The normal dimensional theory or better said the logarithmical dimensional theory is prevelant in both studies and entertainment. It could be described in simplified terms as following. One goes from a dot(0D) and doubles the dimensions. Thus creating two dots and interconnecting them, becoming a line(1D). This again gets doubled and all dimensional corners are connected. Creating a square(2D), which again gets doubled and all corners are interconnected. Creating a cube(3D), which you should imagine getting doubled again and all corners(16 in total now) are interconnected creating a hypercube(4D), also known as a tesseract.
In simple dimensional theory there is a linear approach. The biggest difference between the two models is that in a logarithmical model any level of dimension always contains the lower order dimensions as a copy. Whereas in a linear model they build upon each other. This also eliminates the idea of travelling to other dimensions, as they are all a part of the same super-dimension.
|Dimension||Simple figure||Corner(s)||Angle(s) p/corner|
To imagine the hyper-tetrahedron take a normal tetrahedron(which looks like a pyramid but with a triangle as base) and add within the center an extra corner interconnecting the original 4 corners.
This approach has several interesting real world implications that can be demonstrated. Each figure has the lower dimension simplest figure, as its shadow. E.g. a tetrahedron shaped object always makes a triangle shaped shadow. And a triangle(as seen from a theoretical 2D world) would always have a line as a shadow. And thus a hyper-tetrahedron would always have a tetrahedron as its shadow.
This means that we could theoretically live within a 4D, 5D, or higher dimensional world without ever knowing because everything is a shadow of its upper dimension. With logarithmical dimensional theory there are such weird behaviors as; observed resizing of objects when you move them through their higher dimensional axis. These have never been witnessed in reality. Whereas with a linear approach there is no limitation on possible higher dimensions because there are no implications in any way on its own dimensional level. And thus seem more plausible.