## Origami Holiday Decorations

After a hard work’s term, members of the Math community took a break before the exam week to make some paper holiday decorations. Reading day is meant to be a day for studying and preparing for the final exams. However, it is also a chance to take a deep breath and relax to be ready for the last stretch of the term.

Many opportunities are provided across campus for students to relax. Here at the Department of Mathematics BAMM offered a holiday origami workshop. Students and staff folded paper Christmas wreaths and stars that were then hanged for decoration on the Department’s tree.

## The Mathematics of Origami

Last week, the Department of Mathematics, through the outreach initiative, had a guest for the Recreational Mathematics Seminar. Laura Jimenez is a mathematician and a PhD candidate at the Department of Ecology and Evolutionary Biology at the University of Kansas. In college, she began to develop a passion for origami, and she was able to tie it with mathematics. When she was a master’s student, she worked in mathematical outreach and popularization and gave workshops about the mathematics behind origami.

At the Recreational Mathematics Seminar on Friday, November 15, Laura talked about how not only math is used in proving whether something is foldable or not and finding the folding pattern, but also origami can be a powerful tool for solving mathematical problems, such as solving a third-degree equation.

During the talk, the speaker presented the Huzita axioms, the mathematical principles of paper folding. The axioms list the seven operations that can be achieved by folding paper. The first axiom reads “Given two distinct points p1 and p2, there is a unique fold that passes through both of them.” The second one goes about placing a point into another; in this case the fold created turns out to be the perpendicular bisector to the segment joining the two points. Later on, Laura showed how one can trisect an acute angle by folding paper, in other words, by following the Huzita axioms.

The seminar was followed by a workshop where attendants learned how to fold a cube and a 12-pointed star of modular origami. Some would say that this type of origami is the most mathematical one. Modular origami pieces are made up by several sheets of paper. Each sheet is folded in the same manner, creating a unit with flaps and pockets. Then all pieces are put together by inserting flaps into pockets. This type of origami allows us to build geometrical objects such as the platonic solids.

Laura emphasized how origami is a great tool for teaching geometry.

On Saturday, our guest also ran our Girls Exploring Math Monthly workshop. Students explored the properties of buckyballs, polyhedra with regular pentagons and hexagons as faces. They analyzed the number of vertices, edges, and faces of each type, and came up with a relation between them. When building the origami model of a dodecahedron, the girls also looked at its graph representation and found a Hamiltonian path on it. Then, they used it to decide how to assemble the origami modules so that all the edges connected in every vertex had different colors (using modules of three different colors – it’s not as easy as you might think!).

The Recreational Math Seminar is gaining presence within the Department’s community, showing the most entertaining side of math. Recreational Math is a great way of getting undergraduate students interested in research. We will continue to bring more guests next term.

## Building Real Mathematical Surfaces

This week, Buckeye Aha! Math Moments had a series of events around the work of a special guest. Maria Garcia Monera, from the University of Valencia (Spain), is interested in Topology and Geometry and in designing and building paper models of surfaces.

Even from last week we started preparing for Maria’s visit. We built some of her models and displayed them in the Mathematics Tower lobby for everybody to see. Then on Monday, Maria gave the Recreational Mathematics Seminar. She explained that he technique she uses is based on the work of Felix Klein, Alexander Von Brill, and John Sharp. In an era before computers, where one could render the surfaces on a screen, the motivation for creating these models was aiding students in the visualization of the mathematical objects. The idea behind is finding plane curves contained on surfaces that can be cut in paper and used to reproduce the model.

The seminar was well attended by postdocs, graduate and undergraduate students, and lecturers, becoming the most popular Recreational Mathematics Seminar so far.

On Tuesday, Maria gave a workshop at the Columbus Public Library (Northside Branch) were some children built an orange paper sphere and decorated it as a pumpkin. Then on Wednesday, during Teas, those who attended the seminar (and a few others) put in practice what they had learned, building paper surfaces such as paraboloids, water drops, ellipsoids, and hearts.

Finally, on Thursday Maria worked with a 6th and 7th grade math class at Metro Middle School. She talked about surfaces of revolution and then the children worked in teams to craft spherical pumpkins and hearts.

Recreational Mathematics is about fun and entertainment, but that doesn’t mean there aren’t serious mathematics involved. As the Mathematical Association of America writes “Recreational mathematics is inspired by deep ideas that are hidden in puzzles, games, and other forms of play”. With Maria’s visit, the Department’s community had fun and obtained a tool for helping students visualize surfaces. Hopefully, some were so inspired that they will design their own paper model.