The Golden Ratio and the Number e: Differences, Artistic Applications, and Connections to Water
Introduction
In the world of mathematics, certain numbers transcend their purely numerical nature to become cultural and artistic references.
The golden ratio (φ ≈ 1.618) and the number e (≈ 2.718) are among these fascinating constants which have marked the history of science, the arts and natural philosophy.
Mathematical Definitions and Properties
The Golden Ratio (φ)
The golden ratio is omnipresent in living things, noted φ (phi), it is defined as the positive solution of the equation x² = x + 1, or φ = (1 + √5)/2 ≈ 1.618034. This constant appears naturally in the ratio of the division of a segment according to the "divine proportion": if a segment AB is divided into a point C such that AB/AC = AC/CB, then this ratio is equal to the golden ratio.
Remarkable properties:
- φ² = φ + 1
- 1/φ = φ - 1
- Close connection with the Fibonacci sequence
- Presence in the geometry of the regular pentagon
The Number e (Neper’s Number)
The number e, the base of the natural logarithm, is defined as the limit of (1 + 1/n)ⁿ when n tends to infinity, i.e. e ≈ 2.71828. This fundamental constant of differential and integral calculus governs the phenomena of exponential growth and decay.
Remarkable properties:
- Base of the natural logarithm (ln)
- Derivative of eˣ is eˣ
- Present in the normal law (Gaussian distribution)
- Fundamental in differential equations
Fundamental Differences
Geometric vs. Analytic Nature
The golden ratio has its origins in pure geometry—it arises from visible and tangible ratios of proportions. The number e, on the other hand, arises from mathematical analysis and the dynamic phenomena of growth. We can therefore see that there is a notable difference between the two numbers in their direct applications.
Areas of Application
Golden Ratio: Architecture, visual arts, design, harmonic proportions
Number e: Physics, biology, economics, temporal processes
Aesthetic Perception
The golden ratio is associated with immediate visual harmony, while the number e governs less directly perceptible but fundamental temporal processes in nature.
Applications in the Arts
The Golden Ratio in Artistic Expression
Classical architecture: The Parthenon in Athens features proportions based on the golden ratio in its facade. Notre Dame Cathedral in Paris uses these ratios in its elevations.
Painting: Leonardo da Vinci consciously applied these proportions in "The Mona Lisa" and "The Vitruvian Man." Impressionists like Seurat structured their compositions according to these ratios.
Contemporary design: The Apple logo, the proportions of credit cards, and certain user interfaces intuitively exploit these harmonious proportions.
Music: Some composers like Debussy and Bartók use golden proportions in the temporal structure of their works.
The Number e in Creation
Electronic Music: Amplitude envelopes often follow exponential curves based on e to create natural attacks and decays.
Generative Art: Algorithms using exponential functions create organic patterns and realistic time evolutions.
Photography: The law of reciprocity in photography follows logarithmic relationships related to e.
Relationship with the Water Element: Between Science and Natural Poetry
Ordinary Water: H₂O, Molecular Bohemia
Classical molecular structure: The traditional water molecule (H₂O) struts its stuff with its 104.5° bond angle, like a dancer who never quite finds her balance. If we look for geometric ratios, this angle timidly flirts with our mathematical constants without ever being completely seduced - neither by the golden ratio nor directly by e, but intermolecular interactions follow exponential laws, like a ballet choreographed by the mathematics of the invisible.
Fluid Dynamics: The Navier-Stokes equations that govern the flow of water use exponential and logarithmic functions - the number e thus governs the secret eddies of each drop rolling on a leaf, orchestrating the invisible symphony of fluidity.
Structured Water: H3O2-, the Hexagonal Rebellion
But now a mysterious cousin appears: structured water or hexagonal water H3O2-, where three hydrogen atoms conspire with two oxygen atoms to form something chemically different from "normal" water. This water is causing a scandal in laboratories - some see it as a pseudoscientific hoax, others as a fourth state of matter with regular hexagonal structures. You know which category we fall into at Naturasounds.
Secret Geometry: This structured water is denser, heavier, more viscous, and has higher alkalinity levels than normal water—as if it had decided to take life more seriously than its H₂O sibling. Its hexagonal arrangements evoke cosmic beehives, those perfect geometries where the golden ratio might well nestle in the angles of liquid crystals.
Water that Remembers: Organized in a honeycomb-like sheet that stacks parallel to each other, this EZ (Exclusion Zone) water carries a negative charge and can contain energy. This is truly romantic water that holds electric memories! It is structured like a molecular library, perhaps archiving the geometric secrets of the universe.
Crystallization: When Water Dreams of Geometry
The formation of ice crystals follows complex geometric patterns, sometimes revealing ratios close to the golden ratio within the hexagonal structures. Each snowflake thus becomes an ephemeral mathematical mandala, a frozen meditation on universal harmony.
Natural Aquatic Phenomena: The Symphony of Numbers
Waves and Ripples: Ocean waves behave like true mathematicians of foam, following sinusoidal patterns with exponential damping (e⁻ˣ) as they propagate. Each wave that dies on the sand whispers the song of the number e, that exponential lament of dissipating energy.
Spirals in Water: Whirlpools and eddies create logarithmic spirals often close to golden proportions—true liquid calligraphies where water writes its mathematical poems. These spirals are found fossilized in marine shells, those mother-of-pearl architects who learned their lessons in geometry in prehistoric eddies.
The crystalline memory of springs: Waters reputed to be therapeutic—from the sacred Ganges to the springs of Lourdes—often come from deep underground sources where pressure converts ordinary liquid water into this structured H₃O₂ water. As if the Earth, in its depths, were a patient laboratory that transmutes ordinary water into a geometric elixir.
Evaporation and condensation: These processes follow the exponential laws of our dear e, particularly visible in the drying curves - each drop that evaporates obeys the same equations as population growth or radioactive decay, reminding us that water participates in the great mathematical concert of the universe.
Summary: Complementarity of the Two Constants
In Nature
Nature seems to use the golden ratio for its static spatial structures (leaf arrangement, shell spirals, floral proportions) and the number e for its dynamic processes (population growth, radioactive decay, thermal diffusion).
In Contemporary Art
Contemporary artists are beginning to explore the combination of these two constants: spatial structures based on φ animated by temporalities governed by e. This synthesis opens up new creative avenues that are particularly rich in digital art and interactive installations.
With Water as a Living Metaphor
Water becomes a fascinating element because it embodies this duality between order and chaos, structure and fluidity. Its crystalline structure (ice) reveals geometries sometimes close to the golden ratio, while its fluidity follows the exponential laws of the number e.
But here comes structured water H₃O₂ to disrupt this beautiful dichotomy! This "rebel" water with its hexagonal arrangements could well be the poetic link between our two constants: its perfect geometric structures evoke the harmony of the golden ratio, while its capacity to store and release energy follows the exponential dynamics of the number e.
This water, which remembers, which organizes itself, which stores information in its liquid crystalline arrangements, becomes an artistic medium of incredible richness to explore these two mathematical constants. It reminds us that nature does not separate aesthetics from functionality - it marries them in a molecular dance where each hexagonal arrangement is both beautiful and efficient.
Whether structured water is an established scientific reality or a molecular poetry under investigation, it invites us to consider water not as a simple solvent, but as a liquid architect, a crystalline library, a fluid memory that could well keep the traces of universal mathematical harmony.
Conclusion: Water, Mathematical Muse of the Universe
The golden ratio and the number e represent two complementary facets of mathematical harmony: one spatial and immediately perceptible, the other temporal and procedural. Understanding them enriches our artistic approach and our perception of natural phenomena.
Water—whether the classic H₂O dancing in creative disorder, or the mysterious H₃O₂ structuring itself into a hexagonal library—serves as a metaphorical and physical bridge between these two dimensions of aesthetic and scientific experience. It teaches us that mathematics is not a cold abstraction, but the secret language that matter speaks when it dreams of beauty.
This mathematical and poetic exploration invites us to consider that art, far from being separate from science, draws on the same fundamental structures that govern the universe: from the geometry of crystals to the rhythms of life, from the spirals of water to molecular arrangements, everything participates in this great score where φ and e are the fundamental notes, and water the musician who interprets them with fluid grace, whether in the simplicity of H₂O or the controversial sophistication of H₃O₂.
Because in the end, it doesn't matter whether structured water is based on established science or poetic intuition: it reminds us that nature still holds secrets, and that mathematical wonder remains the surest path to understanding the world that carries us and passes through us.