Icosphere

创建时间：2021/1/26 17:49

• Icosphere
  • Regular polyhedron（正多面体）
    • 定义及分类
    • 相关术语
      • Dual polyhedron
      • Symmetry group
    • 性质
      • Equivalent properties
      • Concentric spheres
      • Symmetry
    • Platonic solid（凸正多面体）
      • Regular dodecahedron(正十二面体)
      • Regular icosahedron(正二十面体)
  • Goldberg polyhedron & Icosphere
    • Goldberg polyhedron
      • Construction
      • truncated icosahedron
    • Geodesic polyhedron
      • Construction
      • Icospheres
  • Ref

Regular polyhedron（正多面体）

定义及分类

在几何学中，正多面体是同时具有等边、等角和等面特性的多面体。 最常见的定义 - 每个面都是全等的正多边形，且每个顶点都是相同数量且相同种类之正多边形的公共顶点。

在中文环境中，一般被大众认知的正多面体通常代表只有五种的凸正多面体，又称为柏拉图立体，其包括了正四面体、立方体、正八面体、正十二面体和正二十面体。然而在定义上，正多面体仅指每个面是正多边形、每条边等长每个角等角且每面全等的多面体，而符合上述定义的多面体不一定是凸多面体，也可能是星形多面体、抽象多面体或扭歪多面体等。

There are 5 finite convex regular polyhedra (the Platonic solids, 柏拉图立体 ), and four regular star polyhedra(星形正多面体) (the Kepler–Poinsot polyhedra), making nine regular polyhedra in all. In addition, there are five regular compounds of the regular polyhedra.

本文整理狭义的三维空间中的正多面体相关知识，而正凸多面体是较常见的，因此这里重点整理正凸多面体的知识。

Each Platonic solid can therefore be denoted by a symbol {p, q} where p is the number of edges (or, equivalently, vertices) of each face, and q is the number of faces (or, equivalently, edges) that meet at each vertex.

图片来源：维基百科

相关术语

Dual polyhedron

In geometry, any polyhedron is associated with a second dual figure, where the vertices of one correspond to the faces of the other and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other.

在几何学中，若一种多面体的每个顶点均能对应到另一种多面体上的每个面的中心，它就是对方的对偶多面体。

根据对偶原则，每种多面体都存在对偶多面体。一种多面体的对偶多面体的对偶多面体等同该种多面体。

对偶多面体有相同的对称性。

Symmetry group

In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. 使物体不变的所有变换的集合。

性质

在几何学中，正多面体是一类对称性可以在其各维度元素的集合上传递的多面体。正多面体通常具有高度对称性，其同时具有边可递，点可递和面可递的性质。

Equivalent properties

• 此多面体的每个顶点都坐落在同一个球上（即存在外接球）

• 每个二面角皆相等

• 所有顶点图（顶点的截面）皆为正多边形

• 所有立体角皆相等

Concentric spheres

正多面体具有三个相关的球体（其他非正多面体至少缺少一种），其球心位于同一个点上：

• 与所有面相切的 内切球

• 与所有棱相切的 棱切球

• 过所有顶点的 外接球

Symmetry

The regular polyhedra are the most symmetrical of all the polyhedra. They lie in just three symmetry groups, which are named after the Platonic solids:

• Tetrahedral

• Octahedral (or cubic)

• Icosahedral (or dodecahedral)

Any shapes with icosahedral or octahedral symmetry will also contain tetrahedral symmetry.

Platonic solid（凸正多面体）

In three-dimensional space, a Platonic solid is a regular, convex polyhedron.

They are named for the ancient Greek philosopher Plato who hypothesized in one of his dialogues, the Timaeus, that the classical elements were made of these regular solids.

1. all its faces are congruent convex regular polygons,

2. none of its faces intersect except at their edges, and

3. the same number of faces meet at each of its vertices.

世界上只存在五种凸正多面体的两种证明见维基百科：https://en.wikipedia.org/wiki/Platonic_solid

Regular dodecahedron(正十二面体)

The convex regular dodecahedron is one of the five regular Platonic solids and can be represented by its Schläfli symbol {5, 3}.

The dual polyhedron is the regular icosahedron {3, 5}, having five equilateral triangles around each vertex.

It has 12 faces, 20 vertices, 30 edges.

Regular icosahedron(正二十面体)

Icosahedron is a regular polyhedron with 20 equilateral triangles.

In geometry, a regular icosahedron is a convex polyhedron with 20 faces, 30 edges and 12 vertices. It is one of the five Platonic solids, and the one with the most faces.

It is the dual of the dodecahedron , which is represented by {5,3}, having three pentagonal faces around each vertex.

the vertices of an icosahedron are the corners of three orthogonal golden rectangles.

Goldberg polyhedron & Icosphere

A Goldberg polyhedron is a dual polyhedron of a geodesic sphere.

Goldberg polyhedron

In mathematics, and more specifically in polyhedral combinatorics, a Goldberg polyhedron is a convex polyhedron made from hexagons and pentagons.

• each face is either a pentagon or hexagon

• exactly three faces meet at each vertex

• they have rotational icosahedral symmetry.

A consequence of Euler’s polyhedron formula is that a Goldberg polyhedron always has exactly 12 pentagonal faces.

图片来源：维基百科

Simple examples of Goldberg polyhedra include the dodecahedron(正十二面体) and truncated icosahedron(足球的形状).

Other forms can be described by taking a chess knight move from one pentagon to the next - first take m steps in one direction, then turn 60° to the left and take n steps. Such a polyhedron is denoted GP(m,n). A dodecahedron is GP(1,0) and a truncated icosahedron is GP(1,1).

T=m^2+mn+n^2 V=20T，E=30T，F = 10T+2 Faces by type 12 {5} and 10(T − 1) {6}

Construction

truncated icosahedron

It has 12 regular pentagonal faces, 20 regular hexagonal faces, 60 vertices and 90 edges.

truncated - This polyhedron can be constructed from an icosahedron with the 12 vertices truncated (cut off) such that one third of each edge is cut off at each of both ends.

The truncated icosahedron (left) compared with an association football.

Geodesic polyhedron

A geodesic polyhedron(测地线多面体) is a convex polyhedron made from triangles.

Geodesic polyhedra are the dual of Goldberg polyhedra.

They usually have icosahedral(二十面体的) symmetry , such that they have 6 triangles at a vertex, except 12 vertices which have 5 triangles. They are the dual of corresponding Goldberg polyhedra with mostly hexagonal faces.

Geodesic polyhedra are a good approximation to a sphere for many purposes.

The most well-known may be the geodesic domes designed by Buckminster Fuller, which geodesic polyhedra are named after. Geodesic grids used in geodesy also have the geometry of geodesic polyhedra. The capsids of some viruses have the shape of geodesic polyhedra, and fullerene molecules have the shape of Goldberg polyhedra. Geodesic polyhedra are available as geometric primitives in the Blender 3D modeling software package, which calls them icospheres: they are an alternative to the UV sphere, having a more regular distribution of vertices than the UV sphere. The Goldberg–Coxeter construction is an expansion of the concepts underlying geodesic polyhedra.

Construction

Geodesic polyhedra are constructed by subdividing faces of simpler polyhedra, and then projecting the new vertices onto the surface of a sphere.

A geodesic polyhedron has straight edges and flat faces that approximate a sphere, but it can also be made as a spherical polyhedron (a tessellation on a sphere) with true geodesic curved edges on the surface of a sphere and spherical triangle faces.

测地线多面体可以看作正二十面体的边细分得到

Icospheres

Icospheres are a type of geodesic dome. In computer graphics, an icosphere is a geometric primitive available in that approximates a sphere as a simplicial polyhedron, formed by subdividing the polygons of a Goldberg polyhedron into their triangular constituents , or equivalently, by subdividing the triangles of a regular icosahedron.

我们采用Class 1的方式来构造，如下图：

An icosphere is then created by splitting each triangle into 4 smaller triangles. This can be done several times, the recursion level is a parameter to the icosphere.

应用案例：http://oskarstalberg.com/game/planet/planet.html

Ref

https://en.wikipedia.org/wiki/Geodesic_polyhedron https://en.wikipedia.org/wiki/Goldberg_polyhedron https://en.wikipedia.org/wiki/Platonic_solid https://en.wikipedia.org/wiki/Dual_polyhedron https://en.wikipedia.org/wiki/Icosahedral_symmetry https://handwiki.org/wiki/Icosphere