Polyhedral Underwater Structures
Authors: Morgan George W. 
Published: October 2008

ABSTRACT
Polyhedral designs with faceted surfaces for underwater vessels, provide a wide range of structural and design possibilities. Faceted surfaces may range from thick slabs of glass to membrane skins stretched over a structural skeleton framework. They permit increased effective shell thickness while minimizing requirement for exact sphericity during the fabrication phase. Potential applications include: underwater oil storage tanks. Instrumentation and observation housings, manned stations and habitat equipment housing, and similar structures.
A feasibility study was conducted at North American Rockwell Corporation to provide basic design information relating to polyhedra in general. These data are presented in nondimensional terms and relate all polyhedral dimensions to a unit radius of the circumscribing sphere. The vertices of all polyhedra investigated are tabulated in both spherical and orthogonal coordinates for immediate design use. The polyhedra investigated include the: tetrahedra, octahedra, dodecahedra, icosahedra, icosadodecahedra truncated tetrahedra, truncated octahedra, truncated dodecahedra, truncated icosahedra, and others.
The data includes: areas of polygon facets, areas of associated spherical surfaces, lengths of chords, numbers of chord members and vertices, and related information.
The basic mathematical properties of polyhedra are included with detailed data on six representative polyhedra. Simplified mathematical solutions are presented. The computer approach to the spatial coordinates is discussed. In addition, transformation equations permit the expression of polyhedral data in terms of surfaces-of-revolution other than the sphere. These include oblate and prolate spheroids, ellipsoids and spheroellipsoids.

INTRODUCTION
Control of the ocean environment is the first requirement in the development of ocean technologies for scientific exploration, offshore oil production and storage, mining, aquaculture or other ocean development. When ocean technology involves either man or equipment, structures can provide either partial or complete isolation from the environment. An example of partial isolation is a pressure-equalized structure which permits man to excape the water but not the pressure. More complete isolation is provided by structures which maintain atmospheric conditions at one-atmosphere pressure, even when submerged to great depths.
The most efficient small structure for maintaining one-atmosphere internal pressure while submerged to significant depths, is the sphere. However, spherical shells developed for larger enclosed volumes atone-atmosphere, are limited by current fabrication and assembly techniques. Alternate structural methods which may by-pass these limitations must be developed. Alternate construction configurations are also economically desirable in structures enclosing large volumes at or near ambient pressure (internal and external pressures approximately equal at operating depths).
This paper investigates alternate structures that combine the advantages of the strength of spheres with widely used materials and fabrication techniques. This paper is Part 1 of the following series:Part 1. Geometrical Properties of Polyhedral VesselsPart 2. Structural Analysis of Polyhedral VesselsPart 3. Optimization of Polyhedral Vessels.
This paper defines the geometrical properties of selected regular and semi-regular polyhedra. Polyhedra of this general type are circumscribed by a spherical surface which contains the vertices of all polygonal faces comprising the surface of the polyhedron. Structures of this general type may be geometrically designed directly from the dimensional data tabulated herein.