Assignment #3 - A Representation of Bell Ringing

Change ringing is a musical art form that emerged in England in the 1600s. It involves ringing a set of bells (tower bells or handbells) one at a time to cycle through different possible orderings. Some key terminology helps in discussing change ringing: a round is a sequence of bells in highest-to-lowest order; a plain change is a technique in which one bell is swapped with its neighbour; and a cross-change involves swapping multiple bells in a single move. For our project, we chose to visualize a plain change: a sequence of changes involving a cross-change followed by a plain change, beginning and ending with a round.

Specifically, we represented a plain hunt on six—a pattern on six bells defined as the course followed by a single bell as it moves from first position to last and back again (Merriam-Webster, 2025). Each 4×4 block of colour in the artwork represents one bell being rung. One thing to notice is that the first and last rows are identical: both are rounds. The artwork also makes the plain changes and cross-changes visible, as only neighbouring colours ever swap.

Another feature worth observing is the surface beading. Each line of beads of the same type and colour traces the path of a bell throughout the plain hunt. The over-under pattern of these beaded strands forms a braid—another elegant way of visualizing the structure of a plain hunt on six bells.

Plain hunt on six bells is not a full extent, meaning it does not cycle through all 720 permutations of six elements. A fun historical fact is that a full extent on eight bells has been rung. Polster and Ross (2007) describe one such performance at the Loughborough Bell Foundry in the U.K., which began at 6:52 a.m. on 27 July 1963 and ended at 12:50 a.m. on 28 July after 17 hours and 58 minutes of continuous ringing.

Change ringing is still alive today. It is practiced in English-speaking countries—Canada, England, Australia, and New Zealand—reminding us that historical traditions are continuously reinterpreted and that music and mathematics have long been interdependent.

References

Arthur T. White. (1983). Ringing the changes. Mathematical Proceedings of the Cambridge Philosophical Society, 94(2), 203–215. Cambridge University Press & Assessment

Gresham College. (2021, January 5). The mathematics of bell ringing [Lecture]. https://www.gresham.ac.uk/watch-now/maths-bellringing Gresham College

Jongrsde, et al. (2015). The mathematics of change ringing. (Bachelor’s thesis / project). Leiden University. Leiden University Math Publications

Nelson, A. C. (2020). The mathematics of bell ringing [Colloquium presentation]. https://annacnelson.github.io/assets/pdf/MathematicsOfBellRinging_2020Talk.pdf

Polster, B., & Ross, M. (2007, October 15). Ringing the changes. The Age / QEDcat. https://www.qedcat.com/archive_cleaned/ringing.html

Polster, B., & Ross, M. (2007). Mathematical Impressions: Change Ringing. Simons Foundation.Ringing Systems Web. (n.d.). Plain Bob Minor – Methods – Blueline. https://rsw.me.uk/blueline/methods/view/Plain_Bob_Minor

Assignment #3

Topic: Crocheting the Permutation of Bells in Change Ringing

General idea: Visualizing the mathematics of bell-ringing through color-coded rows



Goal: Assign colours for the six bells in a Plain Bob Minor and crochet the permutations in rows (approximately 60-120)


Example:

Colour Legend: 1=Red, 2=Orange, 3=Yellow, 4=Green, 5=Blue, 6=Purple

Row 1: 1 2 3 4 5 6

sc in: Red, Orange, Yellow, Green, Blue, Purple

Row 2: 2 1 4 3 6 5

sc in: Orange, Red, Green, Yellow, Purple, Blue

Row 3: 2 4 1 5 3 6

sc in: Orange, Green, Red, Blue, Yellow, Purple

Row 4: 4 2 5 1 6 3

sc in: Green, Orange, Blue, Red, Purple, Yellow

Row 5: 4 5 2 6 1 3

sc in: Green, Blue, Orange, Purple, Red, Yellow



References:

Arthur T. White. (1983). Ringing the changes. Mathematical Proceedings of the Cambridge Philosophical Society, 94(2), 203–215. Cambridge University Press & Assessment

Gresham College. (2021, January 5). The mathematics of bell ringing [Lecture]. https://www.gresham.ac.uk/watch-now/maths-bellringing Gresham College

Jongrsde, et al. (2015). The mathematics of change ringing. (Bachelor’s thesis / project). Leiden University. Leiden University Math Publications

Nelson, A. (2020, February 25). The mathematics of bell ringing [Conference presentation]. GSAC Colloquium. https://annacnelson.github.io/assets/pdf/MathematicsOfBellRinging_2020Talk.pdf Anna C. Nelson

Polster, B., & Ross, M. (2007, October 15). Ringing the changes. The Age / QEDcat. QED Cat+1

Polster, B., & Ross, M. (2007). Mathematical Impressions: Change Ringing. Simons Foundation.Ringing Systems Web. (n.d.). Plain Bob Minor – Methods – Blueline. https://rsw.me.uk/blueline/methods/view/Plain_Bob_Minor

Numbers With Personality

In reading Alice Major’s (2017) delightful paper, I particularly enjoyed the personification of numbers, which imbued them with charming personalities. On page 3, the findings from a study showed that participants perceived numbers ending in 3, 7, and 9 as "less good." This immediately reminded me of a chapter from The Number Devil: A Mathematical Adventure by Hans Magnus Enzensberger. In chapter 3, certain numbers are humorously and dramatically called "primadonnas" in a playful story between a devil and a child.

The concept of personifying numbers—assigning them human-like traits—breaks mathematics out of its typical rigorous, objective, and somewhat disconnected perception. I realized that I employ a similar approach in my own teaching. I often "antagonize" irrational numbers, long decimals, and large numbers that are cumbersome or difficult to factor. I might say, “I don’t like this number; it’s too much for me!” to encourage students to convert unwieldy decimals into more manageable, “prettier” fractions. 

Yet, as I reflect on this and write this post, I wonder if I’m being entirely fair. In striving for equity, perhaps these “othered” numbers deserve more respect and gentle attention rather than playful exile. Maybe, like students themselves, they have quirks and challenges that merit curiosity, patience, and a little kindness—reminding me that even in mathematics, empathy can have a place. 

I later paused on page 5 at the paragraph that begins, “However, it seems that we start learning mathematical skill using a more semantic scaffolding…frontal cortex to the parietal areas.” This section highlights how both linguistic and non-linguistic abilities are linked to supporting the understanding of spatial relationships and the organization of collections of items. A common mathematical norm is its strict presentation and reliance on widely accepted symbols—often Western-dominated—leaving little room for other forms of mathematical knowing. Yet much of mathematics is deeply grounded in language, context, and tangible reasoning. I can see how leveraging linguistic scaffolding and being mindful of the funds of knowledge that my students bring, especially linguistically, provides an opportunity to make abstract concepts more accessible. It also reminds me that storytelling, creativity in art and poetry, and even humor can serve as cognitive tools to help students (and even ourselves) make sense of complex mathematical ideas.

Learning About Persian History

Watching Understanding Iran: What Ancient Persia Reveals about Modern Conflict and stopping at the part (31:40) discussing the Shah’s growing alignment with the West made me reflect on how colonialism and Western influence often shape internal divisions within nations. Ancient Persia was once a powerful empire described in the video as known for its tolerance and ability to integrate diverse religions and cultures. Yet in modern times, the introduction of Western ideologies and values was not met with the same openness. This contrast highlights how colonialism even when indirect can disrupt a nation’s sense of identity and autonomy.

The Shah's Westernization effects were perceived as a dismissal of Persian and Islamic values. It shows how modernization imposed through foreign influence can create tension between progress and preservation. In this way, I learned that the conflict in Iran wasn’t just political but also deeply cultural- a struggle between maintaining heritage and adapting to global power dynamics shaped by centuries of colonial dominance.

Later, at 34:50, the speaker mentioned that East Vahan was once so safe that it didn’t even have a prison, and the Shah could walk through the Bazaar without bodyguards. I found that image really striking — I had never even imagined a time or place where prisons or security for political figures weren’t necessary. It made me think about how much trust and unity must have existed within that society. That sense of peace and stability feels almost unimaginable today. This moment also stood in sharp contrast to the unrest that came later. To me, it represented a period when Iranian society seemed more confident in its identity and less threatened by outside influence. Seeing how fear, division, and violence eventually took hold made me realize how deeply colonial and cultural pressures can disrupt a nation from within — not just politically, but emotionally and socially as well.

Finally, I stopped at 39:11 as the speaker was showing these gorgeous blue tiles and describing how Qur’an verses are harmoniously interwoven with patterns from nature. I don’t know very much about the Qur’an, so I found it fascinating to learn how deeply nature and spirituality are connected in Persian art and architecture. The idea that gardens were seen as reflections of paradise and connections to God really stood out to me - it showed how beauty, faith, and daily life were intertwined in a way that felt peaceful and balanced. Seeing how these artistic traditions have lasted for centuries, even through conflict and foreign influence, made me appreciate how strong cultural identity can be. It reminded me that despite the pressures of modernization and colonial influence, Iran’s artistic and spiritual roots continue to express a deep sense of harmony between humanity, nature, and faith.

Overall, I’ve truly learned a lot from this video and gained a deeper understanding of Iran. It made me realize how deeply history, culture, and faith can shape a nation’s identity. Despite Western and settler influences or periods of conflict, Iran’s enduring artistry and spirituality show a beautiful resilience that continues to define its people and their eternal connection to heritage.

Reflecting on Euclid and Mathematical Beauty

When I took a Euclidean or classical geometry course as an undergraduate, we examined Euclid’s axioms in depth. At that time, it struck me that Euclid was truly the forefather of our modern mathematical system- grounding it in truths I had never previously thought to question. What fascinated me most, and what I later came to appreciate even more through reading about The Elements from the St. Andrew’s site, was how carefully Euclid structured his work- from definitions to postulates to logical axioms. This axiomatic method, where each proposition builds upon the previous in a clear deductive sequence, was my first real encounter with a historical mathematical piece that was a systematic presentation of reasoning itself.

I often think about Euclid’s continuing relevance to our current mathematical framework. Euclidean geometry models the geometry of flat space- the geometry that governs the physical world we interact with daily. In fields like architecture and construction, whether building furniture or entire structures, Euclid’s axioms form the invisible foundation for all spatial reasoning. Although non-Euclidean geometries expand beyond these limits, most of my students will only ever work within the Euclidean framework, since it remains central to the curricular content and to how we understand space in practice.

Something that really stayed with me from our class discussions was learning that Western colonizers and settlers viewed The Elements as second only to the Bible in importance. That was new to me, and it made me wonder whether they revered it not only for its content but also for its systematic way of thinking and its pedagogical value. In many ways, Euclid’s work provided a model for what knowledge and reasoning should look like- structured, logical, and ordered- a mindset that profoundly shaped Western education and, through colonial expansion, much of the modern world.

Whenever I think about the term mathematical beauty, I’m reminded of Francis Su’s (2020) Mathematics for Human Flourishing. Su writes about beauty as a human virtue and explains that the first and most accessible kind of mathematical beauty is sensory beauty. In my previous course (EDCP 553) with Susan, we explored embodied mathematics, which I see as a form of sensory beauty itself. Sensory beauty is the kind we experience with our senses- the intricate fractal patterns in cauliflower, the symmetry of Islamic art, or the luminous stained-glass windows of Notre-Dame in Paris (as in the photo attached that I took this summer). These common notions in Euclidean mathematics seem almost simple, so familiar that I’ve often taken them for granted. Yet it wasn’t until I encountered the “counter-universe” of non-Euclidean geometry that my own mathematical universe felt as though it had shattered. And still, it is precisely through the existence- and the limitations- of Euclidean geometry that mathematicians were able to explore and discover the beauty of the non-Euclidean world. Ultimately, Euclid's enduring legacy reminds me that mathematics is not just a study of numbers or shapes, but of how humans seek order, meaning, and beauty in the world around them.

Reflecting on Solving Mathematical Problems with Ancient Ways

 Throughout the course, it has been both exciting and challenging to solve typical mathematical problems- such as long division- using Babylonian and Egyptian methods. It was especially rewarding to apply Egyptian multiplication and decipher Babylonian symbols, exploring how these techniques were used by ancient civilizations.

Initially, I was a bit apprehensive about this assignment. I knew my instinct would be to approach the problem using modern algebraic notation and formulas that are deeply rooted in my mathematical and geometric foundations. However, as my group began working through the pyramid task- starting with part (a), where we had to draw four spatial diagonals- we realized we could use a method of dissection to create six congruent, square-based, perpendicular pyramids.

We began by discussing the volume of the cube, which naturally led us into a back-and-forth conversation about how that related to the volume of a pyramid. Admittedly, we already knew the "end goal"—the formula for the volume of a pyramid- so that gave us some guidance as we worked backward from the cube to derive the relationship.

In part (b), Yuki drew the oblique pyramids while Shannon created impressive 3D models, which greatly enhanced my understanding. We noticed that the method of “zeroing” the base was a recurring theme in the process, and we became curious about how and why the Egyptians arrived at this concept. An earlier group- Ross, Danielle, and Jayden- had derived the formula for the volume of a truncated pyramid using similar quadratic reasoning, and their work closely mirrored the thinking we used for the full pyramid volume. Their presentation helped deepen my understanding and sparked thoughts about further mathematical extensions, including calculus-based integration and the role of “b” as the top edge in the pyramid’s triangular cross-section.

After their presentation, I really wanted to understand why, in the full pyramid volume formula, the “b” value is simply zeroed out. The textbook explained that in a full pyramid, the top surface area present in a truncated pyramid no longer exists, which seemed to hint at the concept of a limit. As a result, the “b” value is omitted in the full pyramid formula.

Learning from other groups' presentations- especially the geometric dissection problems with Ross, Danielle, and Jayden- was incredibly engaging. Danielle’s 3D model and Shannon’s 3D visualizations stood out to me, making it clear how effective physical models can be in supporting geometric understanding. This experience has made me realize how much I want to incorporate 3D models into my own teaching of volume and surface area. It reinforced the value of hands-on manipulatives and how they can make abstract geometric concepts more tangible and meaningful.

Doing Ancient Problems in Ancient and Modern Ways (Problem 1.2.2, 5000 Years of Geometry – Scriba & Schreiber)

Assignment #1: 

https://docs.google.com/document/d/1WQ4RMwfD78LTYxXzCRlJ4eo0bt2xEsXlwenGioyZuyE/edit?usp=sharing