In principle, one could synthesize polyethylene from cyclohexane.
There are two good reasons why this reaction cannot be performed.
The thermodynamic considerations for ring opening are illustrated by this graph of free energy (hypothetical) of polymerization from cycloalkanes with various ring sizes to polyethylene. (The "two membered ring" actually shows the data for ethylene, CH2=CH2.)
Three and four membered rings are quite strained, so ring opening to polyethylene is favorable if a suitable reaction pathway existed. (Unfortunately, no such reaction is known.) Six membered rings are stable, so they can rarely be polymerized by ring opening. Five and seven membered rings are borderline cases, so sometimes these can be polymerized. Larger rings have transanular steric interactions that cause some strain, so rings of 8 members or more can usually be polymerized. Of course, this graph changes when substitutions are made to the rings (for example, the presence of heteroatoms like O, S, and N, or the introduction of double bonds). However, the graph for the simplest possible rings illustrates the trends in other systems reasonably well.
Even if the ring opening reaction is thermodynamically favorable, there must be reaction chemistry to get there. In most cases, the initiators used for ring opening polymerization are polar or ionic species, so the monomers must have groups that can react. The most common monomers for ring opening polymerization contain some kind of heteroatom in the ring, as in the examples of cyclic ethers, esters, amines, amides, etc. At one extreme, there are examples such as the cyclic phosphazene shown that contain no carbon or hydrogen at all. At the other extreme, just a carbon-carbon double bond will suffice for ring opening metathesis polymerization, as illustrated by norbornene.
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