Self-Packing Allspace-Filling Geometries

950.10 There are a variety of self-packing allspace-filling geometries. Any one of them can be amplified upon in unlimited degree by high-frequency permitted aberrations. For instance, the cube can reoccur in high-frequency multiples with fundamental rectilinear aspectswith a cubical node on the positive face and a corresponding cubical void dimple on the negative facewhich will fill allspace simply because it is a complex of cubes.

954.10A Allspace-filling Hierarchy as Rationally Quantifiable in Whole Volume Units of A or B Modules (Revised)

+:---------------------+:---------+:----------------+:--------------------+
| Synergetics’ | Quanta | Type | Symmetrical or |
| | | | |
| | Module | | |
±---------------------±---------±----------------±--------------------+
| Name | Volume | Polyhedron | Asymmetrical |
±---------------------±---------±----------------±--------------------+
| Mite | | Tetrahedron | Asymmetrical |
±---------------------±---------±----------------±--------------------+
| Syte (3 types) | | Tetrahedron | Asymmetrical |
| | | | |
| Bite | | | |
±---------------------±---------±----------------±--------------------+
| Rite | | Tetrahedron | Asymmetrical |
±---------------------±---------±----------------±--------------------+
| Lite | | Hexahedron | Asymmetrical |
±---------------------±---------±----------------±--------------------+
| Coupler | | Octahedron | Asymmetrical |
±---------------------±---------±----------------±--------------------+
| Cube | | Hexahedron | Simple Symmetrical |
±---------------------±---------±----------------±--------------------+
| Rhombic dodecahedron | | Dodecahedron | Simple Symmetrical |
±---------------------±---------±----------------±--------------------+
| Tetrakaidecahedron | ,432 | Tetradecahedron | Complex Symmetrical |
±---------------------±---------±----------------±--------------------+

Table 943.00 Synergetics Quanta Module Hierarchy

Sun only; no satellites

^k Sun+12 partial sphere satellites

image image image image

950.12 There are eight familiar self-packing allspace-fillers:

1. The cube. (6 faces) Discoverer unknown.

2. The rhombic dodecahedron. (12 faces) Discoverer unknown. This allspace filler is the one that occurs most frequently in nature. Rhombic dodecahedron crystals are frequently found in the floor of mineral-rich deserts.

3. Lord Kelvin’s tetrakaidecahedron. (14 faces)

4. Keith Critchlow’s snub-cornered tetrahedron. (16 faces)

5. The truncated octahedron. (14 faces)

6. The trirectangular tetrahedron. (4 faces) Described by Coxeter, “Regular Polytopes,” p. 71. (See Illus. 950.12 B.)

7. The tetragonal disphenoid. (4 faces) Described by Coxeter, “Regular Polytopes,” p. 71. (See Illus. 950.12 C.)

8. The irregular tetrahedron (Mite). (4 faces) Discovered and described by Fuller. (See Illus. 950.12 A.)