SPIRA MIRABILIS

SPIRA MIRABILIS-from ratios to complex plane

Foreword (EN)

This book closes the series of the first 3 works, which connect the Bible with mathematics: The History of Logarithms, The History and Geometry of Regular Polyhedra and The Geometry of Conics, all written in recent years. My interest in mathematics dates to my childhood, when I marveled at geometry, Pythagoras, radicals and then regular polyhedra and conics, which I carved in glass and sketched in stencils.

Finally, when the almond tree with white hairs began to bloom, I found these hidden treasures in the divine Word, which came from heaven, which I read daily and nourishes my soul. I have been passionate about art since I was little, I would doodle with colored pencils, when I was very lame and Master Donici, my uncle, would take me to landscapes and draw my first portrait.

I tried to combine Beauty, Art with Truth, Mathematics and I always see how they hold hands and climb together on the ladder of heaven. I always find these divine features in the Person of the Lord, like a ray of sunshine reflected by a diamond with thousands of faces.

I started from the measures in the Scriptures, some of the most important, I saw them as logos-measure, fraction, percentage, then I extended them to proportions, to the most beautiful, golden, which defines the harmony of Greek statues, but also the flight of bees around the sun, to derivatives and shadows in paintings. I touched the harmonies of the human body, the portraits of the face, I calculated the proportions of the Greeks and compared them with the portraits of my grandfather and mine. The golden ratio took me to the pentagon, to the golden spiral, then into space to the regular polyhedra, governed by the cube and the same number 5. I enclosed them in a sphere and marveled like Kepler, whom Harmonice Mundi- Universal harmony he believed.

The spirals, golden and logarithmic, classical and differential geometry brought me closer to the verses in the Bible, which define true life, as a spira mirabilis, which rotates uniformly on the circle of life, and grows exponentially, from strength to strength. Here I found the logarithms, hidden under hyperbolas. From the spirals I arrived at the number e, which describes the simple differential equation: growth (geometric progression) in time (arithmetic progression). I stopped at hyperbolic functions, at the chain equation and at the spider’s thread, which blows and sings swayed by the wind.

I walked through history and several heroes illuminated from above have scrolled one by one in my pages. From Fibonacci, Leonardo, to Descartes, Euler and Gauss, their series continue nowadays, when we have the richest information. Euler’s equation and Gauss’s circle revolve around the mysterious number e, like bees around the sun. I found the number e in whirlwinds, sun flowers, in spirals, under hyperbolas, but also in interests in the Lord’s parables. I linked e to the unit circle and pi, I found it again in the regular polygon with 17 sides, which I built with geogebra with 10 exact decimals.”

I took a brief walk through this book, which completes my work in recent years, here in Florida, where every day is sunny. Some of my writings related to Truth and Beauty can be found in Writings, 600 pages-Amazon 2023. I believe that I have seen some glimpses of the logos proportion and thus complete the thoughts about the Logos-Word of John 1.1, when the Person of the Lord became incarnate and walked on earth.

Pavy Beloiu, Florida, January 26, 2026

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About

• Connections between biblical measures and mathematical ratios
• The golden ratio in Greek statues and nature
• Logarithmic spirals and their spiritual meaning
• The number e and exponential growth
• Hyperbolic functions and the catenary equation
• Historical journey from Fibonacci to Gauss
• Complex plane and Euler’s equation
• Construction of the 17-sided regular polygon
• Harmonies of the human body and portrait proportions

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Book Details

• Title: SPIRA MIRABILIS-from ratios to complex plane
• Author: Pavy Beloiu
• Publication Date: January 26, 2026
• Pages: 270
• Languages: English (EN)
• Formats: PDF

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INDEX (EN)

Chapter	Page
Foreword	5
Ratios	7
Ratios-fractions in the Bible	10
Time	11
Space	12
Walking and the axis of time	15
Ratio on axis	18
Medium and extreme ratio - golden number	18
Construction of irrational numbers	23
Some properties related to famous numbers	24
Luca Pacioli	26
Leonardo da Vinci	27
Albrecht Dürer	28
The golden rectangle and the face proportions	30
The golden rectangle and the ear proportions	35
Harmonic analysis of Alexandru Donici’s painting	36
The number 5, the pentagon and proportions	35
History of the Pentagon	42
Construcția pentagonului	44
The Star Pentagon	48
Geometry, harmony and occultism	50
Pitagora	53
Spira mirabilis	57
Spiral equation	66
Other spirals	70
Spiritual interpretation	74
Bernoulli family	81
Bach family	89
Area of the sector in the complex plane - Jakob Bernoulli 1702	91
(blank)	92
Catenary chain equation	93
The connection between the chain equation and the golden ratio	100
Hyperbolic functions	102
Ricatti family	104
The logarithmic spiral in nature	106
Good-evil dualism	110
Spira mirabilis and human nature	111
Sound and the logarithmic scale	113
Spira mirabilis in the 3D equation	114
Descartes and the Cartesian plane	121
Efeseni 3.17-18	133
The derivative and the integral	135
Colin Maclaurin	146
Brook Taylor	150
Spira mirabilis and the complex plane	159
Niccolo Tartaglia and Rafael Bombelli	167
Gottfried Wilhelm Leibniz	168
About imaginary numbers	172
The likeness, or divine logos	174
EBEN EZER and the law of mathematical induction	185
The shadow and its universe	193
Shadow and equilateral hyperbola	200
Spira mirabilis and stella maris	204
The regular polygon with 17 sides - 17-gon	213
Construction with ruler and compass	221
Carl Friedrich Gauss	229
The 17-sided regular polygon easier construction	235
Thoughts on the number e	246
About interest	258
The life of Leonhard Euler	262
Bibliography	270