Space Structures

1D Structures

Wait, "1D"? Aren't these 2 dimensional figures? Not for us. If you were going to travel along these figures you would never be able to turn aside; there is always only one path.

Example	Name	Nodes	Edges	Shapes	Cells	Euler calculation
	triangle	3	3	1	0	1
	square (or quadrangle) (or quadragon) (or quadrilateral)	4	4	1	0	1
	pentagon	5	5	1	0	1
	hexagon	6	6	1	0	1
	septagon	7	7	1	0	1
	octagon	8	8	1	0	1
	nonagon	9	9	1	0	1

The Euler calculation is: Euler = Nodes - Edges + Shapes and there is always just one shape because there is only one path.

Thinking Mathematically

Here's an experiment that you can actually try at home with some of your friends. Have 4 people stand in a circle.

• They don't have to be standing. They could sit down or even lie on the grass.
• There isn't really a circle. That's just an expression. We have no definition of "circle" in our geometry (and the Euclidean circle would require an infinity of people).

Each person is a node. The relationship is adjacency: in other words, Beth is standing to the right of Allan (and Allan is to the left of Beth — same thing). Carli is to the right of Beth and Daniel to the right of Carli and finally Allan is to the right of Daniel. So "to the right (or left) of" relationships are the edges.

The nodes are {Allan, Beth, Carli, Daniel} and the edges are {Allan, Beth}, {Beth, Carli}, {Carli, Daniel}, and {Daniel, Allan}.

You've made a square! Isn't that cool?

Now, here's the experiment: Suppose that another person joins the circle (which still isn't really a circle). What changes when Eve steps between Daniel and Allan?

1. There is 1 new point, Eve.
2. One edge goes away: Allan is no longer to the right of Daniel.
3. Two new edges are added: Eve is to the right of Daniel and Allan is to the right of Eve.
4. The Euler calculation, which was 4 - 4 + 1 = 1, is now 5 - 5 + 1 = 1.

If you are thinking mathematically, you are probably pretty excited by now. You are saying to yourself, "Wow, this is cool! Every time we add one person we also add one edge. And because the Euler calculation adds nodes but subtracts edges — Nodes - Edges + Shapes = (Nodes + 1) - (Edges + 1) + Shapes — the Euler number never, ever changes!"

What would happen, you might be thinking, if most of the people in the world were all in this same circle? Say 6,000,000,000 people. Well, we'd have a six-billion-gon. And then if one more person wanders in and joins the group, we'd add one node. We'd lose one edge, but replace it with 2 new edges. And the Euler calculation would not change.