Space Structures

2D Structures

Wait, "2D"? Aren't these all 3-dimensional solids? No, you are thinking of Euclidean space. In our geometry there is no concept of "solid"; all we see are the nodes and the relationships (edges) between them.

Example	Name	Nodes	Edges	Shapes	Cells	Euler calculation
	tetrahedron	4	6	4	1	1
	square pyramid	5	8	5	1	1
	pentagonal pyramid	6	10	6	1	1
	triangular prism	6	9	5	1	1
	cube	8	12	6	1	1
	nonahedron	9	16	9	1	1

The Euler calculation is Euler = Nodes - Edges + Shapes - Cells

Thought Experiment 1

In this experiment we are going to connect Euclidean geometry with our new geometry. Imagine an ordinary Euclidean cube: right angles, parallel edges, and 6 square shapes. In your thoughts you can turn it around and look at it from different angles so that you have the cube firmly in mind.

Now let's think about the same figure in terms of our geometry. First we need to choose the nodes. Nodes have to be "things" and they need to be countable. We can't use the infinity of Euclidean points that form the Euclidean cube but we can pick specific points. One natural choice is to use the vertices.

Next we need to define the edges. Edges for us are pairs of points — not the same as the line segments which make edges in Euclidean geometry. The line segments do define a relationship between pairs of vertices. We can define our edges as the pairs of nodes which are connected by Euclidean edges.

Now we have defined a structure in our geometry which models the cube in Euclidean geometry. Our structure consists of 8 nodes and 12 edges which form 6 shapes and 1 cell. In other words, a cube. We've mathematically modelled a cube with a cube.

If you are thinking mathematically you may be saying to yourself, "Oh, wow, that is so cool!"

Thought Experiment 2

Here is another thought experiment which you can also do in real life with 7 of your nerdiest friends.

First, divide the 8 people into 2 groups of 4 people each. In the first group, each person should use their RIGHT hand to reach rightward and grab their neighbor's LEFT ankle. This group has now created a square.

In the second group, each person should take their LEFT hand and grasp the RIGHT ankle of their neighbor leftward. Now you have a second square.

Finally, notice that everyone in the first group has has a LEFT hand which hasn't been used for anything. Reach your free LEFT hands across to the other group and grasp a free LEFT ankle.

If you followed the instructions correctly you just built a cube. The people are the 8 nodes; the grasping of ankles are the 12 edges. The 6 shapes are the paths from person to around and between groups.