Certain contemporary problems in geometry have their roots in mathematics covered in elementary school and their further development can be understood gradually throughout high school and university mathematics. One such category of problems is related to the dissection of polygons. Geometric dissection of planar figures is introduced in elementary school when one computes the area of a polygon by cutting it into pieces with disjoint interiors. The method works due to the fact that the area is preserved under geometric dissection. Indeed, the famous Bolyai-Gerwin theorem states that any two polygons with the same area can be cut into polygons and rearranged to form the other. This statement does not hold in three dimensions however. Hilbert's third problem asks for an example of two tetrahedra of the same volume that cannot be cut into tetrahedral pieces that rearrange into the other. The example was given by Hilbert's student, Dehn. Understanding the set-theoretic dissection, in which figures are cut into entirely disjoint pieces, requires higher mathematics and yields surprising results such as the Banach-Tarski paradox that a solid ball B of any size can be finitely dissected and rearranged to form two balls each congruent to B. In this workshop, we will take an inquiry-based hands-on approach to investigate the properties of geometric dissections of polygons, and will prove the Bolyai-Gerwin theorem only assuming high school algebra and geometry. We will also suggest inquiry-based activities to investigate other dissection related problems. Our approach can be used by educators to develop extra curricular materials and hopefully inspire artists to visualize some of these well-known yet intriguing results. “ Audience will need paper, a ruler and a pair of scissors to fully participate in the workshop activities.

We explore how to create hands-on math exploration activities with just sheets of graph paper. We'll try and create some new ones together! Our goal is to understand how to create low-floor, high ceiling, low equipment active learning activities that can be used in a variety of settings, from math circles, to math teacher circles, to undergraduate and even graduate courses. For example, start with the question "what are the possible areas of squares whose vertices are at the corners?" And you get to modular arithmetic, discussions of sums of two squares, etc. You can also discuss various "dynamical systems" starting with a shape and generating new shapes by certain rules, and explore limits/evolution of these procedures. Recommended materials: Graph paper and a pen, or a tablet with graph paper

Modular Origami is the art of constructing shapes, usually polyhedral ones, by putting together easy-to-fold modules or units that securely lock together without cuts or glue. The art enables construction of interesting and complex structures using simple units which look rather uninteresting just by themselves. During the workshop, we will make a simple six-unit modular origami cube with patterns on the faces. We will then discuss other possible constructions with the same kinds of units. If time permits, we can even construct a three-unit hexahedron. However, we will not have time to construct the other possible polyhedral structures during workshop hour, though you will leave with enough basic knowledge and resources to construct on your own at home. Recommended materials: 6 sheets of 4-6” squares of origami paper, preferably in three colors, two sheets of each. Also, have at least about a dozen extra such sheets handy for learning about the additional possible structures. Note that regular origami paper is colored on one side and white on the other. You could use any other paper that has different colors on either side, as long as you can cut them yourselves into precise squares before the workshop begins.

The sixth Huzita-Justin origami axiom, first discovered by Italian mathematician Margherita Beloch, allows for geometric constructions not possible with a compass and straightedge. Some of the problems that stumped early geometers but are solvable with this move include trisecting the angle, doubling the cube, solving cubic equations, and constructing regular heptagons. Utilizing origami, we will demonstrate how this so-called Beloch move is equivalent to finding the mutual tangent to two parabolas and unlock the mystery of a sevenfold Islamic pattern. Recommended materials: Plain printer paper and pencil/pen

The CBMS statement "Active Learning in Post-Secondary Mathematics Education" highlights the importance of "classroom practices that engage students in activities." Hands-on geometry can help students make connections when kinematic and visual activities are linked to visual processing and to mathematics. We'll share explorations we have used in classes ranging from introduction to mathematics, a general education course, to classes on geometry and differential geometry aimed at mathematics majors, including future teachers. Some examples include walking or driving an angle sum, stringing the Pythagorean theorem, and surfing a TNB frame. We use physical models and web-based GeoGebra IGS explorations and p5.js experiences. Participants will have access to the ways we use these in the classroom, including related worksheets and the interactive activities themselves. We’ll also discuss student reactions.
Additional resources

It is vital for preservice teachers to experience innovative and creative mathematics learning and teaching if they are to be successful in their future classrooms. The Geometry as Arts project was a vehicle to explore some mathematical ideas in an elementary preservice teachers’ classroom. This presentation is a qualitative probing slice of a longitudinal study concerned with pre-service teacher beliefs, values, and attitudes toward kinds of pedagogical approaches that engage mathematics learners and lead to desirable outcomes.

In this lecture, I will discuss the process of creating, implementing, and accessing the impact of an Indigenous beadwork assignment in a second-year undergraduate linear algebra course at my university. Emphasis is placed on the process behind the project, including the motivation, context, and relationship building, and I will report our findings. This is joint work with Cathy Mattes.

The medium of mathematical animated GIFs provides an inspiring and easily accessible playground for exploring geometry creatively. Throughout Fall 2020 I wrote programs to draw a series of looping animations I called "Swirly Squares", based on black and white checkerboards. It was a personal exploration of geometric ideas for producing closed loops. At the suggestion of an online friend, I opened my personal project to others, inviting participants to send me short animated segments that I would then stitch together into one long animation I called the "Swirled Series". In this talk I will show some of my personal Swirly Square loops and selections from the Swirled Series. I will also talk about tools for creating similar animations, which should be accessible to anyone with basic programming skills.

We give a visual introduction to several geometric techniques for analyzing data. These include both unsupervised learning (clustering, dimensionality reduction, topic modeling), and supervised learning (k-nearest neighbors, support vector machines), though we don't expect you to know what any of those words mean! The goal is to distill the methods down to visual and oral description without mathematical notation. The performance of data analysis techniques will be illustrated on real-world image and text datasets. Mini-course participants will be encouraged to develop their own purely visual explanations

Participants will be enabled to circulate at will among the posters with each presenter in a virtual breakout rooms.

Poster Sessions

Presenter: Katayoun Movaseghi
https://ubc.zoom.us/j/62694839224

Title: Reducing the 2-Body Problem Using Symplectic Geometry and Its Hamiltonian

Abstract

Context: The Hamiltonian function, H, of a natural physical system, when thought of as a vector field, enriches the study of mechanics and calls for a more formal mathematical approach - symplectic geometry. We are concerned with the Hamiltonian system (M, ω, H) consisting of the phase space (a symplectic manifold), the symplectic form, and the Hamiltonian function, and how symmetries preserve these symplectic objects. This exposition on the basics of symplectic geometry introduces the 2-body problem from celestial mechanics and exploits its symmetries to reduce the number of coordinates in the system from 12 coordinates to 2.

Presenter: Max Hoffman
https://ubc.zoom.us/j/62420316164

Title: Geometry for Student Teachers

Abstract

Context: The poster presents a design research project from the field of higher mathematics education. The aim is the conception of a plane-geometry course for student teachers that on the one hand builds strongly on the skills and knowledge acquired in other courses like Analysis and Linear Algebra. On the other hand, the course is designed to be closely related to the teaching profession. The latter plays an important role in overcoming the second discontinuity in teacher education.

To achieve this goal, we think two innovations to be crucial: First, the use of an instructive and performant axiom system for plane geometry with only two axioms based on the real numbers and metric spaces (plus the parallel axiom). Second, the implementation of an obligatory ePortfolio for the whole semester. The focus of the ePortfolio is on the students’ reflection on connections between the course and their future professional work as school teachers.

The poster will illustrate both, the axiom system and its benefits as well as examples for profession oriented learning activities.

This hands-on workshop will explore the Wurzelschnecke ("root snail"), a simple and elegant spiral construction based on right angle triangles, first attributed to Plato's tutor Theodorus of Cyrene in the 5th C BCE and still studied by mathematicians in our time. We will make and view versions of the Wurzelschnecke at a variety of scales and materials, and play with its possibilities in embodied geometric design in architecture, playground equipment, jewelry, mathematical millinery and more. Our focus will be on its potential use in education, supporting understanding of irrational numbers and trigonometry, and in the geometry of design. I'll share examples from earlier workshops with middle school and high school students, math teachers, grad students and faculty. Workshop participants should have the following materials at hand, if possible: paper, pencil or pen, ruler or straightedge with a square corner, scissors, corrugated cardboard. A protractor and/or carpenter's square are optional but might be helpful.

In this session we will play with several surprising ways of exploring circles — and their properties — using our bodies. This interactive session begins by looking at what happens when we rotate our limbs in very simple movements, and progresses to examining swirling movements popular among contemporary dancers and choreographers. We will explore whole-body circular activities easily done in a very small space and will apply these actions to create movement sequences with the ultimate mathematical prop — an ordinary sheet of paper. Then, learn how it all connects to the curious algebra of quaternions, and see how comprehending the embodiment of the quaternions helps us better understand both the mathematics and the relevant movement arts. No dance experience necessary! Materials needed: • Several sheets of ordinary printer paper • 5 ft/ by 5 ft. area in which to move (non-carpeted area preferred) • A belt and two ordinary (long) socks Karl Schaffer is a dancer and choreographer who co-directs the dance company MoveSpeakSpin, and a math professor at De Anza College.

We begin with a blank sheet of paper and step by step draw layers of lines and circles using a compass and straightedge. From this underlying grid, we can derive a series of related patterns. These are patterns that would have been drawn in the same way across the Islamic Lands for many centuries. They have adorned architecture and objects and illuminated Manuscripts from Spain across to China from the 8th century till now. To begin the session Samira will outline the process, tools and potential outcomes. She will then guide you through drawing a single unit of a repeating pattern step by step so participants may draw along at the same time. Samira will then demonstrate how to draw a slightly more complex pattern derived from the same underlying grid. As this will be at a slightly quicker pace, participants will be sent a PDF step by step guide to aid them to continue drawing a range of related patterns after the session. Materials required: A4 plain paper x 2, a compass, a sharp pencil, a ruler, fineliner pens, sharpener, eraser, colouring pens & pencils. You may already have these tools, but if you wish to buy from Samira’s recommendations, please review the source list of equipment, here: https://www.samiramian.uk/equipment.

In my Introduction I will refer to Sixfold patterns
#42 Islamic Geometric Patterns on an Isometric Grid

I will then go through the construction in this PDF, but will also refer to the following:

Weaving/interlacing
TECHNIQUES #1 - Four Ways to Weave - How to draw Islamic Geometry

Constructing a tiling larger than 2 by 2
#16 - 3 x 3 square tiling - How to draw Islamic Geometry
#17 - 4 x 4 square tiling - How to draw Islamic Geometry
#18 - 5 x 5 square tiling - How to draw Islamic Geometry

The intersection of geometry and art is about as enticing a rabbit hole as Alice could find. More than beauty will draw her in – the symmetries and structure, the sensibilities and values captured in spatial-visual representations can give insight into important mathematical ideas and relationships. Within this intersection is a wonderland for Alice to explore, led by her own curiosities toward new understandings. Many of us have wandered as Alice the student, the teacher, or the mathematician. In this workshop, we will “shift GEAR”, and consider Alice the education researcher and where to her curiosities might lead. Guided by workshop participants’ own wanderings as students, teachers, or mathematicians, we will explore questions such as: How can we reframe curiosities about geometry and art as research questions in education (GARE)? What questions are answerable via education research, and how can this inform teaching / learning in geometry and art (ERGA)? And, what research insights can be gained from collaborations in art, geometry, and education (RAGE)?

The sixth Huzita-Justin origami axiom, first discovered by Italian mathematician Margherita Beloch, allows for geometric constructions not possible with a compass and straightedge. Some of the problems that stumped early geometers but are solvable with this move include trisecting the angle, doubling the cube, solving cubic equations, and constructing regular heptagons. Utilizing origami, we will demonstrate how this so-called Beloch move is equivalent to finding the mutual tangent to two parabolas and unlock the mystery of a sevenfold Islamic pattern. Recommended materials: Plain printer paper and pencil/pen

Pop-up books and cards have been around since the 18th century, and recently have seen a surge in popularity through the elaborate designs of pop-up masters like Robert Sabuda and Matthew Reinhart. But the most stunning and elegant pop-up effect I have encountered is the Pop-Up Spinner card invented by an anonymous Japanese student. How it functions turns out to depend on a geometric theorem concerning linkages proven in an undergraduate thesis. I will explain the connection and prove the theorem.

Strictly speaking, a wallpaper pattern lives in the plane. When we attempt to weave a wallpaper pattern from bands, as in the image shown, the result is no longer a planar object. The symmetry group of this pattern is one of the eighty layer groups. The main point of this presentation is to familiarize the audience with these less well-known groups and highlight their importance in the study of symmetry. For many of the wallpaper groups, there are two ways to extend the action on the plane to an action on space that preserve the base plane. For instance, in the example, the mirror symmetry in the wallpaper group becomes a flip symmetry of the woven pattern in space. This leads to a rich set of examples, which we find beautiful as well as mathematically interesting.

In this presentation, I will give an overview of the Ubiratan D'Ambrosio's concept of ethnomathematics and Elder Albert Marshal's concept of "two-eye seeing." I will address some of the dynamics between these two concepts and illustrate them with two examples. The first example highlights geometry evident in a traditional Haida hat currently on display at the SFU Museum of Anthropology. The second example draws from the work of contemporary Salish artist Dylan Thomas.

Thurston's geometrization conjecture, proved by Perelman in 2003, states that every three-dimensional manifold can be cut into a collection of pieces, each of which has one of eight geometries. These "Thurston geometries" include the euclidean, spherical, and hyperbolic geometries, the products of the two-dimensional spherical and hyperbolic geometries with the euclidean line, and three other, stranger geometries. In this talk, I'll describe joint work with Rémi Coulon, Sabetta Matsumoto, and Steve Trettel, in which we simulate the "inside view" from within each of the Thurston geometries. That is we generate images assuming that light travels along geodesics in the geometry of the manifold. Many of our animations and interactive web-apps are available on our website http://www.3-dimensional.space/

We give a visual introduction to several geometric techniques for analyzing data. These include both unsupervised learning (clustering, dimensionality reduction, topic modeling), and supervised learning (k-nearest neighbors, support vector machines), though we don't expect you to know what any of those words mean! The goal is to distill the methods down to visual and oral description without mathematical notation. The performance of data analysis techniques will be illustrated on real-world image and text datasets. Mini-course participants will be encouraged to develop their own purely visual explanations.