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Hyperbolic Slot

Science Snack
Hyperbolic Slot
How do you put a straight rod through a curved slot?
Hyperbolic Slot
How do you put a straight rod through a curved slot?

If you see a straight rod and a curved slot, common sense says the rod can’t possibly fit through the slot. But if the rod is angled and rotated through space, it describes a three-dimensional shape with a hyperbolic cross section. So if the slot is the exact shape of this hyperbola, you can make the straight rod pass through it.

Tools and Materials
  • Wood rectangle for base, approximately 3 1/2 x 12 x 3/4 inches (9 x 30 x 2 centimeters)—exact dimensions are not critical; a convenient method is to cut a 12-in length of standard 1 x 4 pine lumber, which is 3 1/2 inches wide and 3/4 inches thick
  • Two wood blocks, each 1 1/2 x 3 x 3/4 inches (4 x 7.5 x 2 cm)—exact dimensions are not critical; a convenient method is to cut 3-in long pieces of standard 1 x 2 pine lumber, which is 1 1/2 inches wide and 3/4 inches thick
  • 5-in (12.5-cm) length of wooden dowel, 1/2-in diameter
  • 11-in length of wooden dowel, 3/16-in diameter
  • #6 flat-head wood screw, 1 1/2 in long
  • Two 5-in (12.5-cm) lengths of 1/2-inch Schedule 40 PVC pipe
  • 1/2-inch PVC 90-degree elbow
  • Printed slot template (not shown)
  • 7 x 11-in (18 x 30-cm) piece of foam core (mat board or stiff cardboard can also be used, as well as tempered hardboard (masonite) or ⅛-in acrylic plastic if you have access to a band saw or scroll saw)
  • X-Acto knife or utility knife
  • Drill
  • Drill bits, 1/8 in and 3/16 in
  • Screwdriver
  • Pencil
  • Ruler
  • Hot-glue gun and glue
  • Optional: masking tape

Build the base:

  1. Take the wood board that will serve as a base and lightly draw a line lengthwise down the center.
  2. Drill a 1/8-inch diameter hole all the way through the base, 3 inches (7.5 cm) from one end and on the centerline.
  3. Drill a 1/8-inch diameter hole about 3/4 inch deep in the center of one end of the 1/2-inch dowel. Drill this hole as straight into the center of the dowel as possible.
  4. Screw the 1½-inch screw through the hole in the base, from the bottom side of the base to the top side. The flat head of the screw should end up flush with the bottom side of the base so that the base doesn't wobble. (If tightening the screw doesn't let the head of the screw sink into the soft pine enough without stripping, you may have to “countersink” the screw by slightly enlarging the hole on the bottom with a larger drill bit, allowing the head of the screw to go far enough into the base so that the base can lie flat. Alternatively, you can add small rubber bumpers to the bottom of the base so the base is slightly elevated, thus allowing the screw head to extend slightly below the base without producing a wobble.) The pointed end of the screw should stick straight up through the top of the base. Place the hole end of the 1/2" dowel on the point of the protruding screw and twist the dowel onto the screw until the end of the dowel is tight against the base.

Add the slot:

  1. Download and print the hyperbolic slot template. Lay the printed template on the foam core (or Masonite or other material) and align the right side of the horizontal shaded rectangle on the template with the right edge of the long (11-inch) side of the foam core. Center the template vertically along that edge. You may want to tape it in place.
  2. Using the template as a guide, cut the hyperbolic slot pattern in the foam core, Masonite or other material, using an appropriate cutting tool (knife, bandsaw, scroll saw).
  3. Hot glue one of the wood blocks to the base so that it is offset about 1/8 in from the centerline. Except for the 1/8 in offset, the exact position of the block along the centerline is not critical (see photo below).

  4. Place the foam core on the centerline so that it is against the glued block, and then glue the second wood block on the other side to form a sandwich, with the foam core (or other material) held between the wood blocks. When the glue for both blocks has dried, you will want the foam core to be held firmly in place between the blocks but not so tightly that you can't make small adjustments to its position.

Attach the angled rod:

  1. Drill a 3/16-inch hole all the way through the diameter of one of the ½ inch PVC pipes, 1 in (2.5 cm) from one end.
  2. Insert the 3/16 in wooden dowel through the holes until it is centered. (If the dowel doesn’t remain in place, wrap a rubber band around the upper section of the dowel and then slide the rubber band down until it is up against the PVC at the hole or use a little glue to hold the dowel in place.)
  3. Attach the other end of this PVC pipe to one end of the 90-degree PVC elbow. Attach the second PVC pipe to the other end of the elbow.

  4. Slip the open end of the second PVC pipe over the 1/2-inch dowel protruding from the base.
  5. Use the 3/16-in dowel as a handle to gently rotate the upper PVC pipe until the 3/16-in dowel is close to a 45-degree angle with the horizontal.
  6. Adjust the position of the foam core and the orientation of the 3/16-in dowel arm so the dowel arm can pass smoothly through the slot as the arm assembly is rotated. The adjustment may require some patience and tinkering. If absolutely necessary, you may consider modifying the size of the slot, or cutting some material off the bottom of the foam core.
To Do and Notice

Rotate the arm assembly, and observe the straight rod passing smoothly through the slot. Try to focus your attention on the shape that the rod sweeps out in space as it rotates.

What’s Going On?

As the rod swings around it sweeps out a three-dimensional shape called a hyperboloid. A two-dimensional cross section of the hyperboloid is a shape called a hyperbola. The rod passes through the slot cut into the foam core because the slot has the same hyperbolic shape as the cross-section of the hyperboloid made by the rotating rod.

Going Further

No Template
If you want to learn more about hyperbolas, you can construct the hyperbolic slot without using the prepared template. Hold a manila folder perpendicular to the base, along the centerline of the base between the dowel and the far end of the base. Plot the hyperbolic curve by making marks where the ends of the rotating arm cross the plane of the manila folder, and where the midpoint crosses the plane. Bend a piece of flexible material so that it crosses each of the three points and makes a smooth curve. Cut out this curve from the manila folder. Place the folder between the blocks on the base (shim it with crushed paper or some other improvised material to hold it upright and steady if necessary), and rotate the arm slowly through the slot. Make additional cuts where necessary to allow the wooden arm to move smoothly through the slot without touching. Then either use the manila folder itself for the slot (if it is stiff enough) or transfer the slot pattern to foam board or clear plastic and cut it out with an appropriate tool.

No Slot 
Change the angle of the dowel to vertical and visualize the shape of a slot that would allow the dowel to pass through it. Then position the dowel horizontally and visualize the shape of the slot it would need now.

So What? 
Being able to visualize how the rod goes through the curved slot may help you the next time you have to move a long sofa through a doorway, along a hallway that turns a corner, or up a spiral staircase.

Did You Know? 
A hyperbola can be described by a specific mathematical equation:

\[Y = \frac{1}{x}\]

This Science Snack is part of a collection that showcases female mathematicians and math educators whose work aids or expands our understanding of the phenomena explored in each Snack.

Highlighted Scientist: Dr. Maryam Mirzakhani

Source: Wikimedia Commons

Maryam Mirzakhani (pictured above) was an Iranian mathematician honored with the Fields Medal, the most prestigious award in mathematics. She became both the first woman and the first Iranian to be honored with this award, which is often equated in stature with the Nobel Prize. Mirzakhani specialized in the theoretical mathematics that describes the geometric and dynamic complexities of curved surfaces—spheres, doughnut shapes, and even amoebas. To solve problems, Mirzakhani would draw doodles on sheets of paper and write mathematical formulas around the drawings. In our Science Snack Hyperbolic Slot, you can explore the relationship between a surprising curved shape and a straight line.