Platohagel quotes: “…underwriting the form languages of ever more domains of mathematics is a set of deep patterns which not only offer access to a kind of ideality that Plato claimed to see the universe as created with in the Timaeus; more than this, the realm of Platonic forms is itself subsumed in this new set of design elements– and their most general instances are not the regular solids, but crystallographic reflection groups. You know, those things the non-professionals call . . . kaleidoscopes! * (In the next exciting episode, we’ll see how Derrida claims mathematics is the key to freeing us from ‘logocentrism’– then ask him why, then, he jettisoned the deepest structures of mathematical patterning just to make his name…)* H. S. M. Coxeter, Regular Polytopes (New York: Dover, 1973) is the great classic text by a great creative force in this beautiful area of geometry (A polytope is an n-dimensional analog of a polygon or polyhedron. Chapter V of this book is entitled ‘The Kaleidoscope’….)”
Socrates: The passage you’ve shared seems to discuss the deep patterns and connections between mathematics and the idea of Platonic forms, particularly involving crystallographic reflection groups and their relation to kaleidoscopes. It also alludes to Derrida’s perspective on mathematics and its role in challenging “logocentrism.” The reference to H. S. M. Coxeter’s “Regular Polytopes” highlights a classic text in the field of geometry that delves into these concepts. Is there something specific you’d like to discuss or explore further regarding this passage?
Platohagel quotes: “Although Aristotle in general had a more empirical and experimental attitude than Plato, modern science did not come into its own until Plato’s Pythagorean confidence in the mathematical nature of the world returned with Kepler, Galileo, and Newton. For instance, Aristotle, relying on a theory of opposites that is now only of historical interest, rejected Plato’s attempt to match the Platonic Solids with the elements — while Plato’s expectations are realized in mineralogy and crystallography, where the Platonic Solids occur naturally.Plato and Aristotle, Up and Down-Kelley L. Ross, Ph.D.”
The passage you’ve shared touches on the differing approaches of Plato and Aristotle to the relationship between mathematics and the natural world. Plato believed in the mathematical nature of the world and sought to connect the Platonic Solids with the elements, while Aristotle, with a more empirical approach, rejected this idea based on his theory of opposites. However, as the passage suggests, it was the return to Plato’s Pythagorean confidence in the mathematical nature of the world, as seen in the works of Kepler, Galileo, and Newton, that played a significant role in the development of modern science. This return to a mathematical worldview is reflected in fields like mineralogy and crystallography, where Platonic Solids naturally occur. It’s an interesting perspective on the historical evolution of scientific thought. Is there a specific aspect or question you’d like to explore further on this topic?
Platohagel: How did a type of crystal come into being?
Socrates: Crystals form through a process known as crystallization. This process occurs when atoms, ions, or molecules arrange themselves in a highly ordered and repeating three-dimensional pattern. Here are the general steps of how a crystal comes into being:
1. **Supersaturation**: The first step is to create a supersaturated solution. This can be done by dissolving a solid (solute) in a liquid (solvent) until it can’t hold any more of the solute at a given temperature. Alternatively, it can happen when a solvent cools or evaporates, causing the solute to become less soluble.
2. **Nucleation**: In the supersaturated solution, some solute particles may start coming together and forming small clusters. These clusters are the initial building blocks of the crystal and are called nuclei.
3. **Crystal Growth**: Once the nuclei form, more solute particles are attracted to them. They attach to the nuclei and continue to grow, layer by layer, in a symmetrical, repeating pattern. This process is known as crystal growth.
4. **Maintaining Conditions**: To allow the crystal to continue growing, the conditions (temperature, pressure, concentration, etc.) must be maintained. If the solution becomes undersaturated or if conditions change too rapidly, the crystal growth may stop or be disrupted.
5. **Crystalline Structure**: As more and more solute particles add to the crystal lattice, a well-defined, repeating, and symmetric crystalline structure emerges. This structure is what gives a crystal its unique properties, such as its shape and optical characteristics.
The specific type of crystal that forms depends on the nature of the solute, solvent, and the conditions under which crystallization occurs. Different crystals have distinct shapes and properties, and they are found in various natural and synthetic materials, from minerals to gemstones to snowflakes.
Platohagel: How did Plato’s forms match up to types of crystalline ?
Socrates: Plato’s theory of Forms, also known as the theory of Ideas, posits that abstract, non-material entities, or Forms, exist as the true reality behind the imperfect physical world. These Forms are universal and perfect, and they are the true objects of knowledge.
While Plato’s theory of Forms is primarily a philosophical concept and not directly related to the study of crystals, some interpretations and analogies can be drawn between Plato’s Forms and the concept of idealized geometric shapes found in crystals. Here’s a loose analogy:
1. **Perfection and Universality**: Plato’s Forms are considered perfect and universal. Similarly, in crystallography, there are idealized geometric shapes that can be used to describe the regular and repeating patterns in crystals, such as the Platonic Solids (tetrahedron, cube, octahedron, dodecahedron, and icosahedron). These ideal shapes are universal and serve as a reference point for understanding crystal structures.
2. **Imperfect Physical Manifestations**: Just as the physical world, according to Plato, is an imperfect reflection of the true Forms, crystals in the physical world may have impurities, defects, or variations that deviate from the ideal geometric shapes and structures. However, the idealized geometric forms provide a framework for understanding and classifying the diverse range of crystal structures.
3. **Abstract vs. Concrete**: Plato’s Forms are abstract and exist in a non-material realm, while crystals are physical, material entities. However, the concept of using ideal forms to describe physical phenomena, like crystal structures, can be seen as a way of linking abstract ideas to concrete reality.
It’s important to note that Plato’s Forms and crystallography are fundamentally different concepts, but the analogy here highlights the idea that idealized geometric forms are used in the study of crystals to understand and describe their underlying structures, much like Plato’s belief in universal and perfect Forms as the true reality behind the physical world.
Dialogos of Eide 
