In the show, the box and apple motifs reoccur throughout Mawaru Penguindrum.
What are the significance the two? Does it tie into the plot in any way?
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I haven't seen it myself, but apples can often symbolize sexuality, temptation ("forbidden fruit") , fertility ("bearing fruit" as in your screenshot).
Connections can also be made to the stories of Adam & Eve and Sleeping Beauty.
The circular shape of an apple can also be a depiction of loops, or eternity - which is what it appears to be in Penguindrum:
It is revealed in the finale that the apple is a metaphor for the Penguindrum, which seems to be Kanba's life, shared between the three siblings