History and Geometry of Regular Polyhedra

📕 History and Geometry of Regular Polyhedra

✍️ Introduction

It is a journey into the history of regular polyhedra, from their beginnings to the complicated calculations that led to their inclusion in the same sphere.

🔍 Highlights

How I wrote the geometry of polyhedra

Polyhedra with their sections and construction fascinated me since high school, when I was learning to become an architect. I knew the pentagon and the golden section. I found them in a volume of geometry for architects, in the library of Dinicu Golescu High School in Câmpulung, my hometown. It was dusty and hidden among the shelves, no one asked for it. I was amazed when I saw how the pentagon helped in the construction of Plato's solids, the famous dodecahedron and icosahedron, which fit into the same cube.

I also built them according to Adrian Gheorghiu's thick book, which architects studied in their early years. I would rotate them on all sides, project them and make sections - I had already learned Descriptive Geometry, following Mihai Botez's course and I would play with complicated projections and sections.

I started from the pentagon, found it in my tropical plants in California, then described a brief history of the number 5, and showed it in the Bible and in the pyramid of Cheops. From 5, I get to the pentagon then to the golden ratio.

I move on to history and present some Renaissance masters, who also showed Plato's solids and drew them with shadows and lights. From Pacioli and Leonardo, I get to Dürer and La Francesca, then to Kepler, who gave us the laws of planetary motion. All of them were penetrated by the mysteries that hide behind regular polyhedra.

Kepler went too far and believed that the Universe respected the golden ratio - he referred to the distances between the planets...

The calculations are coming into space and the construction of the 2 older brothers, which are inscribed in the same cube.

I go further and use geogebra for constructions and end up fulfilling my dream, to find the common sphere for all 5!

I went step by step, starting from Euclid, then I saw what sphere encompasses the dodecahedron and the icosahedron.

The other 3, the tetrahedron, the octahedron and the cube quickly and easily fit into the sphere found with geogebra. The last 3 are taught in schools and I did not stop at their properties and construction. The cube encompasses them all and I also stopped at the 3 dimensions, the Bible and the 3D space, introduced by Descartes.

Finally, I did not forget Matila Ghyka, who was and remains famous throughout the world with his books on aesthetics, harmony and the golden ratio.

I hope that readers passionate about geometry will continue my search. I united art with geometry, then with history and I found more in the Bible, in the construction of the tabernacle built by Moses, then in the Temple of Solomon.

📖 Book Details

• Title: History and Geometry of Regular Polyhedra
• Author: Pavy Beloiu
• Publication Date: July 17, 2025
• Pages: 119
• ISBN: 9798292843344
• Formats: Paperback, PDF

🧭 From the Foreword

Foreword

The geometry and polyhedra inscribed in the sphere fascinated me since primary school.

I started at school with the triangle, then the cube and the tetrahedron. Later I discovered the pentagon and the golden section, which go hand in hand. I was amazed by the beauty of constructions with a ruler and compass, and I read how the Greeks knew everything. From geometry in school, I reached the harmony and secrets of proportions, the mean and extreme ratio, then I went further, when I was preparing to become an architect.

Thus, I found in the high school library, The Geometry of Polyhedra, a textbook written by the famous Adrian Gheorghiu, who prepared mathematics for first-year architects. Here I found the stars of polyhedra, the icosahedron and dodecahedron, which start from the cube and are inscribed in the sphere. I constructed them with passion, as well as the conics, which I cut from plastic, like a pane of glass, and polished them to use in my calculation notebooks.

In this modest work, I bring some aspirations that I have carried with me since high school, namely to present Plato's polyhedra and to inscribe them all in the same sphere. I succeeded a few years ago, using Geogebra, which transposed my calculations and thoughts onto paper.

I started with the number 5, the pentagon, and went directly to the elder brothers, the icosahedron and the dodecahedron. I did not deal with the small ones, the tetrahedron and the octahedron, which are taught in schools. Likewise, the cube, I briefly mentioned and used it for spatial constructions. This mysterious cube contains them all and in it are found the 5 solids, discovered by the Greeks and continued in the Renaissance by famous artists and scholars. Leonardo, Dürer, and Kepler are a few daring individuals who delighted in the beauty and mysteries hidden in these polyhedra. Later, Euler and Gauss studied them and formulated theorems.

I also mentioned Matila Ghyka, our famous Romanian, who spread our fame across all continents.

I cannot forget the connection between the number 5 and the Bible, and where I had the opportunity, I also found the deeper, spiritual meaning that lies behind these calculations and geometric forms. These are bold thoughts, and from the different rays that start from my searches, towards beauty and truth, I leave it to the reader to search further, and I hope they reach the same divine image, described in the Scriptures, which surpasses our vision of the absolute and perfection, the Lord Jesus, the Son of God.

You are the most beautiful of men (Psalm 45.2)

I am the Way, the Truth, and the Life (John 14.7)

Pavy Beloiu, Florida, June 2025

🗂️ Selected Topics

👉 For the full chapter list, view the complete index below.

About the book

🧾 INDEX

Preface............................................................................................5
Number 5, the pentagon and proportions......................................6
History of the Pentagon..................................................................8
Construction of the pentagon.......................................................12
Geometry, harmony and occultism...............................................19
Number 5 in the Great Pyramid-Keops........................................25
History of of polyhedra.......................................................….......26
Leonardo and Pacioli..........................................................…........28
Dürer-self-portraits.......................................................................35
Johannes Kepler................................................................….........41
Harmonice Mundi..............................................................….........49
Regular polyhedra-construction and calculations........................56
Dodecahedron...............................................................................57
Icosahedron...................................................................................65
How to inscribe them in the same sphere?...................................73
Euclid's constructions...................................................................83
Practical applications....................................................................97
Duality of the 5 regular polyhedra...............................................102
About the CUBE...........................................................................103
The Cube and the Bible...................................................…..........106
Matila Ghyka-history and destiny..................................…............109
Bibliography....................................................................…............119