Building Archimedean Solids and Platonic Solids: the easy way

Let the lights of the Cosmos generate the geometry of the solids, said Plato

There have been thousands of mathematical pages written on the Archimedean solids and the Platonic solids. However, there has not really been any published work on how to build the Archimedean solids and Platonic solids. This book is the most comprehensive work ever published on how to construct the Archimedean solids and the Platonic solids.

The field of polyhedral geometry, a foundational branch of mathematics, relies on a precise set of terminology to describe the spatial relationships and properties of three-dimensional forms.

Unfortunately, the foundational branch of mathematics for polyhedral geometry does not include the terminology of carpenters who, for centuries, built the Archimedean solids without the aid of mathematics.

Whether you’re building hollow solids or skeletal solids, this book includes miter angles, blade tilt angles, diagrams, and templates to help you make these solids either as hollow solids or as skeletal solids.

This book utilizes geometry to draw out the vertex for each polyhedron, thereby obtaining the dihedral angle and edge bevel angles. No math is required. 

This book is a PDF file containing over 40 table saw sled templates for printing, as well as 350 pages of detailed techniques. Taking the guesswork out of building the solids. 

Let no one ignorant of Geometry enter, said Plato

da Vinic's Polyhedra Φ Project

Golden Rectangle Rhombicosidodecahedron

Φ Rhombicosidodecahedron

Archimedean Solid Fish Tanks

Archimedean Solid Solar Lights with Light Refraction

Roman Dodecahedrons

Platonic Solids as Hollow Solids

Platonic Solids as Skeletal Solids

Archimedean Solids as Skeletal Solids with easy-to-follow table saw sleds

Wooden Polyhedra: Precision Construction Techniques

Mastering Mathematical Beauty Through Advanced Table Saw Work

Introduction

The construction of polyhedra in wood is a testament to the union of mathematical elegance and practical craftsmanship. These intricate forms, defined by their precise geometric properties, present a unique challenge to woodworkers, requiring not only a deep understanding of geometry but also mastery of the tools and techniques that enable flawless execution. This book is designed to guide both novice and advanced woodworkers through the demanding process of building wooden polyhedra, focusing on the meticulous setting of miter and bevel angles on a table saw, the use of specialized jigs, and the essential skills needed for perfect assembly.

Understanding Polyhedral Geometry

Polyhedra are three-dimensional shapes bounded by flat polygonal faces, with the regular vertices and irregular vertices, including the Platonic solids (such as cubes, tetrahedra, and dodecahedra) and Archimedean solids. Each face must meet its neighbors at precise angles, and every edge and vertex must be carefully accounted for in the design and construction process. Before beginning, familiarize yourself with the geometry of your chosen polyhedron—study its faces, edges, and required angles, as this will inform every step of your woodworking plan.

Translating Geometry to Woodworking

The leap from geometric diagram to physical wood requires accurate calculation of both miter angles (the angle at which two pieces meet) and bevel angles (the tilt of the saw blade to match the polyhedral faces). For example, constructing a dodecahedron involves cutting twelve identical pentagonal faces, each meeting at precise dihedral angles. Calculating these angles is crucial; this book includes all the necessary mathematical references and woodworking calculations to determine the exact settings for your table saw.

Utilizing Table Saw Sleds Templates

Table saw sleds are indispensable for polyhedral construction. A well-designed sled stabilizes your workpiece and maintains consistent angles for repeated cuts. For complex polyhedra, the templates included in this book required toggle clamps to hold pieces at unusual angles. Build these table saw sleds from stable materials and verify their accuracy before use.

Techniques for Flawless Assembly

Achieving a seamless fit across the many joints of a polyhedron is the ultimate test of woodworking precision. Here are essential assembly techniques:

⦁ Dry Assembly: Before gluing, assemble all pieces to ensure a proper fit. Minor adjustments can be made with sanding blocks or shooting boards.

⦁ Clamping Strategies: Use band clamps, rubber bands, or custom jigs to hold pieces together without distorting the angles.

⦁ Glue Selection: Choose a wood glue with a reasonable open time to allow for careful positioning. I use Titebond III

⦁ Finishing Touches: After assembly, sand lightly to remove any small imperfections, being careful not to alter the angles at the edges of the polygon faces.

Challenges and Mastery

Polyhedral woodworking is not for the faint of heart. Even a deviation of a few degrees can result in visible gaps that compromise both the structural and aesthetic integrity of the final piece. This book serves as both a challenge and a benchmark for advanced woodworking skills, pushing the limits of a table saw’s accuracy and the craftsman’s attention to detail. Mastery is signified not only by the completion of the polyhedral but by the pursuit of perfection in every joint and surface.

Final Words on this book

Constructing wooden polyhedra is a journey through mathematical art and technical mastery. By following the guidance in this book—calculating angles precisely, mastering saw setups, and executing flawless assembly—you will unlock new levels of woodworking skill and create captivating pieces that celebrate both geometry and craftsmanship.