# The MathHappens Butterfly Puzzle

One thing we love to do at MathHappens is make things.  We make conic section and unit circle models, golden ratio calipers, Scott Kim block sets, and fractal playgrounds, to name a few.  And we also make puzzles!  Puzzles are a great way to combine math and play.  Manipulating shapes, creating patterns, finding things that work and don’t work are all part of what we consider creative math play.

We find inspiration for the things we make in many places- textbooks, classes, conferences, museums, history, the recent solar eclipse.  One person who has inspired some of our puzzles is Marjorie Rice.  In 2019 we had a request from the Thinkery in Austin to feature women in STEM.  That research led us to learn more about Marjorie Rice, and the rest is butterfly-puzzle history!

Marjorie Rice (1923-2017) was a self-taught mathematician who had a keen interest in art.  She was an avid fan of Martin Gardener and closely followed his writings in the 1970s on tiling the plane with convex polygons.  It was believed that all possible convex tilings had been discovered by 1967…until a new one was discovered in 1975.  Marjorie Rice was inspired by this new discovery and set out to find more tessellating polygons.  Within two years she had discovered four new tessellating pentagons, which are part of the 15 convex pentagons we know that tile the plane.

Timeline of convex tessellating pentagon discoveries:

• In 1900 David Hilbert included finding all tessellating pentagons as part of Problem 18 in his list of 23 problems in mathematics to be solved in the next 100 years.
• One convex tessellating pentagon—the Cairo tile—has been used in architecture as far back as the 17th century in India.
• German mathematician Karl Reinhardt began classifying tessellating convex pentagons in 1918 and documented the first five types.
• R. B. Kershner found three more in 1968.
• Richard James discovered a ninth type in 1975.
• In 1976-1977 Marjorie Rice discovered another four types.
• Rolf Stein found a 14th tiling in 1985.
• The most recently discovered 15th tiling was found by Casey Mann, Jennifer McLoud and David Von Derau of the University of Washington Bothell in 2015 using a computer to exhaustively search through a large but finite set of possibilities.
• In 2017 Michael Rao proved that these are the only 15 that exist.

Here is a pdf version with images created by MathHappens intern Megan Do.

How does this relate to our butterfly puzzle, you might ask?  When you see shapes tiling a surface, you tend to see and make patterns with the lines and shapes you see.  Have you ever seen a tile floor made up of diamonds that looks like a bunch of cubes?  Or hexagons arranged (perhaps on a quilt) in a way that you see flowers?

At MathHappens, while looking at and experimenting with the 15 tessellating pentagons, we added images of things we saw on to the pentagons.  Two tessellating Pentagon #7s resemble a cat’s face.  #6 resembles an ice cream cone.  Three #14s look like a bird.  Do you see it?  Marjorie Rice drew bees and butterflies on her pentagons, and from those images we created the Butterfly Puzzle!

Our first several versions of the puzzle we made with hand drawn butterflies.  We have produced many many beautiful butterflies, and many people have enjoyed tessellating the butterflies in the puzzle frame.  Recently, we found a way to take intern Stefany Espinoza’s hand-drawn butterflies and etch them onto the pieces themselves.  This makes coloring the puzzle happen more quickly, meaning we can produce the puzzles faster.  The puzzle also works well with just the etch, and this allows an end recipient to color their own butterflies as they wish.  The result is a beautiful and fun way to play with tessellating pentagons!

Though we have always made the puzzle with the “double” (butterfly) piece you see above, recently our Director of  Math Play Christopher Danielson and intern Brynn DeVann have been experimenting with “half butterflies,” which is made of single pentagon #9s.  There is a new element of play going on now!