Conclusion In your trek through Knowhere you have seen how the structure of space orders things around.  It facilitates individual expression of form on the local level while maintaining overall uniformity of composition on the extended level. . On the one hand, as you saw in Geometry rules, spatial ordering principles permit things to have complementary mirror planes and axes of symmetry, similar surface and volumetric relationships, congruent structural elements, and dual frameworks, to name a few. This allows structures to be transformed into each other via truncation, sectioning and rearrangement, compounding, and stellation operations.  Also structures can be inscribed in each other and they can be packed together to fill space.  All of these attributes are conducive to creating a rich mix of diverse forms that can inhabit space. . On the other hand, the structure of space limits diversity of form by invoking certain other ordering principles that things must abide.   As you probably already know from geometry, two-dimensional polygons and three-dimensional polyhedra must maintain a topological balance between their vertices, edges, and faces.  And, as you saw in Building stability, for polygons, polyhedra, lattices, and plate structures to be inherently stable they must adhere to additional topological constraints.  Another purely geometrical limitation is  that the sum of the interior or dihedral angles of polygons and polyhedra must be constant. And the sum of the angles about a point must equal 360o, for example.  In structural terms, the bearing capacity of beams and columns is limited by spatial considerations such as their moment of inertia and their length.  And the strength of trusses and spaceframes depends significantly on the ratio of their height, or depth, to their span.  All of these factors limit the possible forms that things can take. . Taken as a whole, these abstract ordering principles provide the structural matrix within which the dynamic interaction of physical forces are played out in the real world.  In Structure matters, you saw how the crystal lattice structure accommodates the attractive and repulsive forces of atoms of different size, ligancy, and charge so that they can pack together uniformly into elements and minerals.  Building stability demonstrated that the omni-triangulated two-dimensional truss structure permits the internal forces induced by external loads to be exactly and intimately balanced by aligning them axially.  Likewise, the three-dimensional spaceframe framework can bear a load perpendicular to its plane by dissipating the resulting forces axially among its members by means of lattice action.  The stability of a plate structure is due to the fact that its plates can dissipate internal forces across the entire surface of the plate, and concentrate them at its edges, as counter acting shear stresses.  A lattice tower can dissipate its load throughout its omni-triangulated framework.  And, Gizmoneering showed that machines can be engineered with different combinations of rigid and moving parts to efficiently convert and transmit externally applied forces into mechanical motion that can do work. . All of these structural systems owe their strength, stability, and efficiency in large part to the structural properties of space.  Without these ordering principles existence might well be relegated to either a diverse sand pile of non-uniform particles at one extreme, or a single, uniform entity at the other.  Instead, because of these principles, we enjoy  a great diversity of materials and structures that contribute substantially to our welfare and security.   Indeed space is something else! .
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