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In these pages you will find:

  • Meticulous geometric and mathematical analyses of the 5 Platonic Solids.
  • Meticulous analyses of other important polyhedra. The Rhombic Triacontahedron, for example, is nature's way of nesting the 5 Platonic Solids in a beautiful and elegant fashion.
  • An introduction to the PhiRatio, or Golden Mean, and the Fibonacci series.
  • A thorough scrutiny of the pentagon and the decagon. The pentagon is vitally important in the construction of the icosahedron, the dodecahedron, and other important polyhedra. Knowledge of the above three subjects is essential for those interested in sacred geometry. Otherwise, you are just farting around with stuff that you can't really understand!
  • Some geometric speculation. Note: These pages have not been updated, as have the others above.

I have compiled all of the papers in the first 4 sections into an eBbook, called "A Geometric Analysis of the Five Platonic Solids and Other Semi-Regular Polyhedra (With an Introduction to the Phi Ratio)." The book contains over 140 diagrams, and is aimed at students, teachers, and the general reader. The book also contains comprehensive lookup tables, which the papers do not.

I can assure you that there is not another textbook out there that combines clear, simple explanations and mathematical rigor. The only knowledge required is high school algebra and elementary trigonometry. In the book, I avoid trigonometry unless calculating central and surface angles. If you can understand the Pythagorean Theorem, you can understand almost everything in this book.

Special thanks to Rick Parris for his excellent free software WinGeom, which was used to create the drawings in these pages.

WinGeom is available at

Geometry is the most fundamental subject in the universe, for it describes the relationships within and between all things.

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Geometry Book Cover Image

Can geometry be exciting? It can for those who can appreciate the beauty of numbers and their relationships.

This book contains a meticulous geometric investigation of the 5 Platonic Solids and 5 other important polyhedra, as well as reference charts for each solid. These polyhedra are reflections of Nature herself, and a study of them provides insight into the way the world is structured. Nature is not only beautiful, but highly intelligent. As you explore the polyhedra in this book, this will become apparent over and over again.

Also contains a geometric explanation of the Phi Ratio and Fibonacci series, and a detailed analysis of the pentagon, which forms the basis for many of these solids. The pentagon is composed entirely of Phi relationships.

With over 140 illustrations, this book is perfect for teachers and students of geometry alike. It is a must for those who are serious about sacred geometry.

This book is dedicated to those who can appreciate the logic of numbers and the beauty of nature, for they are both aspects of the same unifying principle.

Note: Requires knowledge of algebra and elementary trigonometry.


"I'm studying to become a carpenter and earlier this year I got a challenge to make an icosahedron from 10cm by 10cm by 10cm cube. Well, it didn't work out at all since I was totally clueless about the angles etc. I just couldn't figure them out by myself.

"After reading your articles I think I understand platonic solids a little better now. Very interesting stuff!

"Here's a few photos I'd like to share with you since you helped me so much. Without you I couldn't have constructed this."

---Otso Silvo

Geometry Papers

Note: The files in the first three sections below have been updated for better readability. They have been designed to display in Internet Explorer, but work in almost all other browsers.

The equations in these papers have been created with Math Type 5.0. Every one of them is a .gif image. Although each image is very small, it may take a bit of time to download the document. You may "zoom in" on any equation and make it bigger by clicking on it.
These papers do not contain the reference charts, as does the book. Also, the links within the pages, to the other pages, do not work -- they are local links within the compiled book.

The Platonic Solids

Semi-Regular Polyhedra

The Division into Mean and Extreme Ratio (or, The Golden Section, or Phi), and the Pentagon

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Having fun with Geometry!

An interesting method of determining square roots:

The Geometric Determination of Square Roots

The Binary Circle/Sphere Pattern and the Binary Spiral:

Properties of The Binary Circle/Sphere Pattern