Moving back to the cool magical super esoteric sacred geometry uber symbol nicknamed E8

The image of the actual thing is 16 MB so I will just link the image map (Called a petrie like the one biologists use to study cells)

https://upload.wikimedia.org/wikipedia/commons/1/14/E8Petrie.svg

That is because it is actually a 3 Dimensional object with multiple facets, like a Diamond
The 2D images lines connecting the points are not over lapping nor intersecting, they are all free between two points

https://en.wikipedia.org/wiki/Lie_group

A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additional properties it must have to be thought of as a "transformation" in the abstract sense, for instance multiplication and the taking of inverses (to allow division), or equivalently, the concept of addition and subtraction. Combining these two ideas, one obtains a continuous group where multiplying points and their inverses is continuous. If the multiplication and taking of inverses are smooth (differentiable) as well, one obtains a Lie group.

Manifold = twisty object
Talking about a different way to look at the problem and work a solution
Abacus for example

https://en.wikipedia.org/wiki/Abacus

Here is an example

https://en.wikipedia.org/wiki/Circle_group

{Picture a round clock with three hands extending all the way out to the edge of the clock
You want to do multiplication, add up all of the angles in the hands of the clock
Sounds weird, but it just shows the circuits built upwards in scale how dimension, shape, structure, organization can take many different methods, forms and shapes along the road to get "there"...whoops we passed it and just got bigger now we are "here"
https://en.wikipedia.org/wiki/E8_(mathematics)

We can think of these as

abstract https://en.wikipedia.org/wiki/Binary_operation

Lie groups provide a natural model for the concept of continuous symmetry