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 1.  If you   want an object to exist at a particular dimensional realm, then you need the node particle or  point that has the adequate amount of elements for that number of dimension.  1.b as with spherical coordinates though we have 3 elements for both Cartesian and spherical to have a point in 3 Dimensional space , we just need to be able to infer or decide what the spatial coordinates are, and assume all spatial dimensional are the same as cartesian. That is to say that Cartesian coordinate system can describe every point which is described by another coordinate system.  Then when you have the point and it's coordinate system. 2. Create the one simplex, as a collection of 2 points. You can see how a vector with direction because it is a line between start and end.  3. Create the two simplex as a collection of 2 one simplices.   You can see in matrix algebra how a pair of line vectors forms a the triangle which represents it's coplan...

  The Four_Simplex (also known as the 5-cell, Pentachoron, or Hyper-tetrahedron) is the simplest possible manifold in four-dimensional space. There is the outer Tetrahedron which exists at only time=0. And We create 4 Tetrahedron by using the 4 faces of the First Tetrahedron, and reducing their area as time progresses, so that there are 5 tetrahedrons formed by the one Tetrahedron shrinking in time.  To understand it from every vantage, we must break it down into its spatial, structural, and temporal properties. 1. The Spatial Vantage: Fundamental Constants Given a regular 4-simplex with edge length $L$ , the basic geometric properties are locked by the following ratios: Altitude ( $H_4$ ): The total vertical distance from the 3D tetrahedral base to the 5th vertex. $$H_4 = \sqrt{\frac{5}{8}}L \approx 0.79056L$$ Hypervolume ( $V_4$ ): The 4D content contained within the 5 cells. $$V_4 = \frac{\sqrt{5}}{96} L^4 \approx 0.02329 L^4$$ Boundary Elements: 5 Vertices (0D) You hav...