You can think of the quad graph as being a combination of the original planar graph with its planar dual.
Now, a (spherical) polyhedron is self-dual if and only if it’s isomorphic to its dual, or equivalently iff there is a part-reversing automorphism of its quad graph.
So that’s what I mean when I say that self-duality is akin to symmetry. Precisely, it is a symmetry of the quad graph. Since our original polyhedra(l graphs) have no symmetry, the only symmetries of the quad graph there could possibly be are part-reversing ones, i.e. dualities.
robinhouston
npub1wj…7nda9
2026-01-20 19:00:37 UTC
in reply to nevent1q…pnt3
Author Public Key
npub1wjc42kn5sa2pek0h5gfygwrs2s6r7cyr7l2mlpna529pkff97jzs97nda9Published at
2026-01-20 19:00:37 UTCEvent JSON
{
"id": "b577fe2fdecb3ad9922cdf27740425a4fdbd26b7a33a1922987dcf9822c86a3f",
"pubkey": "74b1555a7487541cd9f7a21244387054343f6083f7d5bf867da28a1b2525f485",
"created_at": 1768935637,
"kind": 1,
"tags": [
[
"e",
"bbe60e9d48ad23f3a280d5f34c393152f3b734bd5f86d52aec29df28a5ee51fc",
"wss://relay.ditto.pub",
"reply"
],
[
"proxy",
"https://mathstodon.xyz/users/robinhouston/statuses/115928965937771627",
"activitypub"
],
[
"client",
"Mostr",
"31990:6be38f8c63df7dbf84db7ec4a6e6fbbd8d19dca3b980efad18585c46f04b26f9:mostr",
"wss://relay.ditto.pub"
]
],
"content": "You can think of the quad graph as being a combination of the original planar graph with its planar dual.\n\nNow, a (spherical) polyhedron is self-dual if and only if it’s isomorphic to its dual, or equivalently iff there is a part-reversing automorphism of its quad graph.\n\nSo that’s what I mean when I say that self-duality is akin to symmetry. Precisely, it is a symmetry of the quad graph. Since our original polyhedra(l graphs) have no symmetry, the only symmetries of the quad graph there could possibly be are part-reversing ones, i.e. dualities.",
"sig": "8e68a33d49a0f1e7b48ed51b28cbde239f53a5021bd537459bc2ee15d5a62b43c39c8f788005bdec5f60543f8c7f0525e3b7fec02e614aa84d1a0594c067184e"
}