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+#+TITLE: The 4 Ways to Build Things
+#+DATE: <2023-11-30 Thu 14:35>
+#+TAGS: Engineering, Construction, Philosophy
+
+Construction is the combination of stable primitive material objects into more complicated and interesting stable material objects. This should be uncontroversial enough. However, what is surprising is that there seems to be precisely 4 kinds of combination it is possible to use to combine them: blocking, sticking, bonding, joining, fastening, and resting. Furthermore, it is possible to define these concepts rigorously, and prove that they are the only possible means, subject to a relatively unobjectionable axiomatization of what "material object" means.
+
+* The Setup
+
+We assume that any given solid object can be effectively modeled by a collection of classical particles, each constrained to be no more than some specific distance $\epsilon$ from its neighbor (but possibly more constrained as well). This can be viewed as a graph or network, where the vertices represent particles and the edges are lines between each pair of constrained particles. These edges are, of course, of length less than $\epsilon$. Furthermore, we assume that these objects are solid, i.e. that the particles from two different objects are repulsive at short ranges and dense enough that the interiors of the objects never intersect. In order for "interior" to make sense, we also assume that the constraints modeled as such a graph has edges that form a polytope; the interior of this polytope is the interior of the object. This is sufficiently general to reproduce qualitative features of phenomena like flexibility and elasticity, but not general enough to reproduce phenomena like thermal expansion, fractures, or absorption. Last, we assume that it is only the constraints above and external fields that can affect the motion of the particles in any situation.
+
+TODO: diagrams.
+
+This is a reasonably good approximation of homogeneous molecules interacting via bonds that don't break, as atoms are particle-like on the scale of ordinary constructions and are attracted by forces that are short-range repulsive, medium-range attractive, and long-range made irrelevant by kinetic energy (TODO: Lennard-Jones potential diagram).
+
+Given any such object moving under a free Lagrangian and its internal constraints (a system $L$), we want to investigate and classify the resulting systems $L'$ that result from making modifications to this scenario. Specifically, we claim that all $L'$ that constitute systems of this type (particle systems constrained by "bonds" and "solidity") can be realized by some combination of the following 5 particularly natural operations applied to $L$ (restricted):
+
+1. adding particles,
+2. removing particles,
+3. adding inter-particle constraints,
+4. removing inter-particle constraints, and
+5. introducing an external field that affects all the particles.
+
+Additionally, we can define a "distance" between two objects by the minimum number of operation applications it is necessary to perform on one of them so that some accessible state of the result is an accessible state of the other object. This raises interesting algorithmic questions about the easiest way to realize a particular object.
+
+We are also interested whether operations that more closely imitate practically possible construction methods share the universality property: it's rare that it's possible to add or remove particles or constraints in the interior of an object, so if the operations are restricted to particles on the surface of the object, can we still build everything? In a similarly practical vein, we explore the transformations thought of as the combination of separate objects into one or the separation of one object into multiple, and transformations that decrease or increase the number of degrees of freedom
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+Since all (material) constructions consist of finitely many combinations, it suffices to restrict attention to a single combination of two such objects. To make "combination" more precise, we consider (WLOG) the first object having a free Lagrangian, save for its internal constraints, and then consider augmentations of this initial system that maintain the first object
+
+we meanperforming some sort of alteration to the case of free objects far away, such as bring them close together to touch, altering the internal geometry of the objects, or adding an external field into consideration, that reduces the number of degrees of freedom of the objects.
+
+By "sticking," we mean the introduction of new constraints of the same flavor between the exterior particles of two objects, without introducing additional particles.
+
+By "bonding," we mean the introduction of new constraints of the same flavor between the exterior particles of two objects, via the inclusion of additional mediator particles.
+
+By "joining," we mean using the geometry of two objects themselves to interlock them together, using their solidity to constrain them without the introduction of any third object.
+
+By "fastening," we mean introducing a third object whose geometry interlocks the two objects together using its solidity.
+
+By "resting," we mean exploiting an external field to hold the objects against each other.
+
+* The Exhaustiveness Argument.
+
+In the general case, the two objects are far away from each other. They move completely freely, at least locally; this preclueds any possibility of their interiors intersecting or of their particles being connected by chains of constitutive constraints.
+
+To impose additional restrictions, the objects must be close (in the sense that it is possible for infinitesimal motions of the particles consistent with the internal constraints to violate some of the constraints between the two bodies, i.e. either to break a relational constraint chain or to make their interiors intersect).
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+This happens in four cases: either there are additional constraints between the particles of the objects or there are not, or there are additional particles introduced or there are not.
+
+By assumption, the only things that affect the motion of the particles are their internal constraints, their solidity, and external fields.
+
+* Practical Examples
+
+- Wood
+- Metal
+- Rope
+- Fabric
+- Glass
+- Plastic
+- Ceramics