What is a manifold?

Etymology and Naming Confusion

• Several comments note that “manifold” in math and car engines share an Old English/Germanic root (“many-fold”), which can mislead learners who anchor on familiar non-math meanings.
• Some people find etymology helpful for intuition; others say names can be misleading or arbitrary chains of historical choices.

Reception of the Article and Quanta

• Many praise the article as an accessible history-and-concepts piece rather than a dry definition, highlighting its diagrams and storytelling.
• Quanta is widely lauded for non-clickbait, technically serious science writing enabled by philanthropic funding and lack of ads/paywalls.
• A minority find the explanation average or flawed, wanting more on atlases/overlaps and sharper distinctions (e.g., topological vs Riemannian manifolds).

What a Manifold Is (Informal Intuitions and Nuances)

• Intuitive descriptions: “locally looks flat like ℝⁿ,” “you can put a small disc (open set) around any point,” or “double pendulum configuration space is a torus, not a square.”
• Clarifications: a sphere needs multiple charts; global latitude–longitude coordinates have discontinuities, motivating the atlas concept.
• Discussion of spacetime: commenters note that GR spacetime is a 4D pseudo-Riemannian manifold; Minkowski spacetime is the flat special-relativity case.

Pedagogy: Coordinates, Tensors, and Abstraction

• Long subthread on how physicists define tensors via transformation rules vs mathematicians’ coordinate-free multilinear map definition.
• Some argue transformation-based teaching is confusing or circular; others say it’s pragmatic and that the shorthand hides a precise rule.
• Several reminisce about learning relativity/manifolds: better understanding came with more abstract, geometric treatments rather than coordinate-heavy ones.

Applications and Related Concepts

• Brief mentions of Calabi–Yau manifolds (Ricci-flat, used in string theory), with an explanation of Ricci curvature as “volume change” and note that there is no experimental confirmation.
• Discussion of “data manifolds” in ML: often treated as an approximate manifold-plus-noise hypothesis; in practice, ReLU networks break smoothness, but intrinsic low-dimensional structure can still be useful.
• A side question on why cartography rarely uses manifold language: answers cite that manifolds are overkill for simple sphere projections and that cartographic practice predates modern manifold theory.