The distinctive insignia which represents the 36th Infantry Division is called the “T-Patch”. It was first approved by the military on November 12, 1918 but not adopted by the 36th until January 16, 1919.
Here is the original drawing submitted to the National Archives
The original T-Patch was very irregular as noted above, there was not a standard design. The museum has many original WWI era T-Patches in the collection and they show the variety of designs being used by the 36th Infantry Division during 1919 and the early 1920’s The design began to be standardized between WWI and WWII in the 1920’s and 1930’s. As can be seen in the last 2 pictures at the bottom of this post.
The T-Patch on the right of the picture above is the one which would be used from 1940 through 1968 when the 36th Infantry Division was replaced with the 49th Armored Division. When the 36th was reflagged in 2004 the T-Patch returned with that same design, but with different versions, such as a desert tan one, and a black, subdued version.