. ▁A ▁singular ity ▁can ▁be ▁made ▁by ▁ball ing ▁it ▁up , ▁dro pping ▁it ▁on ▁the ▁floor , ▁and ▁fl atten ing ▁it . ▁In ▁some ▁places ▁the ▁flat ▁string ▁will ▁cross ▁itself ▁in ▁an ▁approximate ▁" X " ▁shape . ▁The ▁points ▁on ▁the ▁floor ▁where ▁it ▁does ▁this ▁are ▁one ▁kind ▁of ▁singular ity , ▁the ▁double ▁point : ▁one ▁bit ▁of ▁the ▁floor ▁corresponds ▁to ▁more ▁than ▁one ▁bit ▁of ▁string . ▁Perhaps ▁the ▁string ▁will ▁also ▁touch ▁itself ▁without ▁crossing , ▁like ▁an ▁under lined ▁" U ". ▁This ▁is ▁another ▁kind ▁of ▁singular ity . ▁Un like ▁the ▁double ▁point , ▁it ▁is ▁not ▁stable , ▁in ▁the ▁sense ▁that ▁a ▁small ▁push ▁will ▁lift ▁the ▁bottom ▁of ▁the ▁" U " ▁away ▁from ▁the ▁" underline ". ▁ ▁Vladimir ▁Arnold ▁defines ▁the ▁main ▁goal ▁of ▁singular ity ▁theory ▁as ▁describing ▁how ▁objects ▁depend ▁on ▁parameters , ▁particularly ▁in ▁cases ▁where ▁the ▁properties ▁under go ▁sudden ▁change ▁under ▁a ▁small ▁variation ▁of ▁the ▁parameters . ▁These ▁situations ▁are ▁called ▁per estro ika ▁( ), ▁b if ur c ations ▁or ▁cat ast roph es . ▁Class ifying ▁the ▁types ▁of ▁changes ▁and ▁character izing ▁sets ▁of ▁parameters ▁which ▁give ▁rise ▁to ▁these ▁changes ▁are ▁some ▁of ▁the ▁main ▁mathematical ▁goals . ▁Sing ular ities ▁can ▁occur ▁in ▁a ▁wide ▁range ▁of ▁mathematical ▁objects , ▁from ▁matrices ▁depending ▁on ▁parameters ▁to ▁wave front s . ▁ ▁How ▁singular ities ▁may ▁arise ▁In ▁singular ity ▁theory ▁the ▁general