Hi, Donna!
Polygonal rifling is not new. It was used in muzzle-loading weapons for centuries before other types of rifling were standardized. The advantages then were similar to what they are now: smoother grooves with minimum of sharp eges to conceal fouling and corrosion, easier fabrication, and cleaning (not a factor after modern smokeles power,non-corrosive primers, and jacketed bullets became standard). The old Metford (English) rifling is a polygonal form, and so was the Japanese Arisaka. They have grooves that are partial segments of a circle, rather than the pie-shape or trapazoidal shape of Enfield rifling (most commonly used now). Metford rifling looks worn-out, even when new, because of the absence of sharp, defined lands and grooves.
In recently manufactured firearms, polygonal (cold-formed or hammer-forged, but not broached or cut-rifled) is used by H&K of Germany and Glock of Austria. Their stated advantages were that bullet deformation was less, resulting accuracy better, and with greatly extended barrel life. This was considered important for automatic fire in military small arms. The H&K Model 9 and the earliest Glock pistols were of this style.
Advantages to the manufacturer included lower per-unit costs, and rapid fabrication. The Germans/Austrians of course, pioneered rotary hammer-forging of barrels after WWII, and their machines are top rated.
Glock went to cut-rifling after numerous incidents of bore-leading incidents in their polygonal pistol barrels with resultant "KA-BOOM" blowups.
If I can find my barrel-making reference books, I will scan and send you copies of the articles. Please provide your emai.
HTH
John