In Monday's post, we created a sliceform model of a hyperbolic paraboloid. In today's post, we will create a similar model using skewers. The hyperbolic paraboloid is a ruled surface, which means that you can create it using only straight lines even though it is curved. In fact, the hyperbolic paraboloid is doubly ruled and is one of only three curved surfaces than can be created using two distinct lines passing through each point. The others are the hyperboloid and the flat plane.
The hyperbolic paraboloid that we are going to build is based off of a sculpture by Carlo Sequin called the ribbed hemicube, which consists of 3 hyperbolic paraboloids embedded in a tetrahedron that intersect each other.
Instead of forming all three hyperbolic paraboloids, we are only going to make one of them. But we are going to use both sets of lines since it is a doubly ruled surface. The result will look like this (although you could easily build a recreation of Carlo's sculpture if you continued):
Here's a video showing the surface rotating so you can appreciate its three dimensional nature.
Those of you that have been following Math Craft may notice that it looks similar to the parabolic curves using straight lines.
In fact, this object is really one of the extensions of this object into the third dimension and is formed in a very similar way by marking at even increments and extending lines across these markings.
- Skewers (I used 12 inch)
- Glue (I used superglue)
- Marking pen
Step 1 Cut the Skewers to the Desired Length
I've found that skewers often are not quite evenly cut and are rarely the exact length advertised. I took my approximately 12 inch skewers and cut them down to 10 inches. This has the benefit of removing the sharp points. I used a sharp chisel to cut them but I'm sure you could develop a better method.
Step 2 Make a Regular Tetrahedron
Take six of the skewers.
Use three of them to form an equilateral triangle. Glue together.
Now use this as a template to form two skewers into a 60 degree angle, completing 2/3 of another equilateral triangle.
Connect this to the equilateral triangle. You should be trying to form a regular tetrahedron.
While holding this, connect the last edge. Glue them together. Clamps or another set of hands might be helpful.
Step 3 Mark the Edges of the Tetrahedron in Regular Intervals
Use a ruler and a marking device to mark the edges of the tetrahedron at regular intervals. I used spacings of 1 inch. Note: you could do this step before you built the tetrahedron.
Step 4 Connect the Skewers
Now you are going to use these markings and connect the skewers between the marks. Use one edge—the edge that isn't directly connected to that side. Connect a skewer to the first marks on each of the edges. There will be overlap. I distributed that evenly to each edge. Glue the skewer to the edges.
Keep connecting up skewers by moving one interval mark on each edge. cThe overlap will get greater each time.
Step 5 Use Skewers Going the Other Direction to Doubly Rule the Surface
Now you are going to use two of the other edges to form the other ruling lines. The goal is to have skewers that are always forming right angles at their connections to the skewers that are already there. You will probably have to experiment to pick the correct edges and the correct direction. Once you figure it out, all you do is connect the first mark on one edge to the first mark on the other edge and continue as you did in the previous step.
The finished hyperbolic paraboloid:
Step 6 Remove the Two Extra Tetrahedron Edges
This step is unnecessary, but makes the model look better. I just twist and pull off the extra skewers on the top and bottom of the model. Now it should look like this:
Step 7 Show Off Your Work
If you make the hyperbolic paraboloid or any of the other previous Math Craft projects, please share with us by posting to the corkboard. Perhaps you have some original project or something you've seen on the web that you'd like to share.
If you like these types of projects, let me know in the comments. If you have any other ideas you would like to pursue, let me know in the forum.