Designing a logo with symmetry

Maths and Designing project
This was a class not interested in any ideas that I brought in the class. This project was a turning point. This is what I asked them:
Design a logo with one line of symmetry. You can choose to design: On your life OR A fictional company. Explain why is that a good logo. Some of the samples I received:

Logo for a shoe maker

Logo on life

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