Geometry of four dimensions
Geometry of four dimensions
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四维几何
DOI：
10.32469/10355/15426
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发表时间：
1915
期刊：
影响因子：
--
通讯作者：
H. Manning
中科院分区：
文献类型：
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作者：
H. Manning
我选择了一些定理，并将给出其中一些定理的证明，以便你们知道我们是如何学习这个几何的，以及它有多简单。我假设我们从点开始，把所有的数字都看成是由点组成的。我假定点之间的关系可以称为共线关系，并利用这种关系来定义直线、平面和超平面。直线由两个点确定，平面由三个点而不是一条直线上的点确定，超平面由四个点而不是一个平面上的点确定。例如，超平面由四个点组成，这些点不是一个平面上的点，所有的点与其中任意两个共线，所有的点与通过这个过程得到的任意两个共线。我们的空间是一个超平面超平面中的几何是普通的立体几何。现在为了得到一个四维空间只需要假设有五个点不在一个超平面上。四维空间由我们得到的点组成如果我们在一个超平面上取五个点，所有的点与其中任意两个共线，并且所有的点与通过这个过程得到的任意两个共线。为了方便起见，我假设所有的点都在一个四维空间中，并把这个空间称为超空间。
I have selected a few theorems and will give the proofs of some of them in order that you may know how we study this geometry and just how simple it is.* I will suppose that we have started with points and regard all figures as consisting of points. I assume a relation among points which may be called the collinear relation, and define lines as well as planes and hyperplanes by means of this relation. The line is determined by two points, the plane by three points not points of one line, and the hyperplane by four points not points of one plane. The hyperplane, for example, consists of the points that we get if we take four points not points of one plane, all points collinear with any two of them, and all points collinear with any two obtained by this process. Our space is a hyperplane and geometry in a hyperplane is the ordinary solid geometry. Now in order to get a space of four dimensions it is only necessary to assume the existence of five points not in one hyperplane. Space of four dimensions consists of the points that we get if we take five points not in one hyperplane, all points collinear with any two of them, and all points collinear with any two obtained by this process. For convenience I will assume that all points lie in one space of four dimensions and call this space hyperspace.