障子 and 畳: paper sliding doors and tatami mats
Compare 障子 receiving new paper or transmitting light with 畳 giving off a scent or serving as a sitting surface. Distinguish the mat name たたみ from the folding verb たたむ.
The paper sliding door named 障子
障子 (しょうじ) means a paper sliding door. 「破れた障子に新しい紙を張った。」 means “I covered the torn paper sliding door with new paper.” New paper was applied to the door; the sentence does not describe replacing the entire door.
「朝の光が白い障子を通して入ってきた。」 means “The morning sunlight came through the white paper door.” The white paper door is the path through which the sunlight entered.
The tatami mat named 畳
畳 (たたみ) means a tatami mat. 「和室に新しい畳の香りが広がっていた。」 means “The scent of new tatami mats filled the Japanese-style room.” The scent from new tatami mats filled the Japanese-style room.
「祖父は畳の上に座って新聞を読んでいた。」 means “My grandfather sat on the tatami and read the newspaper.” 畳の上 names the place where the grandfather sat and read.
An object name versus the folding verb
畳 here is the noun たたみ naming a tatami mat. Its reading and sentence role differ from 畳む (たたむ), the verb meaning to fold.
障子 names the paper door covered with new paper or transmitting light. 畳 names the mat giving off a scent or used as a sitting surface. Follow the named door and floor objects.
Can 畳 in 畳の上に座る be read as the folding verb 畳む?
Show the answer and why
No. 畳 here is たたみ, the name of a tatami mat and the surface someone sits on. The verb 畳む is read たたむ and means to fold.