Ariel felt quite pleased that she had been able to convey to Lennie the wonders of two dimensional shapes. And this got her thinking — maybe the same techniques could be applied to more dimensions.

So Ariel took a plane to see her friend Trey, who lives in the third dimensions. She had read in school that there are all sorts of hypothetical shapes in a three dimensional world that cannot exist in the normal two dimensions that she’d lived in all her life. Maybe, she thought, the right visualization technique could help her understand these fabulous shapes.

Trey was very happy to see her. They had first met in his younger days, when he had kept a flat just outside of Plano Texas. He still had fond memories of those times, which despite appearances went very deep.

When Ariel posed the problem to him, Trey was intrigued. How could he combine her visualization techniques with his knowledge of the third dimension, to explain higher dimensional shapes to Ariel?

“Let me think about it overnight,” he said, “and by tomorrow I should have something for you.”

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