. This type of spiral has a distinct mathematical property: for every constant rotation angle (for example one full revolution), the distance from the spiral’s origin (the pole) is multiplied by a fixed factor. This means that as you move along the spiral, the angle between the direction you’re looking and the direction you’re moving remains the same. This consistent angle gives shells their harmonious appearance, as each segment of the spiral is proportionally similar to the next. <\/p>\n\n\n\n

    This spiral can be described with 3 parameters  :<\/p>\n\n\n\n