One of the best written mentions of these patches is found in the book "Let's Go! - The Story of A.S. no. 2448602" by Louis Ranlett.
The author began service with the 77th Division, got commissioned, and then was assigned to the 23rd Infantry, 2nd Division.
It's an excellent read. He had a historian's eye for detail.
An excerpt....from pgs. 233-234
"The G.A.R. men who used to speak to children in grammar school before Memorial Day every year interested me most when they described their wartime experiences in detail. I wanted to know just what they wore and carried and ate, as well as why they fought. That is my warrant for detailing what I took with me into battle.
It's nice to think that if you're killed people will know who you were. I had a privately made silver identification tag on my left wrist along with my wrist watch; about my neck on the very piece of tape on which they hung when I left Camp Upton, were my two issue identification tags of aluminum. By carving alterations with my pocket knife I had kept them up to date. One side read '2448602 -- 2nd Lt.'; the other 'Louis F. Ranlett -- [the following all crossed out ---->] Pvt - Co B - Corp - Sgt. 308 Inf'.
I wore a money-belt that contained my letter of credit, heavy woolen underwear, hand-knitted woolen socks, an issue woolen shirt, the heavy woolen sleeveless sweater that my mother had knitted, my old enlisted man's issue woolen uniform, wrap leggins, field boots. Except for my gold shoulder bars there was no distinguishing my uniform from that of a private. On my left shoulder was sewed the blue diamond-shaped piece of cloth that distinguished every member of the third battalion of the 23rd. The men of the first battalion word blue squares; the men of the second, blue triangles. The battalions of the other regiments used the same system with cloths of different color. Division insignia, that later became such a colorful part of the uniforms of the A.E.F., had not yet been thought of.
Half the soldiers of the Civil War were saved from death by Bibles that turned bullets from their hearts. Every one knows that. Of course I carried my khaki covered Testament, but not in my left breast pocket. My steel mirror was there. In other pockets were my small silk flag, my notebook with a list of names of the men in the platoon, a box of matches, my money, a knife, an indelible pencil. My whistle dangled from my breast-pocket; the satchel of my gas mask was strapped at the alert beneath my chin. My pistol belt carried my canteen, the automatic pistol, four extra loaded magazines, and my first-aid packet. My raincoat hung awkwardly from it at one side. My pack held nothing but my mess-kit, two cans of beans, and two cans of meat. My steel helmet topped off."
From page 244....
"Then I looked at his face. It was green. Through his forehead was a bullet hole! There he knealt, dead, with his eyes open and his hand on his rifle that lay on the low parapet in front of him.
It was not a maneuver after all!
Daly leaned over and took the entrenching shovel from the man's pack. He had none of his own. The act was a wise one. I looked off to the left and saw five other immovable figures. On their shoulders were insignia of the 9th Infantry."
This book is worth having on the shelf...it's one of my favorite Doughboy memoirs....but if it is also been digitized now...and is available at the link below.