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Symmetries
Part: General · Chapter: Symmetries · Edition: 2026-07-12
Keywords: ratio, harmony, symmetry, category theory, information
Glossary:symmetry,ratio,category theory,functor— see GLOSSARY
At a glance
- Symmetry is structure preserved under transformation.
- Ratios and harmonic proportions recur across music, colour, chemistry, and language.
- Category theory abstracts maps that preserve structure (functors).
Online references (live)
Offline snapshots (link-rot shield)
Offline snapshot omitted in this web edition. Offline snapshot · Symmetry · https://en.wikipedia.org/wiki/Symmetry · retrieved 2026-07-12
Offline snapshot omitted in this web edition. Offline snapshot · Ratio · https://en.wikipedia.org/wiki/Ratio · retrieved 2026-07-12
Offline snapshot omitted in this web edition. Offline snapshot · Category theory · https://en.wikipedia.org/wiki/Category_theory · retrieved 2026-07-12
Exploring symmetries between different substances, such as color combinations, musical notes, chemical elements, and language semantic relationships, touches on a fundamental principle that can be generalized through the concept of proportional relationships or ratios. The common ground in theorizing these relationships is found in the concept of harmonic proportions or ratio-based structures that underlie the organization of various phenomena across different domains.
Common Ground: Proportional Relationships
- Mathematics of Proportions and Ratios:
* At the core of this idea is the mathematical concept of a ratio, which is a relationship between two quantities, expressing how many times one quantity contains the other.
* Ratios are fundamental in understanding relationships in various domains because they capture the relative magnitude and structure of elements. - Symmetry and Harmony:
* Symmetry in any system can be understood as a balance or proportionality between its parts. In geometry, symmetry refers to invariance under certain transformations (like rotation or reflection), which can be seen as specific ratios being preserved.
* Harmony, whether in music, color, or even language, is often defined by certain ratios that produce pleasing effects. For example, musical intervals are based on specific frequency ratios (e.g., an octave is a 2:1 ratio), and color harmonies can be derived from specific ratios of wavelengths. - Musical Notes and Harmonic Series:
* The relationships between musical notes are governed by harmonic ratios. The harmonic series describes a sequence of sounds where each sound is a simple multiple (a ratio) of a fundamental frequency.
* Consonance and dissonance in music can be described by how simple or complex the frequency ratios between notes are. - Color Theory:
* In color theory, certain combinations of colors are aesthetically pleasing because they are related by simple ratios. For example, the color wheel arranges colors by their wavelengths, and complementary colors are opposite on the wheel, often related by a simple ratio. - Chemical Elements and Stoichiometry:
* Chemical reactions are governed by stoichiometric ratios, which describe the proportions of elements that combine to form compounds. The idea that certain ratios produce stable compounds is analogous to harmony in music or balance in color.
* The Periodic Table itself reflects a symmetry in the relationships between elements, where properties repeat periodically due to the underlying ratios in atomic structure. - Language and Semantic Structures:
* In language, semantic relationships can often be understood in terms of proportionality or similarity (ratios of meaning or context). For example, analogies are structured around the idea of a proportional relationship between concepts (e.g., A is to B as C is to D).
* The relationships between words and their meanings can be mapped onto a semantic space where distances and angles (geometrically representing ratios) describe similarities and differences in meaning.
Theoretical Frameworks
- Information Theory:
* Information theory can be used to quantify relationships between different elements (whether in language, music, or color) by measuring the entropy or information content. The relative proportions of information can be thought of as ratios that describe the relationship between different signals or messages. - Category Theory:
* Category theory provides a high-level abstract framework where relationships between objects (which could be colors, notes, elements, etc.) are captured by morphisms. The idea of functors preserves the structure between categories, which can be seen as preserving certain proportional relationships or ratios. - Topos Theory:
* Topos theory extends category theory and provides a framework for understanding spaces and logical structures in a highly abstract way. It can be used to describe the relationships and symmetries between different structures by focusing on the relationships between points (objects) and the maps (morphisms) between them.
Conclusion
The common ground across these different domains is the concept of harmonic or proportional relationships. Whether in music, color, chemistry, or language, the idea that relationships are formed by combinations of ratios is a unifying principle that allows for a deep understanding of symmetry and structure across various fields. Theoretical frameworks like information theory, category theory, and topos theory offer tools to formalize and explore these relationships in a rigorous way.
Offline snapshot omitted in this web edition. https://en.wikipedia.org/wiki/Icosahedral_symmetry
Offline snapshot omitted in this web edition. https://en.wikipedia.org/wiki/Translational_symmetry
Offline snapshot omitted in this web edition. https://en.wikipedia.org/wiki/Fractal_dimension
Offline snapshot omitted in this web edition. https://en.wikipedia.org/wiki/Fractal
https://en.wikipedia.org/wiki/Merkabah_mysticism
Offline snapshot omitted in this web edition. https://pt.wikipedia.org/wiki/Sólido_platónico
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Overlapping areas
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Part index
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