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, 
twodimensional polygons and threedimensional 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 360^{o}, 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 
omnitriangulated twodimensional truss structure permits the internal
forces induced by 
external loads to be exactly and intimately balanced by aligning them
axially. Likewise, 
the threedimensional 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 omnitriangulated 
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 nonuniform 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! 
. 