Space Structures

3D Structures

Here are some examples of structures which we classify as 3D. You can easily create more examples by combining our 2D structures.

Example	Name	Nodes	Edges	Shapes	Cells	Euler calculation
	double (stacked) cube	12	20	11	2	1
	double (stacked) triangular prism	9	15	9	2	1
	split cube or adjacent triangular prisms	8	14	9	2	1

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

Hard Thought Experiment

What would happen if we added one edge to a cube? Specifically, let's add a diagonal across one of the square shapes of a cube. That turns the square shape into 2 triangular shapes: We add one edge and that automatically creates an additional shape as well. We no longer have a cube; our new structure has 13 edges and 7 shapes.

Let's do the same thing on another shape. We'll use the shape opposite to the one we just changed. (The "opposite shape" in this experiment means the shape which shares no edge with the one we modified. Each shape of a cube shares an edge with each of 4 other shapes; we might call them adjacent shapes. The sixth shape we are calling the opposite shape.)

Again we add an edge diagonally. Let's say we go in the same direction as we did before. (The "same direction" is a sloppy way of saying that we are pairing nodes which are connected by edges. So if we had picked node A on the top and there is an edge {A,Q} to node Q on the bottom, we are going to work with Q now. It may be easier to see in the pictures below.)

It isn't surprising that the same thing happens with this shape as it did with the first one: The square shape is changed into 2 triangular shapes. Add an edge, add a shape.

But wait! That's not all!

Besides changing 2 existing shape into 4 new shapes, we also created a brand new, never before seen path. The 2 new edges form a circuit with 2 of the old edges. Because this is a "shortest" path we also have a brand new shape.

Example	Description	Nodes	Edges	Shapes	Cells	Euler calculation
	This is the cube we started with.	8	12	6	1	1
	The cube with an added edge on the top. The top shape was a square, but we turn it into 2 triangular shapes. (Remember that we have no concept of "length" or "shape"; the sides in our drawing can shift for better visibility without changing the structure.)	8	13	7	1	1
	The structure after we add another edge. Again, 2 triangular shapes replace the square shape. A new shape appears in the interior of the structure, dividing it into 2 cells.	8	14	9	2	1

The Euler calculation gives the same result through all these changes.

That new shape runs right through the middle of the structure and makes it 3D instead of 2D. Specifically, 2 of the original edges are now each part of 3 shapes. (A 2D structure is defined as having exactly 2 shapes for each edge.)

We can find a 2D structure inside of our new 3D structure. Half of the original cube plus the new shape makes a triangular prism. The other half and the new shape also make a prism. So, our new 3D structure has 2 cells.

We've actually seen this new structure before. Our new structure is the split cube in the table above . The drawings are a little different but the nodes and edges are the same.

Really Hard Thought Experiment

Too easy for you? Here's a thought experiment that requires some real work. We are going to change a cube into an octahedron. Except that in our geometry it doesn't have 8 shapes … so it can't be an octahedron.

Start again with the ordinary Euclidean cube. Last time, we picked the Euclidean vertices to be nodes in our geometry. This time, we we look at the Euclidean faces; the 6 faces will become our 6 nodes. Let's call them Top, Bottom, Right, Left, Front, and Back. Nodes can be anything; in this experiment our nodes will be Euclidean squares.

The next question is how these nodes are related. In the Euclidean cube, Euclidean faces are connected by the Euclidean edges which they have in common. For example, Top is related to Front because they share a line segment. So we will define our edges as the relationship of having a common Euclidean edge: {Top,Front}, {Top,Right}, {Top,Back}, {Top,Left}, {Bottom,Left}, {BottomFront}, {Bottom,Right}, {Bottom,Back}, {Front,Left}, {Left,Back}, {Back,Right}, and {Right,Front}.

What sort of a structure did we create? If you are really good with visualizations, think of shaving off the corners of the cube — no, not shaving but hacking them off, cutting back each corner until you get to the middle of each shape. All 8 corners turn into shapes and the 6 shapes are reduced to mere corners. (There are still 12 edges but now they run crosswise to the way the old edges went.)

If that doesn't work you can imagine each of the Euclidean cube's 6 faces contracting to a single point: The 6 old shapes becomes 6 points and new edges connect them. What happens to the 8 old vertices? They have to expand to compensate for the contraction and so they turn into 8 shapes.

So 6 shapes shrink into 6 vertices while 8 corners grow into 8 shapes. It is an octahedron!

Except that it isn't. Here comes the hard part.

Example	Description	Nodes	Edges	Shapes	Cells	Euler calculation
	octahedron (sort of)	6	12	11	6	-1

The Euler calculation is Euler = Nodes - Edges + Shapes - Cells. What is the Euler calculation showing us?

Consider the node we named Top along with the nodes Front, Right, Back, and Left. These are all connected by edges, {Top,Front}, {Right,Back}, and so on. Together, they form a square pyramid. You can see it in the drawing if you squint just right. (Remember that in our geometry "shape" just means a path — a path that is a circuit and has no available shortcuts. When we define the edges in a structure we define the paths too and all of the shapes.) This pyramid has 5 shapes: 1 square shape and 4 triangular ones.

The same is true of the bottom half of our structure: Bottom, Front, Right, Back, and Left form another square pyramid. (It looks upside down in the drawing.) The square shape is the same as the one in the top pyramid and there are 4 more triangular shapes. So instead of the 8 shapes of an octahedron, we now have 9 shapes. We also have 2 cells.

But wait! That's not all. The edges {Top,Back}, {Back,Bottom}, {Bottom,Front}, and {Front,Top} make another shape. Now we have 10 shapes. Along with Left we form a square pyramid pointing left; similarly to the Right. That's 2 more cells. (We already counted the shapes.)

Thinking the same way, we can identify 2 more pyramids pointing to Front and to Back. With these 2 cells, we have 6 cells, 6 nodes, 12 edges, and 11 shapes.