Flexagons & Other Fun Art and Math Connections
Tully Jr. High School Virginia Killmore How to Make a Mobius Strip
1. Using a ruler and pencil Make a 1”strip on a piece of paper. 2. Cut out the strip
3. Give it a half twist (turn one end over). 4. Tape the ends together.
What To do With a Mobius Strip
• After you make your mobius strip, take a pencil and draw a line along the length. How many sides does the mobius strip have? • Take a pair of scissors and cut the mobius strip along the line you have drawn. What happened? • What do you think will happen if you cut it down the middle again? Try it. Flexagon
A flexagon is a polygon folded from paper with a very remarkable quality; you can turn it inside out to make it reveal hidden surfaces.
History
It all started in 1939 when Arthur H. Stone, a twenty-three-year-old student, was cutting American papers to make them fit into an English file. Left over were some strips of paper that he began to fold in different ways. One of the figures that he made caught his attention. He had folded a strip to a hexagon and joined the ends and when he put three corners together the hexagon turned around and showed a hidden side. He had discovered the tri-hexaflexagon.
Arthur suspected that it should be possible to make more complex hexagons with more hidden sides. The next day he managed to make the hexa-hexa-flexagon which has six sides (four hidden). He showed some friends what he had discovered and soon the whole school was making flexagons and many new were discovered.
Supplies • 3 ½” X 22” Strip of oak tag or heavy paper • Ruler • Pencil • Colored pencils or markers • Instructions
Mark the strip in 4” and 2” intervals as shown below and connect the marks to form ten equilateral triangles and a right triangle at each end of the paper 4 4 4 4 4 2
2 4 4 4 4 4 Cut off the right triangles at each end Lightly fold and score at each line Folding instructions
• Fold as follows • Count 3 triangles from the left and fold them to the back (A) • Count 3 more triangles, and fold the other triangle to the front. (B) • Place the end under the beginning and fold the last triangle over the first triangle. • Glue or tape the last triangle in place (do not tape over any edges)
How To flex
• Pinch together two adjacent triangles, pushing the crease up. Then push the opposite side down and into the center • Flex outward and from the center to revel new face • If it doesn't work try the next Triangle Tessellations
Supplies • Pencil • Marker • Scissors • Tape • 2 sheets of Colored paper – Draw a grid with 3”X3” inch squares on one sheet • 3X5 Card – Cut 2inches off creating a 3”X3” template
1. Cut a wavy line from the bottom left corner to the top right corner of the square 2. Slide the left piece to the right side and tape it down 3. Cut a wavy line from the bottom left corner to the bottom right corner of the square 4. Slide bottom piece to the top and tape it down 5. Trace the template to one of the sheets of colored paper and cut them out. 6. Align a shape with the top left grid Square and glue it down 7. Continue to paste the cut outs in a checkerboard pattern on the grid 8. Erase the grid lines and create details to make recognizable shapes.
There is a great website that has an animation of this process. http://www.jimmcneill.com/demo.html Build a Kaleidocycle
• Start with pattern included in kit • Enlarge it as desired I usually enlarge to fit a 12”X18” sheet of paper • Cut out pattern • Trace the pattern by taping it to oak tag board and drawing over lines using a ruler • Cut and score over each line • Fold and glue as instructed on pattern
Great Web Sites to Explore
• Tessellations http://library.thinkquest.org/16661/ http://www.tessellations.org/index.htm
• MC Escher and Math http://www.mathacademy.com/pr/minitext/escher/index.asp
• Flexagons http://www.mathemagic.org/MathsAndArt/flexagon.htm • http://www.mathematische- basteleien.de/flexagons.htm#Flexagons %20in%20the%20Inte rnet
• Geometry http://ccins.camosun.bc.ca/~jbritton/jbrubbergeom.htm
This project would not have been possible without the generosity of countless teachers and the web.