Didicosm

As a child, her father shows Charlotte the night sky and wants her to realize the truth about the endless worlds and possibilities in the universe. In one of his books, he read about the idea of the universe repeating, but with changes occurring and later uses this thought to rationalize his own suicide. After her mother dies as well, Charlotte is brought to her grandmother and later wants to find the correct topology of the universe, which turns out to be a didicosm (Hantzsche–Wendt manifold). Her own student later comes up with a theoretical explanation involving quantum gravity, concluding this shape is indeed canonical due to being the only platycosm with a finite first homology group. Charlotte returns to her partner thinking that she lives in the best possible universe.

Construction of the Hantzsche–Wendt manifold by (direct or twisted) identification of the surfaces of a cuboid.

While the 3-torus (${\displaystyle T^{3}=\mathbb {R} ^{3}/\mathbb {Z} ^{3}}$), also one of the ten platycosms, can be depicted as space-filling repetition of the exact same cube with same orientation (hence a cube with respective opposite sides identified with same alignment), the didicosm can be depicted as a chessboard-like filling featuring cubes flipped and turned upside down. Both illustrations are featured in the short story.

The first homology of the didicosm is ${\displaystyle \mathbb {Z} _{4}\times \mathbb {Z} _{4}}$. (For the 3-torus it is ${\displaystyle \mathbb {Z} \times \mathbb {Z} \times \mathbb {Z} }$.) The derivation is explained by Greg Egan on his website,[1] which also lists four academic papers taken for the scientific basis of the short story: „Describing the platycosms“ by John Conway and Jean-Paul Rossetti,[2] „The Hantzsche-Wendt Manifold in Cosmic Topology“ by Ralf Aurich and Sven Lustig,[3] „On the coverings of the Hantzsche-Wendt Manifold“ by Grigory Chelnokov and Alexander Mednykh[4] as well as „How Surfaces Intersect in Space“ by J. Scott Carter.[5]

• Didicosm on the website of Greg Egan