This work explores the intimate relationship between two of the most fundamental geometric figures, the regular tetrahedron and the regular hexahedron. Providing a comprehensive review of the literature on these figures along with his own original research, Woodmansee presents new proofs and solutions to classic problems in geometry. This book will appeal to mathematicians, researchers, and students interested in the rich history and vibrant contemporary research in the field of geometry. This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work is in the "public domain in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant.