polyhedron(다면체) vs manifold(다양체)

[석수]

다원체"는 문맥에 따라 영어로 다르게 번역될 수 있습니다.

일반적인 의미인 경우: 만약 다양한 요소나 주체가 존재한다는 의미의 일반 명사(다원성, pluralism)를 표현하고 싶다면 pluralistic 또는 poly- (접두사) 등을 사용할 수 있으며, 구체적인 뜻에 따라 적절한 단어를 선택해야 합니다.

예를 들어, 수학/과학 용어로서의 "다면체"는 polyhedron이라고 합니다.



polyhedron

manifold

A polyhedron is a specific 3D solid with flat polygonal faces, straight edges, and vertices, such as a cube.

A manifold is a more general concept describing a space that is locally like Euclidean space but can be globally different, meaning it can be curved or have a different overall shape.

While some polyhedra are polyhedral manifolds (meaning the collection of faces forms a manifold), not all polyhedra are manifolds, and many manifolds are not polyhedra. 


This video explains what a polyhedron is with visual examples:03:28Math with Mr. JYouTube • 2023. 3. 25.

Polyhedron Definition: A solid, 3D figure with flat faces, straight edges, and vertices.

Faces: Must be polygons (e.g., squares, triangles, hexagons).

Local vs. Global:

Has a specific 3D structure defined by its flat faces and straight edges.

Examples: A cube, pyramid, and prism are all polyhedra.

Non-examples: Spheres, cones, and cylinders are not polyhedra because they have curved surfaces. 


Manifold Definition: A topological space that "locally" resembles Euclidean space (\(R^{n}\)) but can have a different "global" structure.

Faces/Surfaces:

Not restricted to flat polygons; surfaces can be curved. The key is that any small patch on the surface looks like a flat piece of a plane (for a 2-manifold).

Local vs. Global:

Looks like flat Euclidean space up close, but the overall shape can be different. For example, a sphere looks like a flat plane if you only look at a small area.

Examples:

A sphere, a torus (donut shape), and a surface formed by a cube with its top and bottom faces glued together are all 2-manifolds.

Non-examples:

A "suspension" of a polyhedron can fail to be a manifold, particularly at points where the faces are joined in ways that are not locally flat. 

How they relate Polyhedra can be manifolds:

A simple polyhedron where all the faces are properly joined can be considered a "polyhedral manifold," where the "pieces" of the manifold are the polygons that make up the polyhedron.Manifolds can be non-polyhedral:

Many manifolds, like a sphere, have no flat faces at all.A polyhedron can fail to be a manifold:

Some more complex polyhedra might have points or regions where the local structure is not like a flat space, causing them to fail to be a manifold.