A STEM Project: The Magical Mathematical World of the Möbius Strip and the Klein Bottle

The world of mathematics can be magical.  Magicians have performed magic tricks which use mathematical concepts such as the Möbius strip to astound audiences and stimulate the imagination.

Afghan Bands

In the 1800's, there was a magic trick called the "Afghan Bands".  A magician took a long strip of paper and glued the ends together to create a loop.  He repeated the procedure for two more strips of paper. He cut each of the three strips down the center of the strip and the results amazed audiences because they were all different.

What did the magician do to make each result different? The answer was, he put a twist in the paper.  The first strip had no twist and the result was two loops.  The second strip had a half twist and the result was a strip that was twice as long as the original loop.  The third strip had two half twists in the strip and the result was two loops that were intertwined.

To recreate this magic trick, cut six 2 inch wide x 11 inch long strips of paper.  Each Afghan Band needs two of these strips.

Afghan Band #1

Tape two strips of paper together and then tape the ends together to form a loop. Cut the loop lengthwise in half. 

 The result is two loops with an inside and an outside surface.

Afghan Band #2

Tape two strips of paper together and then put a half twist on one end of the paper.  Tape the ends together to form a loop. This is a Möbius strip.

Cut the loop in half lengthwise.

The result is one loop which is twice as long as the original with four half twists or two full twists.

Afghan Band #3

Tape two strips of paper together and then put a half twist on one end of the paper.  

Using the same end, put another half twist on this end of the paper. Tape the ends together to form a loop. 

Cut the loop in half.

The result is two loops that are intertwined with two half twists.

Now that you are amazed by the creation of these Afghan Bands.  Who discovered this mathematical intrigue of twisting a piece of paper? It was a mathematician named Johann Benedict Listing at the University of Leipzig in 1858. He was interested in topology which is the study of surfaces. It was a fellow mathematician, August Ferdinand Möbius, who named the strip of paper with one twist, a Möbius strip. The Afghan Band #2 (see above), before it was cut, was a Möbius strip.

There are many variations of cutting a Möbius strip. Try them for yourself by making Möbius strips with different numbers of half twists. Here is a generalization which can be made about the twist variations that are cut in half down the center line.

Given n is a half twist and bisecting a Möbius strip:

If there are an odd number of half twists, the result will be one band with half the width and twice the length, with (2n + 2) half twists.

If there are an even number of half twists, the result will be two bands with half the width, each band will be of equal length, with n half twists.

For further explorations, try cutting Mobius strips into thirds and varying the number of twists  You will be amazed by the results. Compare them to the results that you achieved when the Mobius strips were cut in half down the center line.

An ordinary sheet of paper has two sides and one edge. A Möbius strip has been described as a "strip without a second side". 

Take a strip of paper, give it a half twist, and tape it together to form a loop. If you take a pen and draw a line along the center of the strip, you will see that the line runs along both sides of the loop. Thus giving the appearance that the Möbius strip has just one side.

However, the second side is still there. The half twist in the paper created a pathway for the opposite side to be connected.

If two bunnies are placed at the edge of a clear Möbius strip and one of the bunnies decides to circumnavigate the edge by hopping. Moving the clear strip to simulate the hopping of the pink bunny until the other white bunny becomes visible through the clear strip. 

 The pink bunny will have hopped half way around the clear strip, one circuit, when the white bunny is visible upside down on the opposite side of the clear strip. The pink bunny will need to continue hopping another circuit in order to get back to its original position. 

It takes two circuits to circumnavigate the entire Möbius strip. This concept is a valuable tool in manufacturing because it allows for conservation of resources. Both sides of a conveyor belt can be used with this concept because a Möbius conveyor belt wears evenly on both sides.

If you combine two mirror images of a Möbius strip, you will get a Klein bottle which is a unique vessel which has only one surface. When water is poured into it, the water goes out the same hole.  The Klein bottled was first described by Christian Felix Klein, a German mathematician in 1882.

 Klein bottle image from Wikipedia.

I have recreated this Klein bottle using paper. 

One end of the U-shaped tube flows into the bottle and the other tube flows through the bottle and comes out the bottom.

  Another view with lid retracted.

When my paper model was cut in half, two Möbius strips were produced.

This cross section view of the Klein bottle is similar to mine above.

If you would like to recreate my Klein Bottle, I am including a PDF file.  Print out the file, cut out the pieces and glue them together.

Here is the PDF for my paper Klein bottle.  I used 65 lb. cardstock.
https://drive.google.com/file/d/1Do29_dDjQREzOe8vibK13nDx0edUj5tE/view?usp=sharing

Here is the .Studio file if you have a Silhouette brand cutting machine.
https://drive.google.com/file/d/1XJCl-3k9Ot8gVfuqnOPemcTlDfQNjy-w/view?usp=sharing

Möbius strips can be explored in many unusual ways. Check out this tasty article which transforms a bagel into a Möbius strip http://www.georgehart.com/bagel/bagel.html

Here is another magic trick with a Möbius strip and a loop.

Cut two 2 inch x 11 inch strips of paper.  Tape one into a loop and the other into a Möbius strip.

Tape the two pieces of paper at a right angle to one another as shown above.

Cut the loop down the center.  The result will be the above figure when untangled. Next cut the white strip (see above, the right is a loop too, sorry for the bad picture) down the middle from the one loop to the other loop.

The result is a square.

Möbius strips are magical!

Making a Möbius Strip With Music Tape for the 15 Note Music Box

A Möbius Strip for a 15 Note Music Box

A Möbius strip is a strip of paper with a twist in it.  The strip of paper is twisted and then taped together to form a continuous loop.  After creating 30 songs in my last blog post https://papercraftetc.blogspot.com/2020/05/30-songs-for-15-note-music-box.html, I made a Möbius strip with one of the songs, Mozart Sonata No. 16 in C. The result was a pleasant sounding song when both sides are played continuously.  The song starts and ends with the same note - C, also known as the key, which allows the song to transition easily between sides.  The key of  C specifies the collection of pitches in the song. When playing the opposite side, it inverts the music playing the same pitches but in a different order so it still sounds good on the opposite side.

The reason why this Möbius strip works is because:

    1. The key is the same.
    2. The pitches are the same. 
    3. The tempo and rhythm are the same.
    4. The chord progression is reversed but still sounds good because of the same key of C and use the same pitches.
    5. There is some symmetry in the original score because there is a sequence of notes that repeats itself on the opposite side. 

Mozart Sonata No. 16 being played on a 15 note music box as a Möbius strip

This Möbius strip is the essence of music improvisation which is the art of creating new music from the original score.  When you write music, you start in a chord progression which sounds good. To the human ear, there are certain tension chords and release chords. The music builds tension and then releases the tension. When you reverse the song, the music is still in the same key. The tension and release might be different on the second side but they still sound good because they are in the correct key. When music is played in the correct key, the correct chords need to be played.  Otherwise, there is dissonance.  Dissonance is the lack of harmony in musical notes.  When the music is played in the correct key, there is no dissonance.

To make the Möbius strip, you can use your own cardstock as indicated in my last blog post. It is song #24 in this post, https://papercraftetc.blogspot.com/2020/05/30-songs-for-15-note-music-box.html

I used the paper tape that came with the music box to create this music. In order to use the paper tape, you need a 24 inch mat.

Here is the .Studio file for the 12 x 24 inch mat so that you can use the original paper tape.

https://drive.google.com/file/d/13hITgEXTOdKdnroBxnE6IcFMt7Om9nF1/view?usp=sharing

Tape two 12 x 12 pieces of cardstock together.

Cut out the two strips from my .Studio file using the Cut by Line feature; it is the red line.  Remove the two strips as shown above and adhere the two original paper tapes that comes with the music box to the location of the removed strips (not shown). Cut out the music using the Cut by Line feature; it is the black line. 

To make the Möbius strip, tape the song together to make one strip.  Insert the song into your music box and twist the end.   Take this end and tape it to the start of the song.  The holes need to be punched out where the tape was used or else these notes will not play.

30 Songs for the 15 Note Music Box

The 15 note music box with paper tape music is a musical instrument which produces a melody programmed on paper tape. The paper tape is fed across a comb. The comb is a like a multi-pronged tuning fork made out of steel with fifteen different lengths of teeth.  Each tooth represents a different note.  The longer the tooth, the lower the note or frequency because the tooth vibrates slower. The shorter the tooth, the higher the note or frequency because the tooth vibrates faster. These vibrations create the sound which create the melody. The holes in the paper tape correspond to different notes. When a hole is encountered in the paper tape, the corresponding note is played because there is a prong sticking up which engages the corresponding tooth on the comb.

Music boxes have been around since the 18th century. Key features of music are pitch (fundamental frequency and harmonics) and timbre (amplitude of harmonic frequencies and resonance of specific frequencies). Each tooth vibrates with a fundamental frequency and multiples of other frequencies (harmonics) which generate a spectrum for each note.  The wood and design of the music box cause it to resonate, or amplify, certain frequencies in this spectrum, which generates its unique musical sound.  While new technology on a computer can attempt to simulate this spectrum, music must be played on a physical instrument in order to generate the correct frequency spectrum.

I made 30 songs for the 15 note music box with the Silhouette.  I especially geared it to my grandchildren who enjoy listening to music. I am sure young and old alike would enjoy playing these tunes. This could be a unique gift to give for a birthday, anniversary or wedding.

15 Note Music Box Playing Beauty and the Beast

Here are the songs that I have included in my .Studio file for you to make with your Silhouette:

1. Under the Sea
2. Pop Goes the Weasel
3. Twinkle, Twinkle Little Star
4. Ode to Joy
5. Beethoven Symphony #6
6. Brahms Lullaby
7. Wheels on the Bus
8. Camptown Races
9. Itsy Bitsy Spider
10. You are my Sunshine
11. Happy Birthday
12. Its a Small World
13. Fly Me to the Moon
14. Super Mario Bros.
15.  Heart and Soul
16.  On Top of Old Smoky
17.  Phantom of the Opera
18.  Beauty and the Beast
19.  Pachabel's Canon in D
20.  Star Spangled Banner
21.  Frere Jacques
22.  Bach - Jesu Joy of Man's Desiring
23.  How Much is the Doggie in the Window
24.  Mozart Sonata No. 16 in C
25.  Oh Susanna
26.  Music Box Dancer
27.  London Bridge is Falling Down
28.  Bach - Minuet in G
29.  Puff the Magic Dragon
30.  Blue Danube

The 30 songs were created on the website https://musicboxmaniacs.com There are links on that website to purchase the 15 note hand crank music box on Amazon.  The wooden music box without any paper tape or hole punch is about $15. I chose the wooden box which included paper tape and a hole punch.  The paper tape and hole punch are not necessary to buy since the Silhouette will be cutting the paper.  I bought the paper tape and hole punch to calculate the size of the holes necessary and the width of the paper. The size of the hole is 0.1 in wide and the width of the paper is 1.616 in. Please note, when purchasing the music box, it must be the 15 note music box for the paper tapes that I have created to work.

The https://musicboxmaniacs.com website has a create feature where you can make your own paper tape music.  I explain how to make the paper tape using your Silhouette software in this blog entry.  https://papercraftetc.blogspot.com/2019/11/how-to-make-paper-tape-music-for-15.html

I used 110 lb. /300 gsm cardstock to make the paper tape music. I purchased the cardstock at Michaels.  The music box needs the thicker paper so that the gears can grab onto the paper.  I recommend using a blade depth of 5 and cutting in 2 passes. I do recommend a new blade and cutting mat.  If the holes are not cut accurately, the music will not play correctly or not at all. I discovered this problem with my automatic blade on my Cameo 3.  The holes were not perfectly round because of a dull/misaligned blade.

I used a Silhouette sketch pan to write the name of each song on the paper tape and an arrow to show the direction in which to feed the music.

Many of the songs in this file are multiple strips of paper that need to be taped together.  I used regular Scotch tape.  The holes that are covered in Scotch tape need to be cut.  I used the hole punch that I received with the purchase of my music box to cut the holes through the Scotch tape.

Here is the PDF.
https://drive.google.com/file/d/1_1fFVvnQHGfJCSy9-3IVwD1cuC-WqZF2/view?usp=sharing


Here is the .Studio file.
https://drive.google.com/file/d/1Tu_huN6bdhUq2r4UZZji6i7KLn89XiRX/view?usp=sharing

If you would like to further your collection of paper tape music, I have made 25 songs for Christmas for the music box in this previous blog post. https://papercraftetc.blogspot.com/2019/10/25-songs-for-christmas-unique-advent.html

Easter Bunny Rubber Band Pop-Up with Envelope

The base of this Easter Bunny is a triamond triangular bicupola from from my blog post https://papercraftetc.blogspot.com/2020/02/a-stem-project-rubber-band-polyhedron.html

Using this as the base, I embellished it to create this pop-up bunny.

Rubber Band Easter Bunny Pop-Up

Video of the Rubber Band Easter Bunny Card in action. 

Here is the PDF. I used 65 lb. cardstock and copy paper for the envelope.

https://drive.google.com/file/d/1PwozQZD1e4PNoOjCsrbnCOfhK5vEQs5n/view?usp=sharing

Here it the .Studio file.

https://drive.google.com/file/d/1dTGoCVQ78Ecbv6HaTv-cxMPrP96gwD15/view?usp=sharing

Glue the bunny pieces together as shown above.

Make the triamond triangular bicupola by folding the pieces as shown.  Apply glue to the largest tab with a circle.

Knot a 1/8 inch rubber band in two locations so that the distance is 1.37 inches between the knots. Slide the rubber band into the hole. I used a Glue Dot to adhere the tail to the tab.  I also applied a piece of tape over the rubber and Glue Dot so that the rubber band will not slip out of the hole.

Apply glue to the tabs as shown above. Close the figure and adhere the tabs to the corresponding ones.

Apply glue to the tab with the small circle.  Adhere it to the corresponding tab.

Slide the rubber band into the hole.  Add a Glue dot to the tail and cover it with tape.

Completed base.

Apply glue as shown to the top of the trapezoid. Adhere the bunny head and the basket. Glue the bunny feet to the bottom half of the based the tail to the back of the bunny (not shown).

Make the envelope by folding the sides and gluing the heart tab. Collapse the bunny and make sure that it fits in the envelope.

Completed envelope and rubber band bunny pop-up.

A STEM Project: A Paper Butterfly That Flutters When The Rubber Band Is Stretched

Rubber Band Butterfly

While I was researching the rubber band polygons from the previous postings, I discovered a few YouTube videos that made a similar butterfly model. I liked the rubber band mechanism but thought I could improve the design. I felt that their top wings were too small and the winding mechanism could be simplified.

I made four models with different wings. After you make a few of my butterfly models, I recommend designing your own wings to determine which type of wing flies the best.

Video of the paper butterfly fluttering when the rubber band is stretched and tossed into the air.

The rubber band has no power until the band is stretched. When the butterfly is released into the air, the stored energy in the rubber band causes the butterfly wings to flutter in a circular motion opposite to the direction that the rubber band was stretched. In other words, the rubber band unwinds to return to its normal state.

Video of the Rubber Band Mechanism Being Wound and Released 

Materials needed to make the rubber band butterflies:

-   20 lb. copy paper or construction paper

-  Scotch Tape to adhere the wings to the winding mechanism

-  2 1/2 inch x 1/16 inch rubber band

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

- 20 gauge craft wire

-wire cutters

-1/8 inch dowel

-cellophane (if you are cutting the model with holes in it)

If you are cutting the models with scissors, here is the PDF:
https://drive.google.com/file/d/150AL32bUHfzu8__XONHkh0e1LtHZoBjN/view?usp=sharing

If you are cutting the models with a Silhouette, here is the .Studio file:
https://drive.google.com/file/d/1yRAlXk6A6YAcZFw58PLA_k6ZbWHESkib/view?usp=sharing

If you are cutting the models with a Cricut, here is the SVG:
https://drive.google.com/file/d/1GIg6PcuJHE_0xqD0yQrgwK9JXxJvN4Zg/view?usp=sharing

Cut the butterfly wings. Cut two pieces of 20 gauge wire that are 6 inches and 9 inches long.

Wrap the 9 inch wire around a 1/8 inch dowel as shown above to form a circle. 

Another view of the wire.

Make another circle by wrapping the wire around the dowel.  Look at the photo above for placement of the circle. The entire mechanism needs to be 2 inches long.

Make a figure 8 with the 2 1/2 inch rubber band.

Fold the rubber band in half.

Slide the rubber band onto the wire until it resides inside the circle.

Twist the ends of the wire together. 

Cut the excess wire off.

Bend the opposite circle, without the rubber band, at a right angle.

Fold the 6 inch piece of wire in half and put a twist in the wire as shown above on the right. The loop created should be 3/8 inch long.

Slide the rubber band onto the 6 inch wire. This will create the crossbar wire.

Squeeze the 6 inch wire together and thread this wire through the center of top circle.  Bend this 6 inch crossbar wire so that it is at a right angle to the rubber band.

Close up of the rubber band mechanism.

Wind the rubber band about 35 times to test the winding mechanism.  The crossbar wire should move freely.  

Cut a piece of tape, adhere it to one side the bottom wing. Cut off any excess tape.  Please note: too much tape adds weight to the model.

Wrap the tape around the wire.

Adhere the tape to the back side of the wing.

Repeat taping the other bottom wing to the winding mechanism.

Cut a piece of tape, adhere it to the bottom side the top wing. Cut off any excess tape. 

Wrap the tape around the crossbar wire as shown above and adhere the tape to the back side of the top wing.  Notice that there is a little wire showing.  

Cut this excess wire off.

Repeat taping and adhering the other top wing to the crossbar wire. 

Completed Rubber Band Butterfly

This butterfly model has cellophane glued over the holes.  The holes need to be covered for aerodynamics.