Do not bend the triple bonds! This familiar undergraduate mantra must be disobeyed if the alkyne group is used as a building block in molecular construction. This Account will describe our exploits in "alkyne origami", that is, folding oligoalkynes into new shapes via cyclization cascades. This research stems from a set of guidelines for the cyclizations of alkynes that we suggested in 2011 ( Gilmore Chem. Rev. 2011 , 111 , 6513 ; Alabugin J. Am. Chem. Soc. 2011 , 133 , 12608 ). The guidelines blended critical analysis of ∼40 years of experimental research with computations into the comprehensive predictions of the relative favorability of dig-cyclizations of anions and radicals. In this Account, we will show how this new understanding has been instrumental in building polyaromatics. In particular, we illustrate the utility of these stereoelectronic models by developing a toolbox of practical, selective, and efficient synthetic transformations. The high energy and high carbon content render alkynes the perfect precursors for the preparation of polyaromatic ribbons and other carbon-rich materials with precisely controlled structure and reactivity. Still, the paradox of alkyne reactivity (alkynes store a lot of energy but are protected kinetically by their relatively strong π-bonds) requires precise use of stereoelectronic factors for lowering the activation barriers for alkyne cyclizations. These factors are drastically different in the "all-exo" and the "all-endo" cyclization cascades of oligoynes. This Account will highlight the interplay between the stereoelectronics of bond formation and topology of acyclic precursor "folding" into a polycyclic ribbon. The topology of folding is simpler for the endo cascades, which are compatible with initiation either at the edge or at the center. In contrast, the exo cascades require precise folding of an oligoalkyne ribbon by starting the cascade exactly at the center of the chain. These differences define the key challenges in the design of these two types of alkyne cyclization cascades. For the endo processes, the folding is simple, but these processes require a strategy ("LUMO Umpolung") for inverting the usual stereoelectronic requirements of alkyne cyclizations. We also show how alkenes can be used as alkyne equivalents in cyclizations coupled with fragmentations and how one can make endo cyclization products without ever going through an endo cyclization. In contrast, each elementary step of the exo cascades benefits from the inherent exo preference for the radical attack, but these cascades require precise initiation by starting exactly at the central alkyne unit of the oligoyne. This strict selectivity requirement led to the development of traceless directing groups capable of supramolecular assistance to the initiation step and self-terminating departure at the end of the cascade. With attention to electronic effects that can stop radical cascades, oligoalkynes can be selectively converted into precisely shaped and functionalized polyaromatic products. The generality of these concepts is further illustrated by the development of radical "peri annulations" at the zigzag edge of acenes.