Measuring Area without Integrating

For our art project related to art history. Ross and I are researching the history of the planimeter. 

We decided that Ross will attempt to create a polar planimeter whilst I try to create the hatchet variant. Admittedly the hatchet planimeter is quite simple in design and as such, I will also be helping Ross ideate some of the mechanisms for his portion.

 Alternatively, a more specialist type of planimeter we would be interested in recreating would be the Adisco Area Measurer (Area Meter). This one is more approachable for secondary students as this one approximates areas with curved rectangles, similar to a Riemann sum, while the other ones utilize Stokes' theorem.

Here are some of the sources I've used:

Eckel, L. H. (1953). The hatchet planimeter. The Forestry Chronicle, 29(3), 273–277. https://doi.org/10.5558/tfc29273-3

Foote, R. L., & Sandifer, E. (2004, January). Area Without Integration: Make Your Own Planimeter. https://persweb.wabash.edu/facstaff/footer/papers/foote.sandifer.reprint.pdf

Green, D. R. (2025, January 15). The hatchet planimeter. Uraone.Com. https://www.uraone.com/PlanimeterVault/hatchet/

How planimeters work. (n.d.). Retrieved December 5, 2025, from https://persweb.wabash.edu/facstaff/footer/planimeter/HowPlanimetersWork.htm

Powerhouse Collection—Adisco area measurer. (n.d.). Retrieved December 5, 2025, from https://collection.powerhouse.com.au/object/383315

Prytz planimeters. (n.d.). Retrieved December 5, 2025, from https://persweb.wabash.edu/facstaff/footer/planimeter/Prytz/Prytz.htm

Staecker, C. (2022, November 22). Lost area-measuring tool. YouTube. https://www.youtube.com/watch?si=JYlrmUObsW5hj-5R&v=xSgf_wxJIlk&feature=youtu.be

Tinker projects. (n.d.). Retrieved December 5, 2025, from https://tinkerprojects.angelfire.com/planimeter.html

 

Numbers with Personality

The first stop for me was Myron's discussion of the Mayan head-variant glyphs. I had never encountered a numeral system where each number had a full portrait, complete with age, hairstyle, or animal markings. Major points out that these faces were not practically necessary, since the Maya already had a perfectly functional bar-and-dot notation. That stopped me because it forced me to question an assumption I hadn’t realized I hold: that number systems evolve toward efficiency. Here, efficiency was never the point. The portraits existed because numbers carried cultural, spiritual, and aesthetic significance beyond quantity. This made me reconsider how much of mathematics is shaped by human meaning-making rather than pure utility.

The second moment that caught me was her section on Ordinal Linguistic Personification, where certain people automatically experience numbers as having personalities. I had known about synesthesia, but not this form. Major describes how “five” might be a motherly figure or “one” might be a tired father, and how these associations are consistent across a person’s lifetime. This stopped me because it suggests numbers occupy a strange cognitive space between language and mathematics. They are symbols of quantity but also words, with all the emotional and social connections that language carries. Even without synesthesia, many people assign moods or traits to specific numbers. Major cites surveys where “eight” is widely described as soft or kind, while primes tend to be seen as excitable or unsettling. That made me reflect on how even in math class, students often talk about numbers as if they have temperaments: “seven is weird,” “four feels stable,” “nine is intimidating.” This isn’t trivial. It hints at how personal and cultural associations shape mathematical thinking long before formal instruction does.

Together, these two stops pushed me to see numbers as far less neutral than I typically imagine them to be. Numbers travel through culture, metaphor, language, superstition, and emotion. As a teacher, this makes me think about how mathematical ideas arrive in the classroom already carrying stories and connotations. Rather than ignoring that, it might be worth acknowledging it, and even using it, to help students build a sense of connection and curiosity about the abstract symbols they work with every day.

History of Iran and Persia

Before watching the video, most of what I knew about Iran came from modern headlines, wars, oil disputes, and political tension. Like many people, I had unintentionally absorbed a very narrow image of the country, one defined almost entirely by modern conflict. So, when the video began outlining the vastness of the Persian Empire, its influence, and its innovations, I was honestly shocked. I had no idea that at one point it encompassed over 40% of the world’s population. It wasn’t just another empire among many, it was a global center of culture, science, and administration. That kind of scale and sophistication completely reframed how I understood Iran’s place in history.

The video’s line, “Today, our image of the Middle East is shaped more by media reports of extremism, political tyranny, and mass migration. And particularly in this age of deepening alienation, UNESCO World Heritage is all the more important”, really stuck with me. It perfectly captured what I was feeling as I watched. The way we view this region today is filtered through fear and politics, not through appreciation of its deep intellectual and artistic heritage. The idea that heritage, the physical and cultural traces of the past, can act as an antidote to alienation feels incredibly powerful. It reminds us that history is about belonging. By preserving and teaching about places like Persepolis, we reconnect with a shared human story that transcends the narrowness of current events.

Another realization I had was around the word “Persian.” I had heard of “ethnic Persians” before, but I never thought about where that identity came from or how far its influence spread. Persia wasn’t just a single kingdom; it was a vast cultural network that valued exchange, between people, languages, and ideas. The empire’s tolerance for different customs and religions, its infrastructure, and its commitment to governance through respect rather than fear, all of that felt unexpectedly modern. It challenged the stereotype of ancient empires as purely militaristic or authoritarian.

What I appreciated most was how the video helped me confront my own preconceptions. It reminded me that history can counter the narratives we’re taught to take for granted. As an educator, that feels especially relevant. Teaching students about civilizations like Persia can help them see that the world is not divided neatly into “us” and “them,” “modern” and “ancient,” or “civilized” and “uncivilized.” Instead, it’s a tapestry of shared growth, curiosity, and creativity. By showing the beauty and complexity of Persian history, we’re also teaching students to look at the present, and each other, with more nuance and empathy.

Euclid and Beauty

I’ve found it fascinating that Euclid’s Elements has lasted for over two thousand years. We don’t know much about Euclid himself, he might’ve even been a team of mathematicians (a little bit like some theories surrounding Shakespeare), but his influence is everywhere. What makes The Elements so timeless, I think, is how it captures the heart of what math really is, starting with a few simple truths and building something incredibly complex and elegant from them.

The postulates and “common notions” are so simple that they almost sound poetic. Take “things which are equal to the same thing are equal to each other.” It sounds obvious, but it’s the foundation for a whole logical universe. That’s the beauty of Euclidean geometry for me, how order and meaning can grow from something small and self-evident. Even now, when we have computers and non-Euclidean geometry and all these new mathematical worlds, Euclid’s work still feels like the purest expression of reasoning.

In class, when we tried to draw leaves using only a straightedge and compass, I was surprised by how much I enjoyed it. I only really knew how to make a perpendicular bisector, so my "leaf" was made entirely of that, and somehow, it still looked good. It reminded me that there’s a kind of beauty that comes from limitation. The same way Daina Taimina uses crochet to bring hyperbolic geometry to life, using Euclid’s tools made me feel connected to math as something creative, not just logical.

If I had to define beauty in Euclid’s geometry, I’d say it’s the combination of simplicity and inevitability. Every line and circle feels like it belongs, like there’s no other way it could have been drawn. Even after all these centuries, The Elements reminds me that math is about finding beauty in the structure of our own thoughts.

2 Pans Puzzle

At first, I had a bit of trouble figuring out how to approach the problem. I didn't really know where to start. Whenever that happens, I write down what I know. So I began with \( a+b+c+d=40, a<b<c<d \). And I thought about it some more and then I realized 2 things: the first being that the smallest weight must be 1 (\(a=1\)) and that you can "add" negatives by putting the weights on the other pan. So then each weight has 3 "states". It can be on the left pan, off, or on the right pan. So then I wrote out some basic expressions: \( b+c+d=39 \Rightarrow b+c+d-a = 38 \). Then the rest of the weights followed from there: \(b=3, c = 9, d = 27\).

 After solving the weights, I had a conversation with Ross about encoding ternary numbers (\(1-40\)) as some combination of the weights. After messing around with "2's complement" (a way to represent negative ternary numbers inspired by 1's complement in binary) and some quick googling, I stumbled onto balanced ternary (https://en.wikipedia.org/wiki/Balanced_ternary) which is ternary but instead of \(2\), there's \(-1\) or sometimes denoted \(T\). After this discovery, Ross said the representation was possible by adding the ternary number to \(1111_3\) then the following encoding emerges \[ 0: \text{on the "negative" pan}\\ 1: \text{not used} \\ 2: \text{on the "positive" pan}\] or this can be converted to balanced ternary by subtracting 1 from each digit then we have:\[ -1(T): \text{on the "negative" pan}\\ 0: \text{not used} \\ 1: \text{on the "positive" pan}\]

This encoding provides a more intuitive interaction to the ternary expressions since, while there are 3 states the weights can take, ternary numbers with 2s can only be represented by adding the next weight up and subtracting the next power down (i.e. \(6= 9-3\)). With this in mind, when we add it to \(1111_3\) it'll carry over a 1 to the next place value leaving a 0/-1/T which represents subtracting the closest power down and adding the next power up. 

After Jimena's comment about finding an algorithm to solve for the locker problem, I would eventually like to find an algorithm which, given a decimal number, can output the combination of weights to properly weigh it. I will leave that for another day.

 The one-pan, 5 weight problem was much easier for me. After going through a couple of the same steps as above, I quickly noticed the relationship between 31(\(2^5-1\)), 5 weights(bits), and only having one pan means no "negative" numbers. As such, the weights are 1,2,4,8, and 16.

 What struck me in working through these problems was how connected they are to fundamental ideas in mathematics: number bases, representation systems, and algorithms. In the future, I’d like to come back to this and write an algorithm that, given any number, can output the combination of weights to balance it.

For teaching, I think this puzzle has a lot of potential. Students could start by physically experimenting with weights on a scale to build intuition, then move into noticing patterns and connections with binary or ternary systems. It could also be extended into coding activities for computer science classes (i.e. writing a program that translates a decimal number into a set of weight placements.) One of the most powerful lessons this puzzle offers is a concrete way to understand different number bases. In the one-pan version, the weights 1, 2, 4, 8, and 16 mirror powers of 2, which naturally connects to binary. In the two-pan version, the weights 1, 3, 9, and 27 link to powers of 3, which connects to ternary and even balanced ternary. What’s valuable here is that students don’t just hear about bases in the abstract—they see and feel how numbers can be built up systematically from powers, whether through addition alone (binary) or through addition and subtraction (balanced ternary). This makes the idea of number bases less about memorizing place-value systems and more about recognizing structure in problem-solving.

Below are some scratch notes from my process.

 

Questions and Surprises from Ancient Egyptian Surveying

What stood out to me most in this reading was the precision the Egyptians seemed to achieve in surveying, especially in balancing distance from the Nile with flood damage to property. My first question is: how could they have had enough accuracy to do this effectively? With measurements based on body parts, like the cubit, which varied from person to person, their margin of error must have been large. It makes me wonder how much trust people placed in these measurements, and how disputes were resolved when land or taxes were on the line.

Another question I have is why they used two different measurements for the cubit: the “royal cubit” and the “short cubit”. What was the practical purpose of maintaining both? Was the royal cubit reserved for official state or religious work, while the short cubit was for everyday use? Or was it more about reinforcing authority by tying measurement itself to the king?

One thing that genuinely surprised me was learning that the government compensated people for flood damage with taxpayer money. It’s interesting to think about this in contrast to today, when private insurance companies often profit from the same kind of risk. It seems almost counterintuitive that an ancient monarchy would have a system of collective compensation, while our modern societies leave that to the market.

 

Reading about Babylonian mathematics made me realize how much I take algebraic notation for granted. I was honestly surprised to learn that the Babylonians even had a concept of algebra at all. I had assumed algebra came much later, with the Greeks. But what the Babylonians show us is that algebraic thinking, reasoning about unknowns and relationships, existed long before algebraic symbols were invented. Instead of using letters like x or y, they described the unknowns through words, images, or geometric interpretations. That makes me wonder if mathematics is really about generalization and abstraction at its core, rather than the symbols we use to express it.

One thing that stood out to me was their way of solving systems of equations by introducing new variables:

• $x=\frac{1}{2}s+w$

• $y=\frac{1}{2}s-w$

At first, this felt counterintuitive. Why replace two variables with two new variables? But the more I thought about it, the more fascinating it became. In some ways, w functions almost like an “error term” or a way of capturing deviation from the mean. By reframing the system in terms of a sum (s) and a difference (w), the Babylonians simplified the relationships without needing our modern notation. It’s a powerful reminder that general mathematical principles can be expressed through clever structure, even without symbols. That said, part of me still resists this approach. It feels strange to assume that x and y must be equidistant from the mean. Still, I can see how powerful it would have been as a systematic way to solve equations in the absence of algebraic symbols.

Thinking about this makes me appreciate that mathematics has always been about patterns, relationships, and structure, not just abstraction through notation. Algebraic symbols make it compact and general, but the Babylonians show us that the ideas themselves have been around much longer than the tools we now use to express them.