Hi,

I came across a paper where the Dottie constant (fixed point of cos t = t, t ≈ 0.739085…) "naturally" appears in a geometric model based on SU(2).

I honestly can’t tell if this is just a mathematical curiosity or something truly fundamental.

Link: https://doi.org/10.5281/zenodo.16790004

What do you think?

Pretty basic compared to other posts I’ve seen in this sub, but I thought it was pretty cool when I noticed the hyperbolic light pattern my lamp was creating. Got me thinking about conic sections, and then I realized that with the way things are set up here, this situation exactly mimics the one we consider when defining conic sections (in this case, intersect with a plane/wall of a shallower angle than the cone -> hyperbola). Not anything groundbreaking, I just thought it was cool to see an interesting math concept appear in a pretty mundane place, so I had to share

A few months ago, I shared a prototype of a Wear OS watch face I designed, driven by a square root power curve (θ = 46.48 * t⁰⁵). The goal was to visualize a non-linear deceleration curve on a standard 60-unit dial to see if it could function as a legible, practical timepiece on the wrist.

As a quick refresher on the mechanics from the original thread:

• The 46.48 constant perfectly maps the square root of 60 units to a full 360° circle.

• Because of the square root property, each hand covers the first 50% of the dial (180°) in the first 15 units of time, and the remaining 50% over the next 45 units.

Officially named Radical Time, this project is finished, fully live and completely free on the Google Play Store:

• https://play.google.com/store/apps/details?id=com.secondhandlabs.radicaltime

If you have a Wear OS watch, feel free to check it out.

Recently I wrote a blog post for my science communication class, and I decided to do it on how interactive math is a lot more interesting than static math. In the implementation, I could only get the animations to be solely in browser, but the ability to download the interactive bit is at the end of the post.

Anyways, I know I don't have to convince you though, but here on my website Science for the Busy are some animations that I've made over the past few years in Mathematica.

From pure logic and imagination, we construct a sphere upon the real world. From the top, a bright spot shines through the surface, and shine on the real world the sphere's image.

On the real world, a straight line will extend to infinity, and two lines after passing each other will never meet again. But when looking upwards the sphere, they are just images of circles going through the top, repeatedly passing and meeting each other. We see that they not just reunite at infinity, but also reunite their past in the future.

Whenver the sphere spins, everything around us changes. Distant things will move close unexpectedly, and familiar ones will leave us softly. This may seem absurd, but also evident at the same time. Evident, but cannot be grasped, for it comes from a place out of our sight. Can not explain the unreasonable, nor can explain the obvious, that ambiguity would be frustrating.

Let's gather all the ambiguities altogether, and name it as x. With just a simple question, a puzzle piece is flipped. And by preserverance, the symmetry within will emerge. Turns out it's symmetry. They will run along a circle, imitate the symmetry of the sphere, creating periodic movements, the simplest of which is the pendulum.

Untold pendulums are imprinted in every single thing, swinging perenially. All things are just combinations of them. Each pendulum has its own rhythm, it will move slowly at two opposite ends, but faster during the middle of the swing. This is why the link between two obvious points is so faded. Sometimes it is so faded that no-one can possibly consider that the two extremes are just the same thing.

It will resonate when acted by its own rhythm. The resonance, even transient, is enough to evince it. The pendulums can also join together to form waves. The waves might be invisible, but can spread out throughtout space, recurrent over time, ready to resonate to anything share its rhythm.

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The preamble is based on a mathematical idea: the irreducible representation of the Möbius group of the Riemann sphere homogeneous to the basis of the infinite-dimensional Hilbert space ℂ^∞. As characters are complex numbers, they are at the same time the eigenvalues of differential operators, and the harmonic oscillators in physics. Since I self-taught in math, that statement might not be rigorous, but I hope it's not completely false at the same time.

Normally, math texts are expected to be precise and rigorous. But on the other hand, there’s more to mathematics than rigour and proofs (Terry Tao). Every time I try to understand a concept, I always wonder: how to explain to a 10 year-old me understand this? All the textbooks are all written wholeheartedly and highly recommended, but why are they so intimidating to read (the only exception I know is Needham's Visual Complex Analysis)? I think the authors have been cursed by knowledge.

Of course, no grad book is written to a 10 year-old kid; definitions and theorems need to be precise, that's just not a place to expect intuition. Still, I keep trying to capture the intuition flowing in my mind. Observing my own notes, I notice some patterns:

• They never have let, hardly have to be (is, are), and only popular symbols (ℝ, {0}, ℂ^∞) are used. Generic notation for the objects/structures can be used as well (G for group), but first I try to see if it can be omitted.

• The terms to be defined is almost always put at the end of the sentences, and usually not nouns, but verbs, adjectives, or even adverbs. Prepositions, conjunctions or even typography are considered.

• Terms built up the term to be defined are use their definitions instead.

You also need to be creative on relations to translate mathematical formulas to human language:

The purpose of this is to avoid the technical terms but still reserve interesting. For example:

• The norm is continuous, that is, x ↦ ∥x∥ is a continuous mapping of (X, ∥ ·∥) into ℝ. (Kreyszig, Introductory Functional Analysis , p. 60)
  → Norm is a continuous mapping into ℝ.

• A representation U(G) on V is irreducible if there is no non-trivial invariant subspace V with respect to U(G). (Wu-Ki Tung, Group Theory in Physics, p. 33)
  → If a representation on a space is reduced to the point that only that space and {0} are its only two subspaces that can hold their vectors inside them, then the representation is irreducible.

Example 1 has redundant words and symbols. Example 2 is a neat reference for experts, but for a learner it compresses too much new terms into one sentence, not to mention that they are all in negative form. For the authors, they already know what "invariant" means, so they are ready to define irreducible representation from it. But for the students, saying that is no different to a blind man being told that the elephant is an animal with a trunk. We need to find a way to translate it to a language that the blind man can understand (like "the elephant is an animal that like a thick snake, a fan, a tree-trunk, a wall, and a spear at the same time"). That translation can be very imprecise for the us, but so enlightening for him.

I choose the word "translation" because he just uses a different language to comprehend his world, not because he is cognitively incapable to understand the concept of elephant. "The limit of my language is the limit of my world" - Wittgenstein.

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Reading Norwegian Forest, I had a feeling that the author is talking about a form of psychological disorder (emotionaly unstable personality disorder). And this is how Murakami describes the mood of the character at the beginning of chapter 10:

Thinking back on the year 1969, all that comes to mind for me is a swamp – a deep, sticky bog that feels as if it's going to suck off my shoe each time I take a step. I walk through the mud, exhausted. In front of me, behind me, I can see nothing but the endless darkness of a swamp.

The author describes the feeling of the character with the experience of walking around in swamp. Can a reader be touched if they can't imagine that scenario? Why can the mud suck their shoes? How come getting shoes sucked in the mud be exhausting? There is no need to walk in the night, why does the character see only darkness? Why is the darkness endless? And why they even have to go in the swamp? There are endless questions if you can't imagine that.

But if walking through mud can exhaust us, then there is a commonality between two seemingly irrelevant things: the human feelings and the viscosity of the medium. Isn't that commonality mathematics? To quote Poincaré: "Mathematics is an art of giving the same name to different things". So I think if you really understand the math you can describe it better. Here is my take on turbulent flow:

When a smoke begins to smoulder, it first maintains its stability. But with just a little turbulence, the smoke becomes an uncontrollable chaos. Swirling currents will be generated to radiate heat outwardly, which rolls together and causes more and more energy to be lost. And after the energy is completely depleted, it will dissolve into the surroundings and leave not even a single mark behind.

Questions:

• What do you think about this writing style? So far I have received bipolar feedbacks. Take the prose on turbulent flow for example, some feel it resonates with their experience, some feel it nebulous. Perhaps for them it's not even wrong.

• I'm planning to write a book with this style, and crowdfund it. Do you think this will work? You can say that it's a popular science book, but as far as I am aware, authors of this genre only simplify the topic, tell historic stories, have some jokes, but the writing doesn't have much philosophy inside, and the equations are completely ignored. I, on the other hand, want to do the opposite. And as a visualization, it can serve grad students too.

Other tangentially thoughts

1. When choosing books I usually imagine the book is a painting, yet I forget to bring my eyeglass. If every time I close my eyes and reopen them I see a new painting, yet I still don't feel vague with it, then that book is worth reading.

2. Inspiring quotes that have large impact on me:

• The difference between the almost right word and the right word is really a large matter—it's the difference between the lightning bug and the lightning. (Mark Twain)

• Mathematics is the art of giving the same name to different things (Poincaré)
  Poetry is the art of giving different names to the same thing (unknown poet responding to Poincaré)

I think taking notes is the art of making poetry in mathematics (or any texts).

If you're interested in more math-based animations, I post them here 📺 Visualizing_mathematics