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When I wrote about this idea with the parallel postulate, I imagined it as a stubborn wrinkle in a sheet.

Geometry classes could explore the strange new worlds that arise from adopting different versions of Euclid’s infamous parallel postulate.

In February, I wrote about Euclid’s parallel postulate, the black sheep of the big, happy family of definitions, postulates, and axioms that make up the foundations of Euclidean geometry.

But what stood out to me the most was that while Carroll spent plenty of time on the parallel postulate, he made no mention in his book of non-Euclidean geometry. It’s not that the parallel postulate was not discussed; several sections include extensive explorations of the topic, and he included two related tables in his text.

Here we have statements that are equivalent to the parallel postulate.