## Wednesday, February 26, 2020

### A STEM Project: Rubber Band Polyhedron Pop-ups - Pyramids

Rubber Band Polyhedron Pop-ups - Pyramids

Rubber Band Pop-Up Pyramids in Action

Rubber band pop-up polyhedrons are intriguing. They are the intersection of mathematics and physics.  The process of compressing the figure stores energy in the rubber band inside the figure. When the collapsed two dimensional shape is allowed to be transformed back into a polyhedron, the stored energy makes the figure pop-up. These pop-ups are a fun way to explore mathematics and physics together.

This is the third blog post for rubber band polyhedron pop-ups.  The first post had nine models of bifrustums and bicupolas.  https://papercraftetc.blogspot.com/2020/02/a-stem-project-rubber-band-polyhedron.html The second blog post had eight models of prisms.  https://papercraftetc.blogspot.com/2020/02/a-stem-project-rubber-band-polyhedron_19.html This blog post has ten models of pyramids.

Pyramids are shapes with sides that are triangles which meet at the top called the apex with a base which is a polygon.  Bipyramids are shapes consisting of two identical pyramids base to base.

Materials needed to make the pop-ups:

65 lb. card stock which can be found at your local craft store in a variety of colors.

-  Aleene's Tacky Glue.  I recommend this glue as it is a quick drying glue which does not warp the paper when used sparingly. I fill a quilling bottle with this glue to get an extra fine point. This is not necessary for this model but I thought I should mention it. If you look at my photos, you will see the extra fine dots of glue.

Glue Dots.  I used 3/8 in. Glue Dots rolled into a ball and attached to the tail of the rubber band so that the rubber band does not slip out of the hole with repeated opening and closing of the model.

-  Scotch Tape to anchor the rubber band with the Glue Dot down to the tab.

1/16 inch rubber bands.  Please note that different rubber bands may have different tensions.  The length that I am giving for each model is just a guide. When making knots in the rubber band, don't pull the knots too tight because the knot acts like a stopper in the hole.

-  Scissors or an electronic paper cutter like a Silhouette or Cricut.

If you are cutting the models with scissors, here is the PDF:

If you are cutting the models with a Silhouette, here is the .Studio file:

If you are cutting the models with a Cricut, here is the SVG:

 1. Triangular Pyramid (Tetrahedron) Faces 4 triangles Edges 6 Vertices 4

Triangular Pyramid

Cut out the triangular pyramid model and bend the tabs as shown in the photo above.

Glue the circle tabs together. Cut and knot a 1.6 inch rubber band. Slide this rubber band onto the circle tab. Add a Glue Dot to the tail of the rubber band.  Cover the Glue Dot and tail with Scotch tape.  Apply glue to the remaining two tabs.

Collapse the figure to make sure the glue adheres correctly.

Slide the rubber band onto the circle tab. Add a Glue Dot to the tail of the rubber band.  Cover the Glue Dot and tail with Scotch tape.

Open and close the model a few times and bend the sides inward and outward to align the model. Completed Triangular Pyramid.

 2. Triangular Bipyramid (Hexahedron) Faces 6 triangles Edges 9 Vertices 5

Triangular Bipyramid

The instructions to make the Triangular Bipyramid are similar to the the Triangular Pyramid instructions. Use a 0.75 inch rubber band.

 3. Regular Pyramid Faces 4 triangles, 1 square Edges 8 Vertices 5

Regular Pyramid

Cut out the triangular pyramid model and bend the tabs as shown in the photo above.

Glue the circle tabs together to form two triangles.

Glue one to the tab and adhere it to the other triangle.

Cut and knot a 1.5 inch rubber band. Slide this rubber band onto the circle tab. Add a Glue Dot to the tail of the rubber band.  Cover the Glue Dot and tail with Scotch tape.

Slide the rubber band onto the other circle tab. Add a Glue Dot to the tail of the rubber band.  Cover the Glue Dot and tail with Scotch tape.

Apply glue to the remaining tab.

Adhere the tab by collapsing the figure.

Completed Regular Pyramid

 4. Octahedron Faces 8 triangles Edges 12 Vertices 6

Octahedron

The instructions to make the Octahedron are similar to the the Triangular Pyramid instructions.
Use a 1.75 inch rubber band.

 5. Pentagonal Pyramid Faces 5 triangles, 1 pentagon Edges 10 Vertices 6

Pentagonal Pyramid

Cut out the pentagonal pyramid model and bend the tabs as shown in the photo above.

Apply glue to the tab as shown above. Knot a rubber band on one side only.  Lay it on the seam.  Adhere the tab to enclose the rubber band.

Thread the rubber band through the pentagon.

Pull the rubber band for a snug fit and make a knot.  Do not cut the rubber band yet.

Apply glue to the one tab and adhere it inside the pentagon.

Apply glue to the other tab and adhere it inside the pentagon.

Compress the figure and make sure that the model pop-ups up correctly.  Adjust the rubber band if needed. To open and close the figure, pull the bottom rubber band outward.

Completed Pentagonal Pyramid

 6. Pentagonal Bipyramid Faces 10 triangles Edges 15 Vertices 7

Two views of a Pentagonal Bipyramid
The instructions to make the Pentagonal Bipyramid are similar to the the Triangular Pyramid instructions. Use a 2 inch rubber band.

 7. Hexagonal Pyramid Faces 6 triangles, 1 hexagon Edges 12 Vertices 7

Hexagonal Pyramid

Cut out the hexagonal pyramid model and bend the tabs as shown in the photo above.

Glue the circle tabs together. Cut and knot a 1.25 inch rubber band. Slide this rubber band onto the circle tab. Add a Glue Dot to the tail of the rubber band.  Cover the Glue Dot and tail with Scotch tape.  Apply glue to the other circle tabs.

Slide the rubber band onto the other circle tab. Add a Glue Dot to the tail of the rubber band.  Cover the Glue Dot and tail with Scotch tape.

Apply glue to the hexagonal base tab. Adhere as shown above.

Apply glue to the other hexagonal base tab and collapse the figure so that the glue adheres.

Completed Hexagonal Pyramid

 8. Hexagonal Bipyramid Faces 12 triangles Edges 18 Vertices 8

Hexagonal Bipyramid

Cut out the hexagonal bipyramid model and bend the tabs as shown in the photo above.

Glue the circle tabs together. Cut and knot two 1.25 inch rubber bands. Slide this rubber band onto each circle tab. Add a Glue Dot to the tail of the rubber bands.  Cover the Glue Dot and tail with Scotch tape.

Glue and adhere one of the tabs to the inside of the pyramid. Apply glue to the other tab as shown.

Collapse the figure and adhere the second tab.

Completed Hexagonal Bipyramid

 9. Decagon Pyramid Faces 10 triangles, 1 decagon Edges 20 Vertices 11

Decagon Pyramid

Cut out the decagon pyramid model and bend the tabs as shown in the photo above.

Apply glue to the tab as shown above. Knot a rubber band on one side only.  Lay it on the seam.  Adhere the tab to enclose the rubber band.

Cut and knot a 2.25 inch rubber band. Slide this rubber band onto the circle tab. Add a Glue Dot to the tail of the rubber band.  Cover the Glue Dot and tail with Scotch tape. Repeat for the other end of the rubber band.

Thread the center rubber band through the hole in the decagon. Apply glue to one of the tabs and adhere to the decagon base. Knot the center rubber band so that it is snug. Do not cut the tail of the rubber band.

Apply glue on the second tab.

Collapse the figure and adhere the second tab.

Open and close the figure by tugging on the knot.  Make sure that it opens and closes correctly.  Cut the tail of the rubber band.

Completed Decagon Pyramid

 10. Decagon Bipyramid Faces 20 triangles Edges 30 Vertices 12

Decagon Bipyramid

Cut out the decagon pyramid model and bend the tabs as shown in the photo above.

Apply glue to the circle tab and adhere. Knot a rubber band on one side only.  Lay it on the seam.  Adhere the second circle tab to enclose the rubber band. Thread the center rubber band through the center of the other decagon but do not knot the end.

Cut and knot two 2.25 inch rubber bands. Slide a rubber band onto the circle tab. Add a Glue Dot to the tail of the rubber band.  Cover the Glue Dot and tail with Scotch tape. Repeat to attach all of the rubber bands.

Attach the decagons together by gluing the tab on one side and adhering to the inside of the decagon. Glue the second tab.

Collapse the figure to make sure that the tabs adhere correctly.  Knot the rubber band. Open and close the figure a few times by tugging on the tails. Cut the tail of the rubber band.

Completed Decagon Bipyramid