The Application Of Quaternions In The Regular Polytopes In 4-Dimension Space

In Euclidean Geometry, there is a kind of extremely regular figure, whose names are polytopes. They have different names in different dimensions of spaces:In the plane, they are called regular polygons; In 3-dimension space, they are called regular polyhedrons. They are always distributing fascination that attracts people to search them. In the long history, effort of mathemati-cians make this area fruitful.In 1795, German mathematician C.F.Gauss give out the famous construe-tion method of regular 17 polygon. The necessary and sufficient condition of the possibility of constructing a regular n polygon is the number of its edge can be decomposed as follows: where pi are different Format Prime Numbers as22+1.In Ancient Greece, Pythagoreans had researched five regular polyhedrons which are Tetrahedron, Cube. Octahedral, Dodecahedron and Icosahedron. In Group Theory, every kind of polyhedron reflects a finite subgroup of Rotation Group of SO3, so they are called Group of Regular Polyhedron.In 1952, Swiss mathematician L. Euler found that in 3-dimension space, the number of the vertexes V, edges E, and faces F of any simple polyhedron satisfy the formulation V -E+F= 2, which is the famous Euler's Polyhedron formulation.In 1883. French mathematician J.H.Poincare proved that Euler's formu- lation can be promoted into higher dimension spaces as follows: where K is a finite simplicial complex, and ar is the number of r-dimension cells. In essence, they are another form of Gauss-Bonnet formulation.In higher dimension. Swiss geometer Ludwig Schlafli proved that in 4-dimension space, there are six convex regular polytopes, which are 5-cell,8-cell,16-cell,24-cell,120-cell and 600-cell. And when the number of dimension n≥5, there arc only three categories, and the numbers of their borders are respectively n+1,2n and 2n.In 1948, English geometer Coxeter, Harold Scott MacDonald introduced the properties of six polytopes in 4-dimension space(which has been given as 5-cell,8-cell,16-cell,24-cell,120-cell and 600-cell), and give their local coordi-nates. His calculation method of coordinates is based on algebra equation and the property of triangle functions. In fact, these questions can be solved more compactly with quaternion.Quaternion is an extension of complex numbers, which was first proposed by Ireland mathematician William Hamilton in 1843. At that time, quater-nion became an important tool of doing researches in geometry and physics via describing the transformation in geometry and Maxwell Formulations in Electromagnetism.In the previous application of quaternion, quaternion is only used to de-scribe the rotation in 3-dimension space, which never reveal the essence of its being a special case of the rotation transformation in 4-dimension space. This article derived the complete formulation of rotation transformation with quaternion via calculating the direction tensor of 2-dimension subspace and the reflection transformation formulation and dispose the relationship between the equator plane of rotation and rotation multiplier.There are there questions to be solved in this article as follows:1. Presenting the reflection and rotation transformations by quaternion. 2.Calculating the coordinates of most complex polytopes.3.Judging whether there is a subset of the vertexes of 120-cell which can be a vertexes of 5-cellThe result of the first question is given as flllows:Let qσbe a quaternion corresponding to a 4-dimension vector寸.p be a 2-dimension subspace,σis a rotation translation with an equator plane p and angleθ.Let R be a matrix of direction tensor:whose i-th row and j-th column element is rij,which satisfiesr122+r132+r142+r232+r242+r342=1.Let The quaternion qσ(v)corresponding to the transformation of寸byσis compared to the presentation of 3-dimension rotation transformation with quaternion,this result can be used to calculat e the rotation angle,axis plane and equator plane.120-cell and 600 cell are the most complicated convex polytopes in 4-dimension space.The relationship of their vertexes is extremely complex. This article calculated the local coordinates of their vertexes.Compared to the previous method,computation and storage are saved to some extent.We reached the result as follows:The point set(±4,0,0,0).((?)+1,(?)-1,0,2).(±1,±1,±1,±1)and the even order of the component of their coordinate constitute the 120 vertexes of convex polytope(3,3,5).The point set(±(4+2(?)),(±2,±2,±2):(±(4+2(?)),±((?)-1)0,±((?)+1)), (±(5+(?)),(±2:0,±(3+(?))),(±(5+(?)),±(1+(?)),±(1+(?),±(1+(?))), (±(2+2(?)),±(1+(?)),±2,±(3+(?)))(士(2+2(?)),±(2+2(?)),0,0), (±(3+3(?)),±(3+3(?)),±(3+3(?)),±((?)-1)). and the even order of the component of their coordinate constitute the 600 vertexes of convex polytope{5,3,3}. This is the result of question 2, and they are introduced in detail in the text.This result is coherent with Coxeter's.The difference is fractions have been converted in to integers, which is better to find more properties.There are many embedding relationships between the six kinds of con-verx regular polytopes, in which 5-cells can be embedded into 120-cclls is most difficult to find. This conclusion can be proved by enumerating. But the num-ber of the vertexs of 120-cells are 600, which makes the calculation impossible without programming. But in this article, we find a transformation can dis-pose the relationship of the vertexes directionally. The conclusion is given as follows:Let RP be a 120-cell, of which V is a vertex. We can find 28 vertexes. which can be equally separated into 7 groups, such that vertexes in every group can constitute the vertexes of 5-cell with V, and such vertexes can not be selected among different groups. This result not only has given positive reply but also given the concrete number.The construct of this article is as this:In the first Chapter, the basic concept and the source of the problems will be elaborated. In the second Chapter, the basic geometric knowledge and the basic knowledge of quaternion will be introduced first, and the first question will be solved there. In the third Chapter the second question will be solved with the base of the result of the first question, and then derive the result of the third question.At last, we give a method of calculating the coordinate in 4-dimension space, with which the computation will be reduced and the demand of hard-ware by animation will be lower, so more influent and clear animation can be produced.In extra, I have views of several proposition in geometry, and their proof will be laid in the appendix.